This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
../../gmp/doc/gmp.texi.

   This manual describes how to install and use the GNU multiple
precision arithmetic library, version 6.1.0.

   Copyright 1991, 1993-2015 Free Software Foundation, Inc.

   Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License, Version
1.3 or any later version published by the Free Software Foundation;
with no Invariant Sections, with the Front-Cover Texts being "A GNU
Manual", and with the Back-Cover Texts being "You have freedom to copy
and modify this GNU Manual, like GNU software".  A copy of the license
is included in *Note GNU Free Documentation License::.

INFO-DIR-SECTION GNU libraries
START-INFO-DIR-ENTRY
* gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.
END-INFO-DIR-ENTRY


File: gmp.info,  Node: Exact Division,  Next: Exact Remainder,  Prev: Block-Wise Barrett Division,  Up: Division Algorithms

15.2.5 Exact Division
---------------------

A so-called exact division is when the dividend is known to be an exact
multiple of the divisor.  Jebelean's exact division algorithm uses this
knowledge to make some significant optimizations (*note References::).

   The idea can be illustrated in decimal for example with 368154
divided by 543.  Because the low digit of the dividend is 4, the low
digit of the quotient must be 8.  This is arrived at from 4*7 mod 10,
using the fact 7 is the modular inverse of 3 (the low digit of the
divisor), since 3*7 == 1 mod 10.  So 8*543=4344 can be subtracted from
the dividend leaving 363810.  Notice the low digit has become zero.

   The procedure is repeated at the second digit, with the next
quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving
325800.  And finally at the third digit with quotient digit 6 (8*7 mod
10), subtracting 6*543=3258 leaving 0.  So the quotient is 678.

   Notice however that the multiplies and subtractions don't need to
extend past the low three digits of the dividend, since that's enough
to determine the three quotient digits.  For the last quotient digit no
subtraction is needed at all.  On a 2NxN division like this one, only
about half the work of a normal basecase division is necessary.

   For an NxM exact division producing Q=N-M quotient limbs, the saving
over a normal basecase division is in two parts.  Firstly, each of the
Q quotient limbs needs only one multiply, not a 2x1 divide and
multiply.  Secondly, the crossproducts are reduced when Q>M to
Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2.  Notice the savings are
complementary.  If Q is big then many divisions are saved, or if Q is
small then the crossproducts reduce to a small number.

   The modular inverse used is calculated efficiently by `binvert_limb'
in `gmp-impl.h'.  This does four multiplies for a 32-bit limb, or six
for a 64-bit limb.  `tune/modlinv.c' has some alternate implementations
that might suit processors better at bit twiddling than multiplying.

   The sub-quadratic exact division described by Jebelean in "Exact
Division with Karatsuba Complexity" is not currently implemented.  It
uses a rearrangement similar to the divide and conquer for normal
division (*note Divide and Conquer Division::), but operating from low
to high.  A further possibility not currently implemented is
"Bidirectional Exact Integer Division" by Krandick and Jebelean which
forms quotient limbs from both the high and low ends of the dividend,
and can halve once more the number of crossproducts needed in a 2NxN
division.

   A special case exact division by 3 exists in `mpn_divexact_by3',
supporting Toom-3 multiplication and `mpq' canonicalizations.  It forms
quotient digits with a multiply by the modular inverse of 3 (which is
`0xAA..AAB') and uses two comparisons to determine a borrow for the next
limb.  The multiplications don't need to be on the dependent chain, as
long as the effect of the borrows is applied, which can help chips with
pipelined multipliers.


File: gmp.info,  Node: Exact Remainder,  Next: Small Quotient Division,  Prev: Exact Division,  Up: Division Algorithms

15.2.6 Exact Remainder
----------------------

If the exact division algorithm is done with a full subtraction at each
stage and the dividend isn't a multiple of the divisor, then low zero
limbs are produced but with a remainder in the high limbs.  For
dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this
remainder r is of the form

     a = q*d + r*b^n

   n represents the number of zero limbs produced by the subtractions,
that being the number of limbs produced for q.  r will be in the range
0<=r<d and can be viewed as a remainder, but one shifted up by a factor
of b^n.

   Carrying out full subtractions at each stage means the same number
of cross products must be done as a normal division, but there's still
some single limb divisions saved.  When d is a single limb some
simplifications arise, providing good speedups on a number of
processors.

   The functions `mpn_divexact_by3', `mpn_modexact_1_odd' and the
internal `mpn_redc_X' functions differ subtly in how they return r,
leading to some negations in the above formula, but all are essentially
the same.

   Clearly r is zero when a is a multiple of d, and this leads to
divisibility or congruence tests which are potentially more efficient
than a normal division.

   The factor of b^n on r can be ignored in a GCD when d is odd, hence
the use of `mpn_modexact_1_odd' by `mpn_gcd_1' and `mpz_kronecker_ui'
etc (*note Greatest Common Divisor Algorithms::).

   Montgomery's REDC method for modular multiplications uses operands
of the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n)
uses the factor of b^n in the exact remainder to reach a product in the
same form (x*y)*b^-n (*note Modular Powering Algorithm::).

   Notice that r generally gives no useful information about the
ordinary remainder a mod d since b^n mod d could be anything.  If
however b^n == 1 mod d, then r is the negative of the ordinary
remainder.  This occurs whenever d is a factor of b^n-1, as for example
with 3 in `mpn_divexact_by3'.  For a 32 or 64 bit limb other such
factors include 5, 17 and 257, but no particular use has been found for
this.


File: gmp.info,  Node: Small Quotient Division,  Prev: Exact Remainder,  Up: Division Algorithms

15.2.7 Small Quotient Division
------------------------------

An NxM division where the number of quotient limbs Q=N-M is small can
be optimized somewhat.

   An ordinary basecase division normalizes the divisor by shifting it
to make the high bit set, shifting the dividend accordingly, and
shifting the remainder back down at the end of the calculation.  This
is wasteful if only a few quotient limbs are to be formed.  Instead a
division of just the top 2*Q limbs of the dividend by the top Q limbs
of the divisor can be used to form a trial quotient.  This requires
only those limbs normalized, not the whole of the divisor and dividend.

   A multiply and subtract then applies the trial quotient to the M-Q
unused limbs of the divisor and N-Q dividend limbs (which includes Q
limbs remaining from the trial quotient division).  The starting trial
quotient can be 1 or 2 too big, but all cases of 2 too big and most
cases of 1 too big are detected by first comparing the most significant
limbs that will arise from the subtraction.  An addback is done if the
quotient still turns out to be 1 too big.

   This whole procedure is essentially the same as one step of the
basecase algorithm done in a Q limb base, though with the trial
quotient test done only with the high limbs, not an entire Q limb
"digit" product.  The correctness of this weaker test can be
established by following the argument of Knuth section 4.3.1 exercise
20 but with the v2*q>b*r+u2 condition appropriately relaxed.


File: gmp.info,  Node: Greatest Common Divisor Algorithms,  Next: Powering Algorithms,  Prev: Division Algorithms,  Up: Algorithms

15.3 Greatest Common Divisor
============================

* Menu:

* Binary GCD::
* Lehmer's Algorithm::
* Subquadratic GCD::
* Extended GCD::
* Jacobi Symbol::


File: gmp.info,  Node: Binary GCD,  Next: Lehmer's Algorithm,  Prev: Greatest Common Divisor Algorithms,  Up: Greatest Common Divisor Algorithms

15.3.1 Binary GCD
-----------------

At small sizes GMP uses an O(N^2) binary style GCD.  This is described
in many textbooks, for example Knuth section 4.5.2 algorithm B.  It
simply consists of successively reducing odd operands a and b using

     a,b = abs(a-b),min(a,b)
     strip factors of 2 from a

   The Euclidean GCD algorithm, as per Knuth algorithms E and A,
repeatedly computes the quotient q = floor(a/b) and replaces a,b by v,
u - q v. The binary algorithm has so far been found to be faster than
the Euclidean algorithm everywhere.  One reason the binary method does
well is that the implied quotient at each step is usually small, so
often only one or two subtractions are needed to get the same effect as
a division.  Quotients 1, 2 and 3 for example occur 67.7% of the time,
see Knuth section 4.5.3 Theorem E.

   When the implied quotient is large, meaning b is much smaller than
a, then a division is worthwhile.  This is the basis for the initial a
mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
and 1x1 cases).  But after that initial reduction, big quotients occur
too rarely to make it worth checking for them.


   The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as
described above.  For two N-bit operands, the algorithm takes about
0.68 iterations per bit.  For optimum performance some attention needs
to be paid to the way the factors of 2 are stripped from a.

   Firstly it may be noted that in twos complement the number of low
zero bits on a-b is the same as b-a, so counting or testing can begin on
a-b without waiting for abs(a-b) to be determined.

   A loop stripping low zero bits tends not to branch predict well,
since the condition is data dependent.  But on average there's only a
few low zeros, so an option is to strip one or two bits arithmetically
then loop for more (as done for AMD K6).  Or use a lookup table to get
a count for several bits then loop for more (as done for AMD K7).  An
alternative approach is to keep just one of a or b odd and iterate

     a,b = abs(a-b), min(a,b)
     a = a/2 if even
     b = b/2 if even

   This requires about 1.25 iterations per bit, but stripping of a
single bit at each step avoids any branching.  Repeating the bit strip
reduces to about 0.9 iterations per bit, which may be a worthwhile
tradeoff.

   Generally with the above approaches a speed of perhaps 6 cycles per
bit can be achieved, which is still not terribly fast with for instance
a 64-bit GCD taking nearly 400 cycles.  It's this sort of time which
means it's not usually advantageous to combine a set of divisibility
tests into a GCD.

   Currently, the binary algorithm is used for GCD only when N < 3.


File: gmp.info,  Node: Lehmer's Algorithm,  Next: Subquadratic GCD,  Prev: Binary GCD,  Up: Greatest Common Divisor Algorithms

15.3.2 Lehmer's algorithm
-------------------------

Lehmer's improvement of the Euclidean algorithms is based on the
observation that the initial part of the quotient sequence depends only
on the most significant parts of the inputs. The variant of Lehmer's
algorithm used in GMP splits off the most significant two limbs, as
suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by
Jebelean (*note References::). The quotients of two double-limb inputs
are collected as a 2 by 2 matrix with single-limb elements. This is
done by the function `mpn_hgcd2'. The resulting matrix is applied to
the inputs using `mpn_mul_1' and `mpn_submul_1'. Each iteration usually
reduces the inputs by almost one limb. In the rare case of a large
quotient, no progress can be made by examining just the most
significant two limbs, and the quotient is computed using plain
division.

   The resulting algorithm is asymptotically O(N^2), just as the
Euclidean algorithm and the binary algorithm. The quadratic part of the
work are the calls to `mpn_mul_1' and `mpn_submul_1'. For small sizes,
the linear work is also significant. There are roughly N calls to the
`mpn_hgcd2' function. This function uses a couple of important
optimizations:

   * It uses the same relaxed notion of correctness as `mpn_hgcd' (see
     next section). This means that when called with the most
     significant two limbs of two large numbers, the returned matrix
     does not always correspond exactly to the initial quotient
     sequence for the two large numbers; the final quotient may
     sometimes be one off.

   * It takes advantage of the fact the quotients are usually small.
     The division operator is not used, since the corresponding
     assembler instruction is very slow on most architectures. (This
     code could probably be improved further, it uses many branches
     that are unfriendly to prediction).

   * It switches from double-limb calculations to single-limb
     calculations half-way through, when the input numbers have been
     reduced in size from two limbs to one and a half.



File: gmp.info,  Node: Subquadratic GCD,  Next: Extended GCD,  Prev: Lehmer's Algorithm,  Up: Greatest Common Divisor Algorithms

15.3.3 Subquadratic GCD
-----------------------

For inputs larger than `GCD_DC_THRESHOLD', GCD is computed via the HGCD
(Half GCD) function, as a generalization to Lehmer's algorithm.

   Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1.
Then HGCD(a,b) returns a transformation matrix T with non-negative
elements, and reduced numbers (c;d) = T^-1 (a;b). The reduced numbers
c,d must be larger than S limbs, while their difference abs(c-d) must
fit in S limbs. The matrix elements will also be of size roughly N/2.

   The HGCD base case uses Lehmer's algorithm, but with the above stop
condition that returns reduced numbers and the corresponding
transformation matrix half-way through. For inputs larger than
`HGCD_THRESHOLD', HGCD is computed recursively, using the divide and
conquer algorithm in "On Scho"nhage's algorithm and subquadratic
integer GCD computation" by Mo"ller (*note References::). The recursive
algorithm consists of these main steps.

   * Call HGCD recursively, on the most significant N/2 limbs. Apply the
     resulting matrix T_1 to the full numbers, reducing them to a size
     just above 3N/2.

   * Perform a small number of division or subtraction steps to reduce
     the numbers to size below 3N/2. This is essential mainly for the
     unlikely case of large quotients.

   * Call HGCD recursively, on the most significant N/2 limbs of the
     reduced numbers. Apply the resulting matrix T_2 to the full
     numbers, reducing them to a size just above N/2.

   * Compute T = T_1 T_2.

   * Perform a small number of division and subtraction steps to
     satisfy the requirements, and return.

   GCD is then implemented as a loop around HGCD, similarly to Lehmer's
algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
`mpn_hgcd2', and applies the resulting matrix to the full numbers, the
sub-quadratic GCD chops off the most significant third of the limbs (the
proportion is a tuning parameter, and 1/3 seems to be more efficient
than, e.g, 1/2), calls `mpn_hgcd', and applies the resulting matrix.
Once the input numbers are reduced to size below `GCD_DC_THRESHOLD',
Lehmer's algorithm is used for the rest of the work.

   The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)),
where M(N) is the time for multiplying two N-limb numbers.


File: gmp.info,  Node: Extended GCD,  Next: Jacobi Symbol,  Prev: Subquadratic GCD,  Up: Greatest Common Divisor Algorithms

15.3.4 Extended GCD
-------------------

The extended GCD function, or GCDEXT, calculates gcd(a,b) and also
cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used
for plain GCD are extended to handle this case. The binary algorithm is
used only for single-limb GCDEXT.  Lehmer's algorithm is used for sizes
up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is
implemented as a loop around HGCD, but with more book-keeping to keep
track of the cofactors. This gives the same asymptotic running time as
for GCD and HGCD, O(M(N)*log(N))

   One difference to plain GCD is that while the inputs a and b are
reduced as the algorithm proceeds, the cofactors x and y grow in size.
This makes the tuning of the chopping-point more difficult. The current
code chops off the most significant half of the inputs for the call to
HGCD in the first iteration, and the most significant two thirds for
the remaining calls. This strategy could surely be improved. Also the
stop condition for the loop, where Lehmer's algorithm is invoked once
the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be
improved by taking into account the current size of the cofactors.


File: gmp.info,  Node: Jacobi Symbol,  Prev: Extended GCD,  Up: Greatest Common Divisor Algorithms

15.3.5 Jacobi Symbol
--------------------

[This section is obsolete.  The current Jacobi code actually uses a very
efficient algorithm.]

   `mpz_jacobi' and `mpz_kronecker' are currently implemented with a
simple binary algorithm similar to that described for the GCDs (*note
Binary GCD::).  They're not very fast when both inputs are large.
Lehmer's multi-step improvement or a binary based multi-step algorithm
is likely to be better.

   When one operand fits a single limb, and that includes
`mpz_kronecker_ui' and friends, an initial reduction is done with
either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
algorithm on a single limb.  The binary algorithm is well suited to a
single limb, and the whole calculation in this case is quite efficient.

   In all the routines sign changes for the result are accumulated
using some bit twiddling, avoiding table lookups or conditional jumps.


File: gmp.info,  Node: Powering Algorithms,  Next: Root Extraction Algorithms,  Prev: Greatest Common Divisor Algorithms,  Up: Algorithms

15.4 Powering Algorithms
========================

* Menu:

* Normal Powering Algorithm::
* Modular Powering Algorithm::


File: gmp.info,  Node: Normal Powering Algorithm,  Next: Modular Powering Algorithm,  Prev: Powering Algorithms,  Up: Powering Algorithms

15.4.1 Normal Powering
----------------------

Normal `mpz' or `mpf' powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is
seen in the exponent, as per Knuth section 4.6.3.  The "left to right"
variant described there is used rather than algorithm A, since it's
just as easy and can be done with somewhat less temporary memory.


File: gmp.info,  Node: Modular Powering Algorithm,  Prev: Normal Powering Algorithm,  Up: Powering Algorithms

15.4.2 Modular Powering
-----------------------

Modular powering is implemented using a 2^k-ary sliding window
algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
(*note References::).  k is chosen according to the size of the
exponent.  Larger exponents use larger values of k, the choice being
made to minimize the average number of multiplications that must
supplement the squaring.

   The modular multiplies and squarings use either a simple division or
the REDC method by Montgomery (*note References::).  REDC is a little
faster, essentially saving N single limb divisions in a fashion similar
to an exact remainder (*note Exact Remainder::).


File: gmp.info,  Node: Root Extraction Algorithms,  Next: Radix Conversion Algorithms,  Prev: Powering Algorithms,  Up: Algorithms

15.5 Root Extraction Algorithms
===============================

* Menu:

* Square Root Algorithm::
* Nth Root Algorithm::
* Perfect Square Algorithm::
* Perfect Power Algorithm::


File: gmp.info,  Node: Square Root Algorithm,  Next: Nth Root Algorithm,  Prev: Root Extraction Algorithms,  Up: Root Extraction Algorithms

15.5.1 Square Root
------------------

Square roots are taken using the "Karatsuba Square Root" algorithm by
Paul Zimmermann (*note References::).

   An input n is split into four parts of k bits each, so with b=2^k we
have n = a3*b^3 + a2*b^2 + a1*b + a0.  Part a3 must be "normalized" so
that either the high or second highest bit is set.  In GMP, k is kept
on a limb boundary and the input is left shifted (by an even number of
bits) to normalize.

   The square root of the high two parts is taken, by recursive
application of the algorithm (bottoming out in a one-limb Newton's
method),

     s1,r1 = sqrtrem (a3*b + a2)

   This is an approximation to the desired root and is extended by a
division to give s,r,

     q,u = divrem (r1*b + a1, 2*s1)
     s = s1*b + q
     r = u*b + a0 - q^2

   The normalization requirement on a3 means at this point s is either
correct or 1 too big.  r is negative in the latter case, so

     if r < 0 then
       r = r + 2*s - 1
       s = s - 1

   The algorithm is expressed in a divide and conquer form, but as
noted in the paper it can also be viewed as a discrete variant of
Newton's method, or as a variation on the schoolboy method (no longer
taught) for square roots two digits at a time.

   If the remainder r is not required then usually only a few high limbs
of r and u need to be calculated to determine whether an adjustment to
s is required.  This optimization is not currently implemented.

   In the Karatsuba multiplication range this algorithm is
O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
limbs.  In the FFT multiplication range this grows to a bound of
O(6*M(N/2)).  In practice a factor of about 1.5 to 1.8 is found in the
Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.

   The algorithm does all its calculations in integers and the resulting
`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'.  The extended
precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.


File: gmp.info,  Node: Nth Root Algorithm,  Next: Perfect Square Algorithm,  Prev: Square Root Algorithm,  Up: Root Extraction Algorithms

15.5.2 Nth Root
---------------

Integer Nth roots are taken using Newton's method with the following
iteration, where A is the input and n is the root to be taken.

              1         A
     a[i+1] = - * ( --------- + (n-1)*a[i] )
              n     a[i]^(n-1)

   The initial approximation a[1] is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just
above the true root.  The iteration converges quadratically when
started from a good approximation.  When n is large more initial bits
are needed to get good convergence.  The current implementation is not
particularly well optimized.


File: gmp.info,  Node: Perfect Square Algorithm,  Next: Perfect Power Algorithm,  Prev: Nth Root Algorithm,  Up: Root Extraction Algorithms

15.5.3 Perfect Square
---------------------

A significant fraction of non-squares can be quickly identified by
checking whether the input is a quadratic residue modulo small integers.

   `mpz_perfect_square_p' first tests the input mod 256, which means
just examining the low byte.  Only 44 different values occur for
squares mod 256, so 82.8% of inputs can be immediately identified as
non-squares.

   On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
for a total 99.25% of inputs identified as non-squares.  On a 64-bit
system 97 is tested too, for a total 99.62%.

   These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
for 64-bits), and such a remainder can be quickly taken just using
additions (see `mpn_mod_34lsub1').

   When nails are in use moduli are instead selected by the `gen-psqr.c'
program and applied with an `mpn_mod_1'.  The same 2^24-1 or 2^48-1
could be done with nails using some extra bit shifts, but this is not
currently implemented.

   In any case each modulus is applied to the `mpn_mod_34lsub1' or
`mpn_mod_1' remainder and a table lookup identifies non-squares.  By
using a "modexact" style calculation, and suitably permuted tables,
just one multiply each is required, see the code for details.  Moduli
are also combined to save operations, so long as the lookup tables
don't become too big.  `gen-psqr.c' does all the pre-calculations.

   A square root must still be taken for any value that passes these
tests, to verify it's really a square and not one of the small fraction
of non-squares that get through (i.e. a pseudo-square to all the tested
bases).

   Clearly more residue tests could be done, `mpz_perfect_square_p' only
uses a compact and efficient set.  Big inputs would probably benefit
from more residue testing, small inputs might be better off with less.
The assumed distribution of squares versus non-squares in the input
would affect such considerations.


File: gmp.info,  Node: Perfect Power Algorithm,  Prev: Perfect Square Algorithm,  Up: Root Extraction Algorithms

15.5.4 Perfect Power
--------------------

Detecting perfect powers is required by some factorization algorithms.
Currently `mpz_perfect_power_p' is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(*Note Nth Root Algorithm::.)

   If a prime divisor p with multiplicity e can be found, then only
roots which are divisors of e need to be considered, much reducing the
work necessary.  To this end divisibility by a set of small primes is
checked.


File: gmp.info,  Node: Radix Conversion Algorithms,  Next: Other Algorithms,  Prev: Root Extraction Algorithms,  Up: Algorithms

15.6 Radix Conversion
=====================

Radix conversions are less important than other algorithms.  A program
dominated by conversions should probably use a different data
representation.

* Menu:

* Binary to Radix::
* Radix to Binary::


File: gmp.info,  Node: Binary to Radix,  Next: Radix to Binary,  Prev: Radix Conversion Algorithms,  Up: Radix Conversion Algorithms

15.6.1 Binary to Radix
----------------------

Conversions from binary to a power-of-2 radix use a simple and fast
O(N) bit extraction algorithm.

   Conversions from binary to other radices use one of two algorithms.
Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
Repeated divisions by b^n are made, where b is the radix and n is the
biggest power that fits in a limb.  But instead of simply using the
remainder r from such divisions, an extra divide step is done to give a
fractional limb representing r/b^n.  The digits of r can then be
extracted using multiplications by b rather than divisions.  Special
case code is provided for decimal, allowing multiplications by 10 to
optimize to shifts and adds.

   Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used.  For an input t, powers b^(n*2^i) of the radix are calculated,
until a power between t and sqrt(t) is reached.  t is then divided by
that largest power, giving a quotient which is the digits above that
power, and a remainder which is those below.  These two parts are in
turn divided by the second highest power, and so on recursively.  When
a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
limbs, the basecase algorithm described above is used.

   The advantage of this algorithm is that big divisions can make use
of the sub-quadratic divide and conquer division (*note Divide and
Conquer Division::), and big divisions tend to have less overheads than
lots of separate single limb divisions anyway.  But in any case the
cost of calculating the powers b^(n*2^i) must first be overcome.

   `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
the same basic thing, the point where it becomes worth doing a big
division to cut the input in half.  `GET_STR_PRECOMPUTE_THRESHOLD'
includes the cost of calculating the radix power required, whereas
`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
case when recursing.

   Since the base case produces digits from least to most significant
but they want to be stored from most to least, it's necessary to
calculate in advance how many digits there will be, or at least be sure
not to underestimate that.  For GMP the number of input bits is
multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
The result is either correct or one too big.

   Examining some of the high bits of the input could increase the
chance of getting the exact number of digits, but an exact result every
time would not be practical, since in general the difference between
numbers 100... and 99... is only in the last few bits and the work to
identify 99...  might well be almost as much as a full conversion.

   The r/b^n scheme described above for using multiplications to bring
out digits might be useful for more than a single limb.  Some brief
experiments with it on the base case when recursing didn't give a
noticeable improvement, but perhaps that was only due to the
implementation.  Something similar would work for the sub-quadratic
divisions too, though there would be the cost of calculating a bigger
radix power.

   Another possible improvement for the sub-quadratic part would be to
arrange for radix powers that balanced the sizes of quotient and
remainder produced, i.e. the highest power would be an b^(n*k)
approximately equal to sqrt(t), not restricted to a 2^i factor.  That
ought to smooth out a graph of times against sizes, but may or may not
be a net speedup.


File: gmp.info,  Node: Radix to Binary,  Prev: Binary to Radix,  Up: Radix Conversion Algorithms

15.6.2 Radix to Binary
----------------------

*This section needs to be rewritten, it currently describes the
algorithms used before GMP 4.3.*

   Conversions from a power-of-2 radix into binary use a simple and fast
O(N) bitwise concatenation algorithm.

   Conversions from other radices use one of two algorithms.  Sizes
below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.  Groups
of n digits are converted to limbs, where n is the biggest power of the
base b which will fit in a limb, then those groups are accumulated into
the result by multiplying by b^n and adding.  This saves
multi-precision operations, as per Knuth section 4.4 part E (*note
References::).  Some special case code is provided for decimal, giving
the compiler a chance to optimize multiplications by 10.

   Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used.  First groups of n digits are converted into limbs.  Then adjacent
limbs are combined into limb pairs with x*b^n+y, where x and y are the
limbs.  Adjacent limb pairs are combined into quads similarly with
x*b^(2n)+y.  This continues until a single block remains, that being
the result.

   The advantage of this method is that the multiplications for each x
are big blocks, allowing Karatsuba and higher algorithms to be used.
But the cost of calculating the powers b^(n*2^i) must be overcome.
`SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000
digits, and on some processors much bigger still.

   `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and
tuned for decimal), though it might be better based on a limb count, so
as to be independent of the base.  But that sort of count isn't used by
the base case and so would need some sort of initial calculation or
estimate.

   The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger
than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that
`mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5,
or more).


File: gmp.info,  Node: Other Algorithms,  Next: Assembly Coding,  Prev: Radix Conversion Algorithms,  Up: Algorithms

15.7 Other Algorithms
=====================

* Menu:

* Prime Testing Algorithm::
* Factorial Algorithm::
* Binomial Coefficients Algorithm::
* Fibonacci Numbers Algorithm::
* Lucas Numbers Algorithm::
* Random Number Algorithms::


File: gmp.info,  Node: Prime Testing Algorithm,  Next: Factorial Algorithm,  Prev: Other Algorithms,  Up: Other Algorithms

15.7.1 Prime Testing
--------------------

The primality testing in `mpz_probab_prime_p' (*note Number Theoretic
Functions::) first does some trial division by small factors and then
uses the Miller-Rabin probabilistic primality testing algorithm, as
described in Knuth section 4.5.4 algorithm P (*note References::).

   For an odd input n, and with n = q*2^k+1 where q is odd, this
algorithm selects a random base x and tests whether x^q mod n is 1 or
-1, or an x^(q*2^j) mod n is 1, for 1<=j<=k.  If so then n is probably
prime, if not then n is definitely composite.

   Any prime n will pass the test, but some composites do too.  Such
composites are known as strong pseudoprimes to base x.  No n is a
strong pseudoprime to more than 1/4 of all bases (see Knuth exercise
22), hence with x chosen at random there's no more than a 1/4 chance a
"probable prime" will in fact be composite.

   In fact strong pseudoprimes are quite rare, making the test much more
powerful than this analysis would suggest, but 1/4 is all that's proven
for an arbitrary n.


File: gmp.info,  Node: Factorial Algorithm,  Next: Binomial Coefficients Algorithm,  Prev: Prime Testing Algorithm,  Up: Other Algorithms

15.7.2 Factorial
----------------

Factorials are calculated by a combination of two algorithms. An idea is
shared among them: to compute the odd part of the factorial; a final
step takes account of the power of 2 term, by shifting.

   For small n, the odd factor of n! is computed with the simple
observation that it is equal to the product of all positive odd numbers
smaller than n times the odd factor of [n/2]!, where [x] is the integer
part of x, and so on recursively. The procedure can be best illustrated
with an example,

     23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^19

   Current code collects all the factors in a single list, with a loop
and no recursion, and compute the product, with no special care for
repeated chunks.

   When n is larger, computation pass trough prime sieving. An helper
function is used, as suggested by Peter Luschny:

                                 n
                               -----
                    n!          | |   L(p,n)
     msf(n) = -------------- =  | |  p
               [n/2]!^2.2^k     p=3

   Where p ranges on odd prime numbers. The exponent k is chosen to
obtain an odd integer number: k is the number of 1 bits in the binary
representation of [n/2]. The function L(p,n) can be defined as zero
when p is composite, and, for any prime p, it is computed with:

               ---
                \    n
     L(p,n) =   /  [---] mod 2   <=  log (n) .
               ---  p^i                p
               i>0

   With this helper function, we are able to compute the odd part of n!
using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion
stops using the small-n algorithm on some [n/2^i].

   Both the above algorithms use binary splitting to compute the
product of many small factors. At first as many products as possible
are accumulated in a single register, generating a list of factors that
fit in a machine word. This list is then split into halves, and the
product is computed recursively.

   Such splitting is more efficient than repeated Nx1 multiplies since
it forms big multiplies, allowing Karatsuba and higher algorithms to be
used.  And even below the Karatsuba threshold a big block of work can
be more efficient for the basecase algorithm.


File: gmp.info,  Node: Binomial Coefficients Algorithm,  Next: Fibonacci Numbers Algorithm,  Prev: Factorial Algorithm,  Up: Other Algorithms

15.7.3 Binomial Coefficients
----------------------------

Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
using C(n,k) = C(n,n-k) if necessary, and then evaluating the following
product simply from i=2 to i=k.

                           k  (n-k+i)
     C(n,k) =  (n-k+1) * prod -------
                          i=2    i

   It's easy to show that each denominator i will divide the product so
far, so the exact division algorithm is used (*note Exact Division::).

   The numerators n-k+i and denominators i are first accumulated into
as many fit a limb, to save multi-precision operations, though for
`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
and n-k+i in general won't fit in a limb at all.


File: gmp.info,  Node: Fibonacci Numbers Algorithm,  Next: Lucas Numbers Algorithm,  Prev: Binomial Coefficients Algorithm,  Up: Other Algorithms

15.7.4 Fibonacci Numbers
------------------------

The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
calculating isolated F[n] or F[n],F[n-1] values efficiently.

   For small n, a table of single limb values in `__gmp_fib_table' is
used.  On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
to F[93].  For convenience the table starts at F[-1].

   Beyond the table, values are generated with a binary powering
algorithm, calculating a pair F[n] and F[n-1] working from high to low
across the bits of n.  The formulas used are

     F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
     F[2k-1] =   F[k]^2 + F[k-1]^2

     F[2k] = F[2k+1] - F[2k-1]

   At each step, k is the high b bits of n.  If the next bit of n is 0
then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
and the process repeated until all bits of n are incorporated.  Notice
these formulas require just two squares per bit of n.

   It'd be possible to handle the first few n above the single limb
table with simple additions, using the defining Fibonacci recurrence
F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
be faster for only about 10 or 20 values of n, and including a block of
code for just those doesn't seem worthwhile.  If they really mattered
it'd be better to extend the data table.

   Using a table avoids lots of calculations on small numbers, and
makes small n go fast.  A bigger table would make more small n go fast,
it's just a question of balancing size against desired speed.  For GMP
the code is kept compact, with the emphasis primarily on a good
powering algorithm.

   `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
interested in F[n].  In this case the last step of the algorithm can
become one multiply instead of two squares.  One of the following two
formulas is used, according as n is odd or even.

     F[2k]   = F[k]*(F[k]+2F[k-1])

     F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k

   F[2k+1] here is the same as above, just rearranged to be a multiply.
For interest, the 2*(-1)^k term both here and above can be applied
just to the low limb of the calculation, without a carry or borrow into
further limbs, which saves some code size.  See comments with
`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.


File: gmp.info,  Node: Lucas Numbers Algorithm,  Next: Random Number Algorithms,  Prev: Fibonacci Numbers Algorithm,  Up: Other Algorithms

15.7.5 Lucas Numbers
--------------------

`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
Fibonacci numbers with the following simple formulas.

     L[k]   =   F[k] + 2*F[k-1]
     L[k-1] = 2*F[k] -   F[k-1]

   `mpz_lucnum_ui' is only interested in L[n], and some work can be
saved.  Trailing zero bits on n can be handled with a single square
each.

     L[2k] = L[k]^2 - 2*(-1)^k

   And the lowest 1 bit can be handled with one multiply of a pair of
Fibonacci numbers, similar to what `mpz_fib_ui' does.

     L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k


File: gmp.info,  Node: Random Number Algorithms,  Prev: Lucas Numbers Algorithm,  Up: Other Algorithms

15.7.6 Random Numbers
---------------------

For the `urandomb' functions, random numbers are generated simply by
concatenating bits produced by the generator.  As long as the generator
has good randomness properties this will produce well-distributed N bit
numbers.

   For the `urandomm' functions, random numbers in a range 0<=R<N are
generated by taking values R of ceil(log2(N)) bits each until one
satisfies R<N.  This will normally require only one or two attempts,
but the attempts are limited in case the generator is somehow
degenerate and produces only 1 bits or similar.

   The Mersenne Twister generator is by Matsumoto and Nishimura (*note
References::).  It has a non-repeating period of 2^19937-1, which is a
Mersenne prime, hence the name of the generator.  The state is 624
words of 32-bits each, which is iterated with one XOR and shift for each
32-bit word generated, making the algorithm very fast.  Randomness
properties are also very good and this is the default algorithm used by
GMP.

   Linear congruential generators are described in many text books, for
instance Knuth volume 2 (*note References::).  With a modulus M and
parameters A and C, an integer state S is iterated by the formula S <-
A*S+C mod M.  At each step the new state is a linear function of the
previous, mod M, hence the name of the generator.

   In GMP only moduli of the form 2^N are supported, and the current
implementation is not as well optimized as it could be.  Overheads are
significant when N is small, and when N is large clearly the multiply
at each step will become slow.  This is not a big concern, since the
Mersenne Twister generator is better in every respect and is therefore
recommended for all normal applications.

   For both generators the current state can be deduced by observing
enough output and applying some linear algebra (over GF(2) in the case
of the Mersenne Twister).  This generally means raw output is
unsuitable for cryptographic applications without further hashing or
the like.


File: gmp.info,  Node: Assembly Coding,  Prev: Other Algorithms,  Up: Algorithms

15.8 Assembly Coding
====================

The assembly subroutines in GMP are the most significant source of
speed at small to moderate sizes.  At larger sizes algorithm selection
becomes more important, but of course speedups in low level routines
will still speed up everything proportionally.

   Carry handling and widening multiplies that are important for GMP
can't be easily expressed in C.  GCC `asm' blocks help a lot and are
provided in `longlong.h', but hand coding low level routines invariably
offers a speedup over generic C by a factor of anything from 2 to 10.

* Menu:

* Assembly Code Organisation::
* Assembly Basics::
* Assembly Carry Propagation::
* Assembly Cache Handling::
* Assembly Functional Units::
* Assembly Floating Point::
* Assembly SIMD Instructions::
* Assembly Software Pipelining::
* Assembly Loop Unrolling::
* Assembly Writing Guide::


File: gmp.info,  Node: Assembly Code Organisation,  Next: Assembly Basics,  Prev: Assembly Coding,  Up: Assembly Coding

15.8.1 Code Organisation
------------------------

The various `mpn' subdirectories contain machine-dependent code, written
in C or assembly.  The `mpn/generic' subdirectory contains default code,
used when there's no machine-specific version of a particular file.

   Each `mpn' subdirectory is for an ISA family.  Generally 32-bit and
64-bit variants in a family cannot share code and have separate
directories.  Within a family further subdirectories may exist for CPU
variants.

   In each directory a `nails' subdirectory may exist, holding code with
nails support for that CPU variant.  A `NAILS_SUPPORT' directive in each
file indicates the nails values the code handles.  Nails code only
exists where it's faster, or promises to be faster, than plain code.
There's no effort put into nails if they're not going to enhance a
given CPU.


File: gmp.info,  Node: Assembly Basics,  Next: Assembly Carry Propagation,  Prev: Assembly Code Organisation,  Up: Assembly Coding

15.8.2 Assembly Basics
----------------------

`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
overall GMP performance.  All multiplications and divisions come down to
repeated calls to these.  `mpn_add_n', `mpn_sub_n', `mpn_lshift' and
`mpn_rshift' are next most important.

   On some CPUs assembly versions of the internal functions
`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
mainly through avoiding function call overheads.  They can also
potentially make better use of a wide superscalar processor, as can
bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'.

   The restrictions on overlaps between sources and destinations (*note
Low-level Functions::) are designed to facilitate a variety of
implementations.  For example, knowing `mpn_add_n' won't have partly
overlapping sources and destination means reading can be done far ahead
of writing on superscalar processors, and loops can be vectorized on a
vector processor, depending on the carry handling.


File: gmp.info,  Node: Assembly Carry Propagation,  Next: Assembly Cache Handling,  Prev: Assembly Basics,  Up: Assembly Coding

15.8.3 Carry Propagation
------------------------

The problem that presents most challenges in GMP is propagating carries
from one limb to the next.  In functions like `mpn_addmul_1' and
`mpn_add_n', carries are the only dependencies between limb operations.

   On processors with carry flags, a straightforward CISC style `adc' is
generally best.  AMD K6 `mpn_addmul_1' however is an example of an
unusual set of circumstances where a branch works out better.

   On RISC processors generally an add and compare for overflow is
used.  This sort of thing can be seen in `mpn/generic/aors_n.c'.  Some
carry propagation schemes require 4 instructions, meaning at least 4
cycles per limb, but other schemes may use just 1 or 2.  On wide
superscalar processors performance may be completely determined by the
number of dependent instructions between carry-in and carry-out for
each limb.

   On vector processors good use can be made of the fact that a carry
bit only very rarely propagates more than one limb.  When adding a
single bit to a limb, there's only a carry out if that limb was
`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
`mpn/cray/add_n.c' is an example of this, it adds all limbs in
parallel, adds one set of carry bits in parallel and then only rarely
needs to fall through to a loop propagating further carries.

   On the x86s, GCC (as of version 2.95.2) doesn't generate
particularly good code for the RISC style idioms that are necessary to
handle carry bits in C.  Often conditional jumps are generated where
`adc' or `sbb' forms would be better.  And so unfortunately almost any
loop involving carry bits needs to be coded in assembly for best
results.


File: gmp.info,  Node: Assembly Cache Handling,  Next: Assembly Functional Units,  Prev: Assembly Carry Propagation,  Up: Assembly Coding

15.8.4 Cache Handling
---------------------

GMP aims to perform well both on operands that fit entirely in L1 cache
and those which don't.

   Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
large operands, so L2 and main memory performance is important for them.
`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
basecases, so L1 performance matters most for them, unless assembly
versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
case the remaining uses are mostly for larger operands.

   For L2 or main memory operands, memory access times will almost
certainly be more than the calculation time.  The aim therefore is to
maximize memory throughput, by starting a load of the next cache line
while processing the contents of the previous one.  Clearly this is
only possible if the chip has a lock-up free cache or some sort of
prefetch instruction.  Most current chips have both these features.

   Prefetching sources combines well with loop unrolling, since a
prefetch can be initiated once per unrolled loop (or more than once if
the loop covers more than one cache line).

   On CPUs without write-allocate caches, prefetching destinations will
ensure individual stores don't go further down the cache hierarchy,
limiting bandwidth.  Of course for calculations which are slow anyway,
like `mpn_divrem_1', write-throughs might be fine.

   The distance ahead to prefetch will be determined by memory latency
versus throughput.  The aim of course is to have data arriving
continuously, at peak throughput.  Some CPUs have limits on the number
of fetches or prefetches in progress.

   If a special prefetch instruction doesn't exist then a plain load
can be used, but in that case care must be taken not to attempt to read
past the end of an operand, since that might produce a segmentation
violation.

   Some CPUs or systems have hardware that detects sequential memory
accesses and initiates suitable cache movements automatically, making
life easy.


File: gmp.info,  Node: Assembly Functional Units,  Next: Assembly Floating Point,  Prev: Assembly Cache Handling,  Up: Assembly Coding

15.8.5 Functional Units
-----------------------

When choosing an approach for an assembly loop, consideration is given
to what operations can execute simultaneously and what throughput can
thereby be achieved.  In some cases an algorithm can be tweaked to
accommodate available resources.

   Loop control will generally require a counter and pointer updates,
costing as much as 5 instructions, plus any delays a branch introduces.
CPU addressing modes might reduce pointer updates, perhaps by allowing
just one updating pointer and others expressed as offsets from it, or
on CISC chips with all addressing done with the loop counter as a
scaled index.

   The final loop control cost can be amortised by processing several
limbs in each iteration (*note Assembly Loop Unrolling::).  This at
least ensures loop control isn't a big fraction the work done.

   Memory throughput is always a limit.  If perhaps only one load or
one store can be done per cycle then 3 cycles/limb will the top speed
for "binary" operations like `mpn_add_n', and any code achieving that
is optimal.

   Integer resources can be freed up by having the loop counter in a
float register, or by pressing the float units into use for some
multiplying, perhaps doing every second limb on the float side (*note
Assembly Floating Point::).

   Float resources can be freed up by doing carry propagation on the
integer side, or even by doing integer to float conversions in integers
using bit twiddling.


File: gmp.info,  Node: Assembly Floating Point,  Next: Assembly SIMD Instructions,  Prev: Assembly Functional Units,  Up: Assembly Coding

15.8.6 Floating Point
---------------------

Floating point arithmetic is used in GMP for multiplications on CPUs
with poor integer multipliers.  It's mostly useful for `mpn_mul_1',
`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
`mpn_mul_basecase' on both 32-bit and 64-bit machines.

   With IEEE 53-bit double precision floats, integer multiplications
producing up to 53 bits will give exact results.  Breaking a 64x64
multiplication into eight 16x32->48 bit pieces is convenient.  With
some care though six 21x32->53 bit products can be used, if one of the
lower two 21-bit pieces also uses the sign bit.

   For the `mpn_mul_1' family of functions on a 64-bit machine, the
invariant single limb is split at the start, into 3 or 4 pieces.
Inside the loop, the bignum operand is split into 32-bit pieces.  Fast
conversion of these unsigned 32-bit pieces to floating point is highly
machine-dependent.  In some cases, reading the data into the integer
unit, zero-extending to 64-bits, then transferring to the floating
point unit back via memory is the only option.

   Converting partial products back to 64-bit limbs is usually best
done as a signed conversion.  Since all values are smaller than 2^53,
signed and unsigned are the same, but most processors lack unsigned
conversions.


   Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or
`mpn_addmul_1' with a 64-bit limb.  The single limb operand V is split
into four 16-bit parts.  The multi-limb operand U is split in the loop
into two 32-bit parts.

                     +---+---+---+---+
                     |v48|v32|v16|v00|    V operand
                     +---+---+---+---+

                     +-------+---+---+
                 x   |  u32  |  u00  |    U operand (one limb)
                     +---------------+

     ---------------------------------

                         +-----------+
                         | u00 x v00 |    p00    48-bit products
                         +-----------+
                     +-----------+
                     | u00 x v16 |        p16
                     +-----------+
                 +-----------+
                 | u00 x v32 |            p32
                 +-----------+
             +-----------+
             | u00 x v48 |                p48
             +-----------+
                 +-----------+
                 | u32 x v00 |            r32
                 +-----------+
             +-----------+
             | u32 x v16 |                r48
             +-----------+
         +-----------+
         | u32 x v32 |                    r64
         +-----------+
     +-----------+
     | u32 x v48 |                        r80
     +-----------+

   p32 and r32 can be summed using floating-point addition, and
likewise p48 and r48.  p00 and p16 can be summed with r64 and r80 from
the previous iteration.

   For each loop then, four 49-bit quantities are transferred to the
integer unit, aligned as follows,

     |-----64bits----|-----64bits----|
                        +------------+
                        | p00 + r64' |    i00
                        +------------+
                    +------------+
                    | p16 + r80' |        i16
                    +------------+
                +------------+
                | p32 + r32  |            i32
                +------------+
            +------------+
            | p48 + r48  |                i48
            +------------+

   The challenge then is to sum these efficiently and add in a carry
limb, generating a low 64-bit result limb and a high 33-bit carry limb
(i48 extends 33 bits into the high half).


File: gmp.info,  Node: Assembly SIMD Instructions,  Next: Assembly Software Pipelining,  Prev: Assembly Floating Point,  Up: Assembly Coding

15.8.7 SIMD Instructions
------------------------

The single-instruction multiple-data support in current microprocessors
is aimed at signal processing algorithms where each data point can be
treated more or less independently.  There's generally not much support
for propagating the sort of carries that arise in GMP.

   SIMD multiplications of say four 16x16 bit multiplies only do as much
work as one 32x32 from GMP's point of view, and need some shifts and
adds besides.  But of course if say the SIMD form is fully pipelined
and uses less instruction decoding then it may still be worthwhile.

   On the x86 chips, MMX has so far found a use in `mpn_rshift' and
`mpn_lshift', and is used in a special case for 16-bit multipliers in
the P55 `mpn_mul_1'.  SSE2 is used for Pentium 4 `mpn_mul_1',
`mpn_addmul_1', and `mpn_submul_1'.


File: gmp.info,  Node: Assembly Software Pipelining,  Next: Assembly Loop Unrolling,  Prev: Assembly SIMD Instructions,  Up: Assembly Coding

15.8.8 Software Pipelining
--------------------------

Software pipelining consists of scheduling instructions around the
branch point in a loop.  For example a loop might issue a load not for
use in the present iteration but the next, thereby allowing extra
cycles for the data to arrive from memory.

   Naturally this is wanted only when doing things like loads or
multiplies that take several cycles to complete, and only where a CPU
has multiple functional units so that other work can be done in the
meantime.

   A pipeline with several stages will have a data value in progress at
each stage and each loop iteration moves them along one stage.  This is
like juggling.

   If the latency of some instruction is greater than the loop time
then it will be necessary to unroll, so one register has a result ready
to use while another (or multiple others) are still in progress.
(*note Assembly Loop Unrolling::).


File: gmp.info,  Node: Assembly Loop Unrolling,  Next: Assembly Writing Guide,  Prev: Assembly Software Pipelining,  Up: Assembly Coding

15.8.9 Loop Unrolling
---------------------

Loop unrolling consists of replicating code so that several limbs are
processed in each loop.  At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for
example alternately using one register combination and then another.
Judicious use of `m4' macros can help avoid lots of duplication in the
source code.

   Any amount of unrolling can be handled with a loop counter that's
decremented by N each time, stopping when the remaining count is less
than the further N the loop will process.  Or by subtracting N at the
start, the termination condition becomes when the counter C is less
than 0 (and the count of remaining limbs is C+N).

   Alternately for a power of 2 unroll the loop count and remainder can
be established with a shift and mask.  This is convenient if also
making a computed jump into the middle of a large loop.

   The limbs not a multiple of the unrolling can be handled in various
ways, for example

   * A simple loop at the end (or the start) to process the excess.
     Care will be wanted that it isn't too much slower than the
     unrolled part.

   * A set of binary tests, for example after an 8-limb unrolling, test
     for 4 more limbs to process, then a further 2 more or not, and
     finally 1 more or not.  This will probably take more code space
     than a simple loop.

   * A `switch' statement, providing separate code for each possible
     excess, for example an 8-limb unrolling would have separate code
     for 0 remaining, 1 remaining, etc, up to 7 remaining.  This might
     take a lot of code, but may be the best way to optimize all cases
     in combination with a deep pipelined loop.

   * A computed jump into the middle of the loop, thus making the first
     iteration handle the excess.  This should make times smoothly
     increase with size, which is attractive, but setups for the jump
     and adjustments for pointers can be tricky and could become quite
     difficult in combination with deep pipelining.


File: gmp.info,  Node: Assembly Writing Guide,  Prev: Assembly Loop Unrolling,  Up: Assembly Coding

15.8.10 Writing Guide
---------------------

This is a guide to writing software pipelined loops for processing limb
vectors in assembly.

   First determine the algorithm and which instructions are needed.
Code it without unrolling or scheduling, to make sure it works.  On a
3-operand CPU try to write each new value to a new register, this will
greatly simplify later steps.

   Then note for each instruction the functional unit and/or issue port
requirements.  If an instruction can use either of two units, like U0
or U1 then make a category "U0/U1".  Count the total using each unit
(or combined unit), and count all instructions.

   Figure out from those counts the best possible loop time.  The goal
will be to find a perfect schedule where instruction latencies are
completely hidden.  The total instruction count might be the limiting
factor, or perhaps a particular functional unit.  It might be possible
to tweak the instructions to help the limiting factor.

   Suppose the loop time is N, then make N issue buckets, with the
final loop branch at the end of the last.  Now fill the buckets with
dummy instructions using the functional units desired.  Run this to
make sure the intended speed is reached.

   Now replace the dummy instructions with the real instructions from
the slow but correct loop you started with.  The first will typically
be a load instruction.  Then the instruction using that value is placed
in a bucket an appropriate distance down.  Run the loop again, to check
it still runs at target speed.

   Keep placing instructions, frequently measuring the loop.  After a
few you will need to wrap around from the last bucket back to the top
of the loop.  If you used the new-register for new-value strategy above
then there will be no register conflicts.  If not then take care not to
clobber something already in use.  Changing registers at this time is
very error prone.

   The loop will overlap two or more of the original loop iterations,
and the computation of one vector element result will be started in one
iteration of the new loop, and completed one or several iterations
later.

   The final step is to create feed-in and wind-down code for the loop.
A good way to do this is to make a copy (or copies) of the loop at the
start and delete those instructions which don't have valid antecedents,
and at the end replicate and delete those whose results are unwanted
(including any further loads).

   The loop will have a minimum number of limbs loaded and processed,
so the feed-in code must test if the request size is smaller and skip
either to a suitable part of the wind-down or to special code for small
sizes.


File: gmp.info,  Node: Internals,  Next: Contributors,  Prev: Algorithms,  Up: Top

16 Internals
************

*This chapter is provided only for informational purposes and the
various internals described here may change in future GMP releases.
Applications expecting to be compatible with future releases should use
only the documented interfaces described in previous chapters.*

* Menu:

* Integer Internals::
* Rational Internals::
* Float Internals::
* Raw Output Internals::
* C++ Interface Internals::


File: gmp.info,  Node: Integer Internals,  Next: Rational Internals,  Prev: Internals,  Up: Internals

16.1 Integer Internals
======================

`mpz_t' variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated.  The fields are as follows.

`_mp_size'
     The number of limbs, or the negative of that when representing a
     negative integer.  Zero is represented by `_mp_size' set to zero,
     in which case the `_mp_d' data is unused.

`_mp_d'
     A pointer to an array of limbs which is the magnitude.  These are
     stored "little endian" as per the `mpn' functions, so `_mp_d[0]'
     is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the
     most significant.  Whenever `_mp_size' is non-zero, the most
     significant limb is non-zero.

     Currently there's always at least one limb allocated, so for
     instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui'
     can fetch `_mp_d[0]' unconditionally (though its value is then
     only wanted if `_mp_size' is non-zero).

`_mp_alloc'
     `_mp_alloc' is the number of limbs currently allocated at `_mp_d',
     and naturally `_mp_alloc >= ABS(_mp_size)'.  When an `mpz' routine
     is about to (or might be about to) increase `_mp_size', it checks
     `_mp_alloc' to see whether there's enough space, and reallocates
     if not.  `MPZ_REALLOC' is generally used for this.

   The various bitwise logical functions like `mpz_and' behave as if
negative values were twos complement.  But sign and magnitude is always
used internally, and necessary adjustments are made during the
calculations.  Sometimes this isn't pretty, but sign and magnitude are
best for other routines.

   Some internal temporary variables are setup with `MPZ_TMP_INIT' and
these have `_mp_d' space obtained from `TMP_ALLOC' rather than the
memory allocation functions.  Care is taken to ensure that these are
big enough that no reallocation is necessary (since it would have
unpredictable consequences).

   `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is
usually a `long'.  This is done to make the fields just 32 bits on some
64 bits systems, thereby saving a few bytes of data space but still
providing plenty of range.


File: gmp.info,  Node: Rational Internals,  Next: Float Internals,  Prev: Integer Internals,  Up: Internals

16.2 Rational Internals
=======================

`mpq_t' variables represent rationals using an `mpz_t' numerator and
denominator (*note Integer Internals::).

   The canonical form adopted is denominator positive (and non-zero),
no common factors between numerator and denominator, and zero uniquely
represented as 0/1.

   It's believed that casting out common factors at each stage of a
calculation is best in general.  A GCD is an O(N^2) operation so it's
better to do a few small ones immediately than to delay and have to do
a big one later.  Knowing the numerator and denominator have no common
factors can be used for example in `mpq_mul' to make only two cross
GCDs necessary, not four.

   This general approach to common factors is badly sub-optimal in the
presence of simple factorizations or little prospect for cancellation,
but GMP has no way to know when this will occur.  As per *Note
Efficiency::, that's left to applications.  The `mpq_t' framework might
still suit, with `mpq_numref' and `mpq_denref' for direct access to the
numerator and denominator, or of course `mpz_t' variables can be used
directly.


File: gmp.info,  Node: Float Internals,  Next: Raw Output Internals,  Prev: Rational Internals,  Up: Internals

16.3 Float Internals
====================

Efficient calculation is the primary aim of GMP floats and the use of
whole limbs and simple rounding facilitates this.

   `mpf_t' floats have a variable precision mantissa and a single
machine word signed exponent.  The mantissa is represented using sign
and magnitude.

        most                   least
     significant            significant
        limb                   limb

                                 _mp_d
      |---- _mp_exp --->           |
       _____ _____ _____ _____ _____
      |_____|_____|_____|_____|_____|
                        . <------------ radix point

       <-------- _mp_size --------->

The fields are as follows.

`_mp_size'
     The number of limbs currently in use, or the negative of that when
     representing a negative value.  Zero is represented by `_mp_size'
     and `_mp_exp' both set to zero, and in that case the `_mp_d' data
     is unused.  (In the future `_mp_exp' might be undefined when
     representing zero.)

`_mp_prec'
     The precision of the mantissa, in limbs.  In any calculation the
     aim is to produce `_mp_prec' limbs of result (the most significant
     being non-zero).

`_mp_d'
     A pointer to the array of limbs which is the absolute value of the
     mantissa.  These are stored "little endian" as per the `mpn'
     functions, so `_mp_d[0]' is the least significant limb and
     `_mp_d[ABS(_mp_size)-1]' the most significant.

     The most significant limb is always non-zero, but there are no
     other restrictions on its value, in particular the highest 1 bit
     can be anywhere within the limb.

     `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being
     for convenience (see below).  There are no reallocations during a
     calculation, only in a change of precision with `mpf_set_prec'.

`_mp_exp'
     The exponent, in limbs, determining the location of the implied
     radix point.  Zero means the radix point is just above the most
     significant limb.  Positive values mean a radix point offset
     towards the lower limbs and hence a value >= 1, as for example in
     the diagram above.  Negative exponents mean a radix point further
     above the highest limb.

     Naturally the exponent can be any value, it doesn't have to fall
     within the limbs as the diagram shows, it can be a long way above
     or a long way below.  Limbs other than those included in the
     `{_mp_d,_mp_size}' data are treated as zero.

   The `_mp_size' and `_mp_prec' fields are `int', although the
`mp_size_t' type is usually a `long'.  The `_mp_exp' field is usually
`long'.  This is done to make some fields just 32 bits on some 64 bits
systems, thereby saving a few bytes of data space but still providing
plenty of precision and a very large range.


The following various points should be noted.

Low Zeros
     The least significant limbs `_mp_d[0]' etc can be zero, though
     such low zeros can always be ignored.  Routines likely to produce
     low zeros check and avoid them to save time in subsequent
     calculations, but for most routines they're quite unlikely and
     aren't checked.

Mantissa Size Range
     The `_mp_size' count of limbs in use can be less than `_mp_prec' if
     the value can be represented in less.  This means low precision
     values or small integers stored in a high precision `mpf_t' can
     still be operated on efficiently.

     `_mp_size' can also be greater than `_mp_prec'.  Firstly a value is
     allowed to use all of the `_mp_prec+1' limbs available at `_mp_d',
     and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves
     `_mp_size' unchanged and so the size can be arbitrarily bigger than
     `_mp_prec'.

Rounding
     All rounding is done on limb boundaries.  Calculating `_mp_prec'
     limbs with the high non-zero will ensure the application requested
     minimum precision is obtained.

     The use of simple "trunc" rounding towards zero is efficient,
     since there's no need to examine extra limbs and increment or
     decrement.

Bit Shifts
     Since the exponent is in limbs, there are no bit shifts in basic
     operations like `mpf_add' and `mpf_mul'.  When differing exponents
     are encountered all that's needed is to adjust pointers to line up
     the relevant limbs.

     Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit
     shifts, but the choice is between an exponent in limbs which
     requires shifts there, or one in bits which requires them almost
     everywhere else.

Use of `_mp_prec+1' Limbs
     The extra limb on `_mp_d' (`_mp_prec+1' rather than just
     `_mp_prec') helps when an `mpf' routine might get a carry from its
     operation.  `mpf_add' for instance will do an `mpn_add' of
     `_mp_prec' limbs.  If there's no carry then that's the result, but
     if there is a carry then it's stored in the extra limb of space and
     `_mp_size' becomes `_mp_prec+1'.

     Whenever `_mp_prec+1' limbs are held in a variable, the low limb
     is not needed for the intended precision, only the `_mp_prec' high
     limbs.  But zeroing it out or moving the rest down is unnecessary.
     Subsequent routines reading the value will simply take the high
     limbs they need, and this will be `_mp_prec' if their target has
     that same precision.  This is no more than a pointer adjustment,
     and must be checked anyway since the destination precision can be
     different from the sources.

     Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs
     if available.  This ensures that a variable which has `_mp_size'
     equal to `_mp_prec+1' will get its full exact value copied.
     Strictly speaking this is unnecessary since only `_mp_prec' limbs
     are needed for the application's requested precision, but it's
     considered that an `mpf_set' from one variable into another of the
     same precision ought to produce an exact copy.

Application Precisions
     `__GMPF_BITS_TO_PREC' converts an application requested precision
     to an `_mp_prec'.  The value in bits is rounded up to a whole limb
     then an extra limb is added since the most significant limb of
     `_mp_d' is only non-zero and therefore might contain only one bit.

     `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
     extra limb from `_mp_prec' before converting to bits.  The net
     effect of reading back with `mpf_get_prec' is simply the precision
     rounded up to a multiple of `mp_bits_per_limb'.

     Note that the extra limb added here for the high only being
     non-zero is in addition to the extra limb allocated to `_mp_d'.
     For example with a 32-bit limb, an application request for 250
     bits will be rounded up to 8 limbs, then an extra added for the
     high being only non-zero, giving an `_mp_prec' of 9.  `_mp_d' then
     gets 10 limbs allocated.  Reading back with `mpf_get_prec' will
     take `_mp_prec' subtract 1 limb and multiply by 32, giving 256
     bits.

     Strictly speaking, the fact the high limb has at least one bit
     means that a float with, say, 3 limbs of 32-bits each will be
     holding at least 65 bits, but for the purposes of `mpf_t' it's
     considered simply to be 64 bits, a nice multiple of the limb size.


File: gmp.info,  Node: Raw Output Internals,  Next: C++ Interface Internals,  Prev: Float Internals,  Up: Internals

16.4 Raw Output Internals
=========================

`mpz_out_raw' uses the following format.

     +------+------------------------+
     | size |       data bytes       |
     +------+------------------------+

   The size is 4 bytes written most significant byte first, being the
number of subsequent data bytes, or the twos complement negative of
that when a negative integer is represented.  The data bytes are the
absolute value of the integer, written most significant byte first.

   The most significant data byte is always non-zero, so the output is
the same on all systems, irrespective of limb size.

   In GMP 1, leading zero bytes were written to pad the data bytes to a
multiple of the limb size.  `mpz_inp_raw' will still accept this, for
compatibility.

   The use of "big endian" for both the size and data fields is
deliberate, it makes the data easy to read in a hex dump of a file.
Unfortunately it also means that the limb data must be reversed when
reading or writing, so neither a big endian nor little endian system
can just read and write `_mp_d'.


File: gmp.info,  Node: C++ Interface Internals,  Prev: Raw Output Internals,  Up: Internals

16.5 C++ Interface Internals
============================

A system of expression templates is used to ensure something like
`a=b+c' turns into a simple call to `mpz_add' etc.  For `mpf_class' the
scheme also ensures the precision of the final destination is used for
any temporaries within a statement like `f=w*x+y*z'.  These are
important features which a naive implementation cannot provide.

   A simplified description of the scheme follows.  The true scheme is
complicated by the fact that expressions have different return types.
For detailed information, refer to the source code.

   To perform an operation, say, addition, we first define a "function
object" evaluating it,

     struct __gmp_binary_plus
     {
       static void eval(mpf_t f, const mpf_t g, const mpf_t h)
       {
         mpf_add(f, g, h);
       }
     };

And an "additive expression" object,

     __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
     operator+(const mpf_class &f, const mpf_class &g)
     {
       return __gmp_expr
         <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
     }

   The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
to encapsulate any possible kind of expression into a single template
type.  In fact even `mpf_class' etc are `typedef' specializations of
`__gmp_expr'.

   Next we define assignment of `__gmp_expr' to `mpf_class'.

     template <class T>
     mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
     {
       expr.eval(this->get_mpf_t(), this->precision());
       return *this;
     }

     template <class Op>
     void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
     (mpf_t f, mp_bitcnt_t precision)
     {
       Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
     }

   where `expr.val1' and `expr.val2' are references to the expression's
operands (here `expr' is the `__gmp_binary_expr' stored within the
`__gmp_expr').

   This way, the expression is actually evaluated only at the time of
assignment, when the required precision (that of `f') is known.
Furthermore the target `mpf_t' is now available, thus we can call
`mpf_add' directly with `f' as the output argument.

   Compound expressions are handled by defining operators taking
subexpressions as their arguments, like this:

     template <class T, class U>
     __gmp_expr
     <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
     operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
     {
       return __gmp_expr
         <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
         (expr1, expr2);
     }

   And the corresponding specializations of `__gmp_expr::eval':

     template <class T, class U, class Op>
     void __gmp_expr
     <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
     (mpf_t f, mp_bitcnt_t precision)
     {
       // declare two temporaries
       mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
       Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
     }

   The expression is thus recursively evaluated to any level of
complexity and all subexpressions are evaluated to the precision of `f'.


File: gmp.info,  Node: Contributors,  Next: References,  Prev: Internals,  Up: Top

Appendix A Contributors
***********************

Torbjo"rn Granlund wrote the original GMP library and is still the main
developer.  Code not explicitly attributed to others, was contributed by
Torbjo"rn.  Several other individuals and organizations have contributed
GMP.  Here is a list in chronological order on first contribution:

   Gunnar Sjo"din and Hans Riesel helped with mathematical problems in
early versions of the library.

   Richard Stallman helped with the interface design and revised the
first version of this manual.

   Brian Beuning and Doug Lea helped with testing of early versions of
the library and made creative suggestions.

   John Amanatides of York University in Canada contributed the function
`mpz_probab_prime_p'.

   Paul Zimmermann wrote the REDC-based mpz_powm code, the
Scho"nhage-Strassen FFT multiply code, and the Karatsuba square root
code.  He also improved the Toom3 code for GMP 4.2.  Paul sparked the
development of GMP 2, with his comparisons between bignum packages.
The ECMNET project Paul is organizing was a driving force behind many
of the optimizations in GMP 3.  Paul also wrote the new GMP 4.3 nth
root code (with Torbjo"rn).

   Ken Weber (Kent State University, Universidade Federal do Rio Grande
do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact',
`mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil)
grant 301314194-2.

   Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
configure.  He has also made valuable suggestions and tested numerous
intermediary releases.

   Joachim Hollman was involved in the design of the `mpf' interface,
and in the `mpz' design revisions for version 2.

   Bennet Yee contributed the initial versions of `mpz_jacobi' and
`mpz_legendre'.

   Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
`mpn/m68k/rshift.S' (now in `.asm' form).

   Robert Harley of Inria, France and David Seal of ARM, England,
suggested clever improvements for population count.  Robert also wrote
highly optimized Karatsuba and 3-way Toom multiplication functions for
GMP 3, and contributed the ARM assembly code.

   Torsten Ekedahl of the Mathematical department of Stockholm
University provided significant inspiration during several phases of
the GMP development.  His mathematical expertise helped improve several
algorithms.

   Linus Nordberg wrote the new configure system based on autoconf and
implemented the new random functions.

   Kevin Ryde worked on a large number of things: optimized x86 code,
m4 asm macros, parameter tuning, speed measuring, the configure system,
function inlining, divisibility tests, bit scanning, Jacobi symbols,
Fibonacci and Lucas number functions, printf and scanf functions, perl
interface, demo expression parser, the algorithms chapter in the
manual, `gmpasm-mode.el', and various miscellaneous improvements
elsewhere.

   Kent Boortz made the Mac OS 9 port.

   Steve Root helped write the optimized alpha 21264 assembly code.

   Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
`istream' input routines.

   Jason Moxham rewrote `mpz_fac_ui'.

   Pedro Gimeno implemented the Mersenne Twister and made other random
number improvements.

   Niels Mo"ller wrote the sub-quadratic GCD, extended GCD and jacobi
code, the quadratic Hensel division code, and (with Torbjo"rn) the new
divide and conquer division code for GMP 4.3.  Niels also helped
implement the new Toom multiply code for GMP 4.3 and implemented helper
functions to simplify Toom evaluations for GMP 5.0.  He wrote the
original version of mpn_mulmod_bnm1, and he is the main author of the
mini-gmp package used for gmp bootstrapping.

   Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply
strategy, and found the optimal strategies for evaluation and
interpolation in Toom multiplication.

   Marco Bodrato helped implement the new Toom multiply code for GMP
4.3 and implemented most of the new Toom multiply and squaring code for
5.0.  He is the main author of the current mpn_mulmod_bnm1,
mpn_mullo_n, and mpn_sqrlo.  Marco also wrote the functions mpn_invert
and mpn_invertappr, and improved the speed of integer root extraction.
He is the author of the current combinatorial functions: binomial,
factorial, multifactorial, primorial.

   David Harvey suggested the internal function `mpn_bdiv_dbm1',
implementing division relevant to Toom multiplication.  He also worked
on fast assembly sequences, in particular on a fast AMD64
`mpn_mul_basecase'. He wrote the internal middle product functions
`mpn_mulmid_basecase', `mpn_toom42_mulmid', `mpn_mulmid_n' and related
helper routines.

   Martin Boij wrote `mpn_perfect_power_p'.

   Marc Glisse improved `gmpxx.h': use fewer temporaries (faster),
specializations of `numeric_limits' and `common_type', C++11 features
(move constructors, explicit bool conversion, UDL), make the conversion
from `mpq_class' to `mpz_class' explicit, optimize operations where one
argument is a small compile-time constant, replace some heap
allocations by stack allocations.  He also fixed the eofbit handling of
C++ streams, and removed one division from `mpq/aors.c'.

   David S Miller wrote assembly code for SPARC T3 and T4.

   Mark Sofroniou cleaned up the types of mul_fft.c, letting it work
for huge operands.

   Ulrich Weigand ported GMP to the powerpc64le ABI.

   (This list is chronological, not ordered after significance.  If you
have contributed to GMP but are not listed above, please tell
<gmp-devel@gmplib.org> about the omission!)

   The development of floating point functions of GNU MP 2, were
supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
project POSSO (POlynomial System SOlving).

   The development of GMP 2, 3, and 4.0 was supported in part by the
IDA Center for Computing Sciences.

   The development of GMP 4.3, 5.0, and 5.1 was supported in part by
the Swedish Foundation for Strategic Research.

   Thanks go to Hans Thorsen for donating an SGI system for the GMP
test system environment.


File: gmp.info,  Node: References,  Next: GNU Free Documentation License,  Prev: Contributors,  Up: Top

Appendix B References
*********************

B.1 Books
=========

   * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study
     in Analytic Number Theory and Computational Complexity", Wiley,
     1998.

   * Richard Crandall and Carl Pomerance, "Prime Numbers: A
     Computational Perspective", 2nd edition, Springer-Verlag, 2005.
     `http://www.math.dartmouth.edu/~carlp/'

   * Henri Cohen, "A Course in Computational Algebraic Number Theory",
     Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
     `http://www.math.u-bordeaux.fr/~cohen/'

   * Donald E. Knuth, "The Art of Computer Programming", volume 2,
     "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
     `http://www-cs-faculty.stanford.edu/~knuth/taocp.html'

   * John D. Lipson, "Elements of Algebra and Algebraic Computing", The
     Benjamin Cummings Publishing Company Inc, 1981.

   * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
     "Handbook of Applied Cryptography",
     `http://www.cacr.math.uwaterloo.ca/hac/'

   * Richard M. Stallman and the GCC Developer Community, "Using the
     GNU Compiler Collection", Free Software Foundation, 2008,
     available online `https://gcc.gnu.org/onlinedocs/', and in the GCC
     package `https://ftp.gnu.org/gnu/gcc/'

B.2 Papers
==========

   * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP
     Square Root", Journal of Automated Reasoning, volume 29, 2002, pp.
     225-252.  Also available online as INRIA Research Report 4475,
     June 2002, `http://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf'

   * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
     Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
     `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022'

   * Torbjo"rn Granlund and Peter L. Montgomery, "Division by Invariant
     Integers using Multiplication", in Proceedings of the SIGPLAN
     PLDI'94 Conference, June 1994.  Also available
     `https://gmplib.org/~tege/divcnst-pldi94.pdf'.

   * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by
     invariant integers", IEEE Transactions on Computers, 11 June 2010.
     `https://gmplib.org/~tege/division-paper.pdf'

   * Torbjo"rn Granlund and Niels Mo"ller, "Division of integers large
     and small", to appear.

   * Tudor Jebelean, "An algorithm for exact division", Journal of
     Symbolic Computation, volume 15, 1993, pp. 169-180.  Research
     report version available
     `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'

   * Tudor Jebelean, "Exact Division with Karatsuba Complexity -
     Extended Abstract", RISC-Linz technical report 96-31,
     `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz'

   * Tudor Jebelean, "Practical Integer Division with Karatsuba
     Complexity", ISSAC 97, pp. 339-341.  Technical report available
     `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz'

   * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
     ISSAC 93, pp. 111-116.  Technical report version available
     `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz'

   * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for
     Finding the GCD of Long Integers", Journal of Symbolic
     Computation, volume 19, 1995, pp. 145-157.  Technical report
     version also available
     `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz'

   * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
     Division", Journal of Symbolic Computation, volume 21, 1996, pp.
     441-455.  Early technical report version also available
     `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz'

   * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
     623-dimensionally equidistributed uniform pseudorandom number
     generator", ACM Transactions on Modelling and Computer Simulation,
     volume 8, January 1998, pp. 3-30.  Available online
     `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz'
     (or .pdf)

   * R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
     Proceedings of the 13th Annual IEEE Symposium on Switching and
     Automata Theory, October 1972, pp. 90-96.  Reprinted as "Fast
     Modular Transforms", Journal of Computer and System Sciences,
     volume 8, number 3, June 1974, pp. 366-386.

   * Niels Mo"ller, "On Scho"nhage's algorithm and subquadratic integer
     GCD   computation", in Mathematics of Computation, volume 77,
     January 2008, pp.    589-607.

   * Peter L. Montgomery, "Modular Multiplication Without Trial
     Division", in Mathematics of Computation, volume 44, number 170,
     April 1985.

   * Arnold Scho"nhage and Volker Strassen, "Schnelle Multiplikation
     grosser Zahlen", Computing 7, 1971, pp. 281-292.

   * Kenneth Weber, "The accelerated integer GCD algorithm", ACM
     Transactions on Mathematical Software, volume 21, number 1, March
     1995, pp. 111-122.

   * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
     3805, November 1999,
     `http://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf'

   * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
     Implementations",
     `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'

   * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
     IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
     Reprinted as "More on Multiplying and Squaring Large Integers",
     IEEE Transactions on Computers, volume 43, number 8, August 1994,
     pp. 899-908.


File: gmp.info,  Node: GNU Free Documentation License,  Next: Concept Index,  Prev: References,  Up: Top

Appendix C GNU Free Documentation License
*****************************************

                     Version 1.3, 3 November 2008

     Copyright (C) 2000-2002, 2007, 2008 Free Software Foundation, Inc.
     `http://fsf.org/'

     Everyone is permitted to copy and distribute verbatim copies
     of this license document, but changing it is not allowed.

  0. PREAMBLE

     The purpose of this License is to make a manual, textbook, or other
     functional and useful document "free" in the sense of freedom: to
     assure everyone the effective freedom to copy and redistribute it,
     with or without modifying it, either commercially or
     noncommercially.  Secondarily, this License preserves for the
     author and publisher a way to get credit for their work, while not
     being considered responsible for modifications made by others.

     This License is a kind of "copyleft", which means that derivative
     works of the document must themselves be free in the same sense.
     It complements the GNU General Public License, which is a copyleft
     license designed for free software.

     We have designed this License in order to use it for manuals for
     free software, because free software needs free documentation: a
     free program should come with manuals providing the same freedoms
     that the software does.  But this License is not limited to
     software manuals; it can be used for any textual work, regardless
     of subject matter or whether it is published as a printed book.
     We recommend this License principally for works whose purpose is
     instruction or reference.

  1. APPLICABILITY AND DEFINITIONS

     This License applies to any manual or other work, in any medium,
     that contains a notice placed by the copyright holder saying it
     can be distributed under the terms of this License.  Such a notice
     grants a world-wide, royalty-free license, unlimited in duration,
     to use that work under the conditions stated herein.  The
     "Document", below, refers to any such manual or work.  Any member
     of the public is a licensee, and is addressed as "you".  You
     accept the license if you copy, modify or distribute the work in a
     way requiring permission under copyright law.

     A "Modified Version" of the Document means any work containing the
     Document or a portion of it, either copied verbatim, or with
     modifications and/or translated into another language.

     A "Secondary Section" is a named appendix or a front-matter section
     of the Document that deals exclusively with the relationship of the
     publishers or authors of the Document to the Document's overall
     subject (or to related matters) and contains nothing that could
     fall directly within that overall subject.  (Thus, if the Document
     is in part a textbook of mathematics, a Secondary Section may not
     explain any mathematics.)  The relationship could be a matter of
     historical connection with the subject or with related matters, or
     of legal, commercial, philosophical, ethical or political position
     regarding them.

     The "Invariant Sections" are certain Secondary Sections whose
     titles are designated, as being those of Invariant Sections, in
     the notice that says that the Document is released under this
     License.  If a section does not fit the above definition of
     Secondary then it is not allowed to be designated as Invariant.
     The Document may contain zero Invariant Sections.  If the Document
     does not identify any Invariant Sections then there are none.

     The "Cover Texts" are certain short passages of text that are
     listed, as Front-Cover Texts or Back-Cover Texts, in the notice
     that says that the Document is released under this License.  A
     Front-Cover Text may be at most 5 words, and a Back-Cover Text may
     be at most 25 words.

     A "Transparent" copy of the Document means a machine-readable copy,
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     otherwise Transparent file format whose markup, or absence of
     markup, has been arranged to thwart or discourage subsequent
     modification by readers is not Transparent.  An image format is
     not Transparent if used for any substantial amount of text.  A
     copy that is not "Transparent" is called "Opaque".

     Examples of suitable formats for Transparent copies include plain
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     standard-conforming simple HTML, PostScript or PDF designed for
     human modification.  Examples of transparent image formats include
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     can be read and edited only by proprietary word processors, SGML or
     XML for which the DTD and/or processing tools are not generally
     available, and the machine-generated HTML, PostScript or PDF
     produced by some word processors for output purposes only.

     The "Title Page" means, for a printed book, the title page itself,
     plus such following pages as are needed to hold, legibly, the
     material this License requires to appear in the title page.  For
     works in formats which do not have any title page as such, "Title
     Page" means the text near the most prominent appearance of the
     work's title, preceding the beginning of the body of the text.

     The "publisher" means any person or entity that distributes copies
     of the Document to the public.

     A section "Entitled XYZ" means a named subunit of the Document
     whose title either is precisely XYZ or contains XYZ in parentheses
     following text that translates XYZ in another language.  (Here XYZ
     stands for a specific section name mentioned below, such as
     "Acknowledgements", "Dedications", "Endorsements", or "History".)
     To "Preserve the Title" of such a section when you modify the
     Document means that it remains a section "Entitled XYZ" according
     to this definition.

     The Document may include Warranty Disclaimers next to the notice
     which states that this License applies to the Document.  These
     Warranty Disclaimers are considered to be included by reference in
     this License, but only as regards disclaiming warranties: any other
     implication that these Warranty Disclaimers may have is void and
     has no effect on the meaning of this License.

  2. VERBATIM COPYING

     You may copy and distribute the Document in any medium, either
     commercially or noncommercially, provided that this License, the
     copyright notices, and the license notice saying this License
     applies to the Document are reproduced in all copies, and that you
     add no other conditions whatsoever to those of this License.  You
     may not use technical measures to obstruct or control the reading
     or further copying of the copies you make or distribute.  However,
     you may accept compensation in exchange for copies.  If you
     distribute a large enough number of copies you must also follow
     the conditions in section 3.

     You may also lend copies, under the same conditions stated above,
     and you may publicly display copies.

  3. COPYING IN QUANTITY

     If you publish printed copies (or copies in media that commonly
     have printed covers) of the Document, numbering more than 100, and
     the Document's license notice requires Cover Texts, you must
     enclose the copies in covers that carry, clearly and legibly, all
     these Cover Texts: Front-Cover Texts on the front cover, and
     Back-Cover Texts on the back cover.  Both covers must also clearly
     and legibly identify you as the publisher of these copies.  The
     front cover must present the full title with all words of the
     title equally prominent and visible.  You may add other material
     on the covers in addition.  Copying with changes limited to the
     covers, as long as they preserve the title of the Document and
     satisfy these conditions, can be treated as verbatim copying in
     other respects.

     If the required texts for either cover are too voluminous to fit
     legibly, you should put the first ones listed (as many as fit
     reasonably) on the actual cover, and continue the rest onto
     adjacent pages.

     If you publish or distribute Opaque copies of the Document
     numbering more than 100, you must either include a
     machine-readable Transparent copy along with each Opaque copy, or
     state in or with each Opaque copy a computer-network location from
     which the general network-using public has access to download
     using public-standard network protocols a complete Transparent
     copy of the Document, free of added material.  If you use the
     latter option, you must take reasonably prudent steps, when you
     begin distribution of Opaque copies in quantity, to ensure that
     this Transparent copy will remain thus accessible at the stated
     location until at least one year after the last time you
     distribute an Opaque copy (directly or through your agents or
     retailers) of that edition to the public.

     It is requested, but not required, that you contact the authors of
     the Document well before redistributing any large number of
     copies, to give them a chance to provide you with an updated
     version of the Document.

  4. MODIFICATIONS

     You may copy and distribute a Modified Version of the Document
     under the conditions of sections 2 and 3 above, provided that you
     release the Modified Version under precisely this License, with
     the Modified Version filling the role of the Document, thus
     licensing distribution and modification of the Modified Version to
     whoever possesses a copy of it.  In addition, you must do these
     things in the Modified Version:

       A. Use in the Title Page (and on the covers, if any) a title
          distinct from that of the Document, and from those of
          previous versions (which should, if there were any, be listed
          in the History section of the Document).  You may use the
          same title as a previous version if the original publisher of
          that version gives permission.

       B. List on the Title Page, as authors, one or more persons or
          entities responsible for authorship of the modifications in
          the Modified Version, together with at least five of the
          principal authors of the Document (all of its principal
          authors, if it has fewer than five), unless they release you
          from this requirement.

       C. State on the Title page the name of the publisher of the
          Modified Version, as the publisher.

       D. Preserve all the copyright notices of the Document.

       E. Add an appropriate copyright notice for your modifications
          adjacent to the other copyright notices.

       F. Include, immediately after the copyright notices, a license
          notice giving the public permission to use the Modified
          Version under the terms of this License, in the form shown in
          the Addendum below.

       G. Preserve in that license notice the full lists of Invariant
          Sections and required Cover Texts given in the Document's
          license notice.

       H. Include an unaltered copy of this License.

       I. Preserve the section Entitled "History", Preserve its Title,
          and add to it an item stating at least the title, year, new
          authors, and publisher of the Modified Version as given on
          the Title Page.  If there is no section Entitled "History" in
          the Document, create one stating the title, year, authors,
          and publisher of the Document as given on its Title Page,
          then add an item describing the Modified Version as stated in
          the previous sentence.

       J. Preserve the network location, if any, given in the Document
          for public access to a Transparent copy of the Document, and
          likewise the network locations given in the Document for
          previous versions it was based on.  These may be placed in
          the "History" section.  You may omit a network location for a
          work that was published at least four years before the
          Document itself, or if the original publisher of the version
          it refers to gives permission.

       K. For any section Entitled "Acknowledgements" or "Dedications",
          Preserve the Title of the section, and preserve in the
          section all the substance and tone of each of the contributor
          acknowledgements and/or dedications given therein.

       L. Preserve all the Invariant Sections of the Document,
          unaltered in their text and in their titles.  Section numbers
          or the equivalent are not considered part of the section
          titles.

       M. Delete any section Entitled "Endorsements".  Such a section
          may not be included in the Modified Version.

       N. Do not retitle any existing section to be Entitled
          "Endorsements" or to conflict in title with any Invariant
          Section.

       O. Preserve any Warranty Disclaimers.

     If the Modified Version includes new front-matter sections or
     appendices that qualify as Secondary Sections and contain no
     material copied from the Document, you may at your option
     designate some or all of these sections as invariant.  To do this,
     add their titles to the list of Invariant Sections in the Modified
     Version's license notice.  These titles must be distinct from any
     other section titles.

     You may add a section Entitled "Endorsements", provided it contains
     nothing but endorsements of your Modified Version by various
     parties--for example, statements of peer review or that the text
     has been approved by an organization as the authoritative
     definition of a standard.

     You may add a passage of up to five words as a Front-Cover Text,
     and a passage of up to 25 words as a Back-Cover Text, to the end
     of the list of Cover Texts in the Modified Version.  Only one
     passage of Front-Cover Text and one of Back-Cover Text may be
     added by (or through arrangements made by) any one entity.  If the
     Document already includes a cover text for the same cover,
     previously added by you or by arrangement made by the same entity
     you are acting on behalf of, you may not add another; but you may
     replace the old one, on explicit permission from the previous
     publisher that added the old one.

     The author(s) and publisher(s) of the Document do not by this
     License give permission to use their names for publicity for or to
     assert or imply endorsement of any Modified Version.

  5. COMBINING DOCUMENTS

     You may combine the Document with other documents released under
     this License, under the terms defined in section 4 above for
     modified versions, provided that you include in the combination
     all of the Invariant Sections of all of the original documents,
     unmodified, and list them all as Invariant Sections of your
     combined work in its license notice, and that you preserve all
     their Warranty Disclaimers.

     The combined work need only contain one copy of this License, and
     multiple identical Invariant Sections may be replaced with a single
     copy.  If there are multiple Invariant Sections with the same name
     but different contents, make the title of each such section unique
     by adding at the end of it, in parentheses, the name of the
     original author or publisher of that section if known, or else a
     unique number.  Make the same adjustment to the section titles in
     the list of Invariant Sections in the license notice of the
     combined work.

     In the combination, you must combine any sections Entitled
     "History" in the various original documents, forming one section
     Entitled "History"; likewise combine any sections Entitled
     "Acknowledgements", and any sections Entitled "Dedications".  You
     must delete all sections Entitled "Endorsements."

  6. COLLECTIONS OF DOCUMENTS

     You may make a collection consisting of the Document and other
     documents released under this License, and replace the individual
     copies of this License in the various documents with a single copy
     that is included in the collection, provided that you follow the
     rules of this License for verbatim copying of each of the
     documents in all other respects.

     You may extract a single document from such a collection, and
     distribute it individually under this License, provided you insert
     a copy of this License into the extracted document, and follow
     this License in all other respects regarding verbatim copying of
     that document.

  7. AGGREGATION WITH INDEPENDENT WORKS

     A compilation of the Document or its derivatives with other
     separate and independent documents or works, in or on a volume of
     a storage or distribution medium, is called an "aggregate" if the
     copyright resulting from the compilation is not used to limit the
     legal rights of the compilation's users beyond what the individual
     works permit.  When the Document is included in an aggregate, this
     License does not apply to the other works in the aggregate which
     are not themselves derivative works of the Document.

     If the Cover Text requirement of section 3 is applicable to these
     copies of the Document, then if the Document is less than one half
     of the entire aggregate, the Document's Cover Texts may be placed
     on covers that bracket the Document within the aggregate, or the
     electronic equivalent of covers if the Document is in electronic
     form.  Otherwise they must appear on printed covers that bracket
     the whole aggregate.

  8. TRANSLATION

     Translation is considered a kind of modification, so you may
     distribute translations of the Document under the terms of section
     4.  Replacing Invariant Sections with translations requires special
     permission from their copyright holders, but you may include
     translations of some or all Invariant Sections in addition to the
     original versions of these Invariant Sections.  You may include a
     translation of this License, and all the license notices in the
     Document, and any Warranty Disclaimers, provided that you also
     include the original English version of this License and the
     original versions of those notices and disclaimers.  In case of a
     disagreement between the translation and the original version of
     this License or a notice or disclaimer, the original version will
     prevail.

     If a section in the Document is Entitled "Acknowledgements",
     "Dedications", or "History", the requirement (section 4) to
     Preserve its Title (section 1) will typically require changing the
     actual title.

  9. TERMINATION

     You may not copy, modify, sublicense, or distribute the Document
     except as expressly provided under this License.  Any attempt
     otherwise to copy, modify, sublicense, or distribute it is void,
     and will automatically terminate your rights under this License.

     However, if you cease all violation of this License, then your
     license from a particular copyright holder is reinstated (a)
     provisionally, unless and until the copyright holder explicitly
     and finally terminates your license, and (b) permanently, if the
     copyright holder fails to notify you of the violation by some
     reasonable means prior to 60 days after the cessation.

     Moreover, your license from a particular copyright holder is
     reinstated permanently if the copyright holder notifies you of the
     violation by some reasonable means, this is the first time you have
     received notice of violation of this License (for any work) from
     that copyright holder, and you cure the violation prior to 30 days
     after your receipt of the notice.

     Termination of your rights under this section does not terminate
     the licenses of parties who have received copies or rights from
     you under this License.  If your rights have been terminated and
     not permanently reinstated, receipt of a copy of some or all of
     the same material does not give you any rights to use it.

 10. FUTURE REVISIONS OF THIS LICENSE

     The Free Software Foundation may publish new, revised versions of
     the GNU Free Documentation License from time to time.  Such new
     versions will be similar in spirit to the present version, but may
     differ in detail to address new problems or concerns.  See
     `https://www.gnu.org/copyleft/'.

     Each version of the License is given a distinguishing version
     number.  If the Document specifies that a particular numbered
     version of this License "or any later version" applies to it, you
     have the option of following the terms and conditions either of
     that specified version or of any later version that has been
     published (not as a draft) by the Free Software Foundation.  If
     the Document does not specify a version number of this License,
     you may choose any version ever published (not as a draft) by the
     Free Software Foundation.  If the Document specifies that a proxy
     can decide which future versions of this License can be used, that
     proxy's public statement of acceptance of a version permanently
     authorizes you to choose that version for the Document.

 11. RELICENSING

     "Massive Multiauthor Collaboration Site" (or "MMC Site") means any
     World Wide Web server that publishes copyrightable works and also
     provides prominent facilities for anybody to edit those works.  A
     public wiki that anybody can edit is an example of such a server.
     A "Massive Multiauthor Collaboration" (or "MMC") contained in the
     site means any set of copyrightable works thus published on the MMC
     site.

     "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
     license published by Creative Commons Corporation, a not-for-profit
     corporation with a principal place of business in San Francisco,
     California, as well as future copyleft versions of that license
     published by that same organization.

     "Incorporate" means to publish or republish a Document, in whole or
     in part, as part of another Document.

     An MMC is "eligible for relicensing" if it is licensed under this
     License, and if all works that were first published under this
     License somewhere other than this MMC, and subsequently
     incorporated in whole or in part into the MMC, (1) had no cover
     texts or invariant sections, and (2) were thus incorporated prior
     to November 1, 2008.

     The operator of an MMC Site may republish an MMC contained in the
     site under CC-BY-SA on the same site at any time before August 1,
     2009, provided the MMC is eligible for relicensing.


ADDENDUM: How to use this License for your documents
====================================================

To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and license
notices just after the title page:

       Copyright (C)  YEAR  YOUR NAME.
       Permission is granted to copy, distribute and/or modify this document
       under the terms of the GNU Free Documentation License, Version 1.3
       or any later version published by the Free Software Foundation;
       with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
       Texts.  A copy of the license is included in the section entitled ``GNU
       Free Documentation License''.

   If you have Invariant Sections, Front-Cover Texts and Back-Cover
Texts, replace the "with...Texts." line with this:

         with the Invariant Sections being LIST THEIR TITLES, with
         the Front-Cover Texts being LIST, and with the Back-Cover Texts
         being LIST.

   If you have Invariant Sections without Cover Texts, or some other
combination of the three, merge those two alternatives to suit the
situation.

   If your document contains nontrivial examples of program code, we
recommend releasing these examples in parallel under your choice of
free software license, such as the GNU General Public License, to
permit their use in free software.


File: gmp.info,  Node: Concept Index,  Next: Function Index,  Prev: GNU Free Documentation License,  Up: Top

Concept Index
*************

 [index ]
* Menu:

* #include:                              Headers and Libraries.
                                                              (line   6)
* --build:                               Build Options.       (line  52)
* --disable-fft:                         Build Options.       (line 313)
* --disable-shared:                      Build Options.       (line  45)
* --disable-static:                      Build Options.       (line  45)
* --enable-alloca:                       Build Options.       (line 274)
* --enable-assert:                       Build Options.       (line 319)
* --enable-cxx:                          Build Options.       (line 226)
* --enable-fat:                          Build Options.       (line 161)
* --enable-profiling <1>:                Build Options.       (line 323)
* --enable-profiling:                    Profiling.           (line   6)
* --exec-prefix:                         Build Options.       (line  32)
* --host:                                Build Options.       (line  66)
* --prefix:                              Build Options.       (line  32)
* -finstrument-functions:                Profiling.           (line  66)
* 2exp functions:                        Efficiency.          (line  43)
* 68000:                                 Notes for Particular Systems.
                                                              (line  94)
* 80x86:                                 Notes for Particular Systems.
                                                              (line 150)
* ABI <1>:                               Build Options.       (line 168)
* ABI:                                   ABI and ISA.         (line   6)
* About this manual:                     Introduction to GMP. (line  57)
* AC_CHECK_LIB:                          Autoconf.            (line  11)
* AIX <1>:                               Notes for Particular Systems.
                                                              (line   7)
* AIX:                                   ABI and ISA.         (line 178)
* Algorithms:                            Algorithms.          (line   6)
* alloca:                                Build Options.       (line 274)
* Allocation of memory:                  Custom Allocation.   (line   6)
* AMD64:                                 ABI and ISA.         (line  44)
* Anonymous FTP of latest version:       Introduction to GMP. (line  37)
* Application Binary Interface:          ABI and ISA.         (line   6)
* Arithmetic functions <1>:              Rational Arithmetic. (line   6)
* Arithmetic functions <2>:              Float Arithmetic.    (line   6)
* Arithmetic functions:                  Integer Arithmetic.  (line   6)
* ARM:                                   Notes for Particular Systems.
                                                              (line  20)
* Assembly cache handling:               Assembly Cache Handling.
                                                              (line   6)
* Assembly carry propagation:            Assembly Carry Propagation.
                                                              (line   6)
* Assembly code organisation:            Assembly Code Organisation.
                                                              (line   6)
* Assembly coding:                       Assembly Coding.     (line   6)
* Assembly floating Point:               Assembly Floating Point.
                                                              (line   6)
* Assembly loop unrolling:               Assembly Loop Unrolling.
                                                              (line   6)
* Assembly SIMD:                         Assembly SIMD Instructions.
                                                              (line   6)
* Assembly software pipelining:          Assembly Software Pipelining.
                                                              (line   6)
* Assembly writing guide:                Assembly Writing Guide.
                                                              (line   6)
* Assertion checking <1>:                Debugging.           (line  79)
* Assertion checking:                    Build Options.       (line 319)
* Assignment functions <1>:              Assigning Integers.  (line   6)
* Assignment functions <2>:              Initializing Rationals.
                                                              (line   6)
* Assignment functions <3>:              Simultaneous Float Init & Assign.
                                                              (line   6)
* Assignment functions <4>:              Simultaneous Integer Init & Assign.
                                                              (line   6)
* Assignment functions:                  Assigning Floats.    (line   6)
* Autoconf:                              Autoconf.            (line   6)
* Basics:                                GMP Basics.          (line   6)
* Binomial coefficient algorithm:        Binomial Coefficients Algorithm.
                                                              (line   6)
* Binomial coefficient functions:        Number Theoretic Functions.
                                                              (line 124)
* Binutils strip:                        Known Build Problems.
                                                              (line  28)
* Bit manipulation functions:            Integer Logic and Bit Fiddling.
                                                              (line   6)
* Bit scanning functions:                Integer Logic and Bit Fiddling.
                                                              (line  40)
* Bit shift left:                        Integer Arithmetic.  (line  38)
* Bit shift right:                       Integer Division.    (line  62)
* Bits per limb:                         Useful Macros and Constants.
                                                              (line   7)
* Bug reporting:                         Reporting Bugs.      (line   6)
* Build directory:                       Build Options.       (line  19)
* Build notes for binary packaging:      Notes for Package Builds.
                                                              (line   6)
* Build notes for particular systems:    Notes for Particular Systems.
                                                              (line   6)
* Build options:                         Build Options.       (line   6)
* Build problems known:                  Known Build Problems.
                                                              (line   6)
* Build system:                          Build Options.       (line  52)
* Building GMP:                          Installing GMP.      (line   6)
* Bus error:                             Debugging.           (line   7)
* C compiler:                            Build Options.       (line 179)
* C++ compiler:                          Build Options.       (line 250)
* C++ interface:                         C++ Class Interface. (line   6)
* C++ interface internals:               C++ Interface Internals.
                                                              (line   6)
* C++ istream input:                     C++ Formatted Input. (line   6)
* C++ ostream output:                    C++ Formatted Output.
                                                              (line   6)
* C++ support:                           Build Options.       (line 226)
* CC:                                    Build Options.       (line 179)
* CC_FOR_BUILD:                          Build Options.       (line 213)
* CFLAGS:                                Build Options.       (line 179)
* Checker:                               Debugging.           (line 115)
* checkergcc:                            Debugging.           (line 122)
* Code organisation:                     Assembly Code Organisation.
                                                              (line   6)
* Compaq C++:                            Notes for Particular Systems.
                                                              (line  25)
* Comparison functions <1>:              Integer Comparisons. (line   6)
* Comparison functions <2>:              Float Comparison.    (line   6)
* Comparison functions:                  Comparing Rationals. (line   6)
* Compatibility with older versions:     Compatibility with older versions.
                                                              (line   6)
* Conditions for copying GNU MP:         Copying.             (line   6)
* Configuring GMP:                       Installing GMP.      (line   6)
* Congruence algorithm:                  Exact Remainder.     (line  30)
* Congruence functions:                  Integer Division.    (line 137)
* Constants:                             Useful Macros and Constants.
                                                              (line   6)
* Contributors:                          Contributors.        (line   6)
* Conventions for parameters:            Parameter Conventions.
                                                              (line   6)
* Conventions for variables:             Variable Conventions.
                                                              (line   6)
* Conversion functions <1>:              Converting Integers. (line   6)
* Conversion functions <2>:              Converting Floats.   (line   6)
* Conversion functions:                  Rational Conversions.
                                                              (line   6)
* Copying conditions:                    Copying.             (line   6)
* CPPFLAGS:                              Build Options.       (line 205)
* CPU types <1>:                         Introduction to GMP. (line  24)
* CPU types:                             Build Options.       (line 108)
* Cross compiling:                       Build Options.       (line  66)
* Cryptography functions, low-level:     Low-level Functions. (line 507)
* Custom allocation:                     Custom Allocation.   (line   6)
* CXX:                                   Build Options.       (line 250)
* CXXFLAGS:                              Build Options.       (line 250)
* Cygwin:                                Notes for Particular Systems.
                                                              (line  57)
* Darwin:                                Known Build Problems.
                                                              (line  51)
* Debugging:                             Debugging.           (line   6)
* Demonstration programs:                Demonstration Programs.
                                                              (line   6)
* Digits in an integer:                  Miscellaneous Integer Functions.
                                                              (line  23)
* Divisibility algorithm:                Exact Remainder.     (line  30)
* Divisibility functions:                Integer Division.    (line 137)
* Divisibility testing:                  Efficiency.          (line  91)
* Division algorithms:                   Division Algorithms. (line   6)
* Division functions <1>:                Rational Arithmetic. (line  24)
* Division functions <2>:                Integer Division.    (line   6)
* Division functions:                    Float Arithmetic.    (line  33)
* DJGPP <1>:                             Notes for Particular Systems.
                                                              (line  57)
* DJGPP:                                 Known Build Problems.
                                                              (line  18)
* DLLs:                                  Notes for Particular Systems.
                                                              (line  70)
* DocBook:                               Build Options.       (line 346)
* Documentation formats:                 Build Options.       (line 339)
* Documentation license:                 GNU Free Documentation License.
                                                              (line   6)
* DVI:                                   Build Options.       (line 342)
* Efficiency:                            Efficiency.          (line   6)
* Emacs:                                 Emacs.               (line   6)
* Exact division functions:              Integer Division.    (line 112)
* Exact remainder:                       Exact Remainder.     (line   6)
* Example programs:                      Demonstration Programs.
                                                              (line   6)
* Exec prefix:                           Build Options.       (line  32)
* Execution profiling <1>:               Build Options.       (line 323)
* Execution profiling:                   Profiling.           (line   6)
* Exponentiation functions <1>:          Float Arithmetic.    (line  41)
* Exponentiation functions:              Integer Exponentiation.
                                                              (line   6)
* Export:                                Integer Import and Export.
                                                              (line  45)
* Expression parsing demo:               Demonstration Programs.
                                                              (line  15)
* Extended GCD:                          Number Theoretic Functions.
                                                              (line  43)
* Factor removal functions:              Number Theoretic Functions.
                                                              (line 104)
* Factorial algorithm:                   Factorial Algorithm. (line   6)
* Factorial functions:                   Number Theoretic Functions.
                                                              (line 112)
* Factorization demo:                    Demonstration Programs.
                                                              (line  25)
* Fast Fourier Transform:                FFT Multiplication.  (line   6)
* Fat binary:                            Build Options.       (line 161)
* FFT multiplication <1>:                FFT Multiplication.  (line   6)
* FFT multiplication:                    Build Options.       (line 313)
* Fibonacci number algorithm:            Fibonacci Numbers Algorithm.
                                                              (line   6)
* Fibonacci sequence functions:          Number Theoretic Functions.
                                                              (line 132)
* Float arithmetic functions:            Float Arithmetic.    (line   6)
* Float assignment functions <1>:        Simultaneous Float Init & Assign.
                                                              (line   6)
* Float assignment functions:            Assigning Floats.    (line   6)
* Float comparison functions:            Float Comparison.    (line   6)
* Float conversion functions:            Converting Floats.   (line   6)
* Float functions:                       Floating-point Functions.
                                                              (line   6)
* Float initialization functions <1>:    Simultaneous Float Init & Assign.
                                                              (line   6)
* Float initialization functions:        Initializing Floats. (line   6)
* Float input and output functions:      I/O of Floats.       (line   6)
* Float internals:                       Float Internals.     (line   6)
* Float miscellaneous functions:         Miscellaneous Float Functions.
                                                              (line   6)
* Float random number functions:         Miscellaneous Float Functions.
                                                              (line  27)
* Float rounding functions:              Miscellaneous Float Functions.
                                                              (line   9)
* Float sign tests:                      Float Comparison.    (line  34)
* Floating point mode:                   Notes for Particular Systems.
                                                              (line  34)
* Floating-point functions:              Floating-point Functions.
                                                              (line   6)
* Floating-point number:                 Nomenclature and Types.
                                                              (line  21)
* fnccheck:                              Profiling.           (line  77)
* Formatted input:                       Formatted Input.     (line   6)
* Formatted output:                      Formatted Output.    (line   6)
* Free Documentation License:            GNU Free Documentation License.
                                                              (line   6)
* FreeBSD:                               Notes for Particular Systems.
                                                              (line  43)
* frexp <1>:                             Converting Integers. (line  43)
* frexp:                                 Converting Floats.   (line  24)
* FTP of latest version:                 Introduction to GMP. (line  37)
* Function classes:                      Function Classes.    (line   6)
* FunctionCheck:                         Profiling.           (line  77)
* GCC Checker:                           Debugging.           (line 115)
* GCD algorithms:                        Greatest Common Divisor Algorithms.
                                                              (line   6)
* GCD extended:                          Number Theoretic Functions.
                                                              (line  43)
* GCD functions:                         Number Theoretic Functions.
                                                              (line  26)
* GDB:                                   Debugging.           (line  58)
* Generic C:                             Build Options.       (line 152)
* GMP Perl module:                       Demonstration Programs.
                                                              (line  35)
* GMP version number:                    Useful Macros and Constants.
                                                              (line  12)
* gmp.h:                                 Headers and Libraries.
                                                              (line   6)
* gmpxx.h:                               C++ Interface General.
                                                              (line   8)
* GNU Debugger:                          Debugging.           (line  58)
* GNU Free Documentation License:        GNU Free Documentation License.
                                                              (line   6)
* GNU strip:                             Known Build Problems.
                                                              (line  28)
* gprof:                                 Profiling.           (line  41)
* Greatest common divisor algorithms:    Greatest Common Divisor Algorithms.
                                                              (line   6)
* Greatest common divisor functions:     Number Theoretic Functions.
                                                              (line  26)
* Hardware floating point mode:          Notes for Particular Systems.
                                                              (line  34)
* Headers:                               Headers and Libraries.
                                                              (line   6)
* Heap problems:                         Debugging.           (line  24)
* Home page:                             Introduction to GMP. (line  33)
* Host system:                           Build Options.       (line  66)
* HP-UX:                                 ABI and ISA.         (line  77)
* HPPA:                                  ABI and ISA.         (line  77)
* I/O functions <1>:                     I/O of Integers.     (line   6)
* I/O functions <2>:                     I/O of Floats.       (line   6)
* I/O functions:                         I/O of Rationals.    (line   6)
* i386:                                  Notes for Particular Systems.
                                                              (line 150)
* IA-64:                                 ABI and ISA.         (line 116)
* Import:                                Integer Import and Export.
                                                              (line  11)
* In-place operations:                   Efficiency.          (line  57)
* Include files:                         Headers and Libraries.
                                                              (line   6)
* info-lookup-symbol:                    Emacs.               (line   6)
* Initialization functions <1>:          Initializing Floats. (line   6)
* Initialization functions <2>:          Random State Initialization.
                                                              (line   6)
* Initialization functions <3>:          Simultaneous Float Init & Assign.
                                                              (line   6)
* Initialization functions <4>:          Simultaneous Integer Init & Assign.
                                                              (line   6)
* Initialization functions <5>:          Initializing Rationals.
                                                              (line   6)
* Initialization functions:              Initializing Integers.
                                                              (line   6)
* Initializing and clearing:             Efficiency.          (line  21)
* Input functions <1>:                   Formatted Input Functions.
                                                              (line   6)
* Input functions <2>:                   I/O of Rationals.    (line   6)
* Input functions <3>:                   I/O of Floats.       (line   6)
* Input functions:                       I/O of Integers.     (line   6)
* Install prefix:                        Build Options.       (line  32)
* Installing GMP:                        Installing GMP.      (line   6)
* Instruction Set Architecture:          ABI and ISA.         (line   6)
* instrument-functions:                  Profiling.           (line  66)
* Integer:                               Nomenclature and Types.
                                                              (line   6)
* Integer arithmetic functions:          Integer Arithmetic.  (line   6)
* Integer assignment functions <1>:      Assigning Integers.  (line   6)
* Integer assignment functions:          Simultaneous Integer Init & Assign.
                                                              (line   6)
* Integer bit manipulation functions:    Integer Logic and Bit Fiddling.
                                                              (line   6)
* Integer comparison functions:          Integer Comparisons. (line   6)
* Integer conversion functions:          Converting Integers. (line   6)
* Integer division functions:            Integer Division.    (line   6)
* Integer exponentiation functions:      Integer Exponentiation.
                                                              (line   6)
* Integer export:                        Integer Import and Export.
                                                              (line  45)
* Integer functions:                     Integer Functions.   (line   6)
* Integer import:                        Integer Import and Export.
                                                              (line  11)
* Integer initialization functions <1>:  Initializing Integers.
                                                              (line   6)
* Integer initialization functions:      Simultaneous Integer Init & Assign.
                                                              (line   6)
* Integer input and output functions:    I/O of Integers.     (line   6)
* Integer internals:                     Integer Internals.   (line   6)
* Integer logical functions:             Integer Logic and Bit Fiddling.
                                                              (line   6)
* Integer miscellaneous functions:       Miscellaneous Integer Functions.
                                                              (line   6)
* Integer random number functions:       Integer Random Numbers.
                                                              (line   6)
* Integer root functions:                Integer Roots.       (line   6)
* Integer sign tests:                    Integer Comparisons. (line  28)
* Integer special functions:             Integer Special Functions.
                                                              (line   6)
* Interix:                               Notes for Particular Systems.
                                                              (line  65)
* Internals:                             Internals.           (line   6)
* Introduction:                          Introduction to GMP. (line   6)
* Inverse modulo functions:              Number Theoretic Functions.
                                                              (line  70)
* IRIX <1>:                              ABI and ISA.         (line 141)
* IRIX:                                  Known Build Problems.
                                                              (line  38)
* ISA:                                   ABI and ISA.         (line   6)
* istream input:                         C++ Formatted Input. (line   6)
* Jacobi symbol algorithm:               Jacobi Symbol.       (line   6)
* Jacobi symbol functions:               Number Theoretic Functions.
                                                              (line  79)
* Karatsuba multiplication:              Karatsuba Multiplication.
                                                              (line   6)
* Karatsuba square root algorithm:       Square Root Algorithm.
                                                              (line   6)
* Kronecker symbol functions:            Number Theoretic Functions.
                                                              (line  91)
* Language bindings:                     Language Bindings.   (line   6)
* Latest version of GMP:                 Introduction to GMP. (line  37)
* LCM functions:                         Number Theoretic Functions.
                                                              (line  64)
* Least common multiple functions:       Number Theoretic Functions.
                                                              (line  64)
* Legendre symbol functions:             Number Theoretic Functions.
                                                              (line  82)
* libgmp:                                Headers and Libraries.
                                                              (line  22)
* libgmpxx:                              Headers and Libraries.
                                                              (line  27)
* Libraries:                             Headers and Libraries.
                                                              (line  22)
* Libtool:                               Headers and Libraries.
                                                              (line  33)
* Libtool versioning:                    Notes for Package Builds.
                                                              (line   9)
* License conditions:                    Copying.             (line   6)
* Limb:                                  Nomenclature and Types.
                                                              (line  31)
* Limb size:                             Useful Macros and Constants.
                                                              (line   7)
* Linear congruential algorithm:         Random Number Algorithms.
                                                              (line  25)
* Linear congruential random numbers:    Random State Initialization.
                                                              (line  32)
* Linking:                               Headers and Libraries.
                                                              (line  22)
* Logical functions:                     Integer Logic and Bit Fiddling.
                                                              (line   6)
* Low-level functions:                   Low-level Functions. (line   6)
* Low-level functions for cryptography:  Low-level Functions. (line 507)
* Lucas number algorithm:                Lucas Numbers Algorithm.
                                                              (line   6)
* Lucas number functions:                Number Theoretic Functions.
                                                              (line 143)
* MacOS X:                               Known Build Problems.
                                                              (line  51)
* Mailing lists:                         Introduction to GMP. (line  44)
* Malloc debugger:                       Debugging.           (line  30)
* Malloc problems:                       Debugging.           (line  24)
* Memory allocation:                     Custom Allocation.   (line   6)
* Memory management:                     Memory Management.   (line   6)
* Mersenne twister algorithm:            Random Number Algorithms.
                                                              (line  17)
* Mersenne twister random numbers:       Random State Initialization.
                                                              (line  13)
* MINGW:                                 Notes for Particular Systems.
                                                              (line  57)
* MIPS:                                  ABI and ISA.         (line 141)
* Miscellaneous float functions:         Miscellaneous Float Functions.
                                                              (line   6)
* Miscellaneous integer functions:       Miscellaneous Integer Functions.
                                                              (line   6)
* MMX:                                   Notes for Particular Systems.
                                                              (line 156)
* Modular inverse functions:             Number Theoretic Functions.
                                                              (line  70)
* Most significant bit:                  Miscellaneous Integer Functions.
                                                              (line  34)
* MPN_PATH:                              Build Options.       (line 327)
* MS Windows:                            Notes for Particular Systems.
                                                              (line  57)
* MS-DOS:                                Notes for Particular Systems.
                                                              (line  57)
* Multi-threading:                       Reentrancy.          (line   6)
* Multiplication algorithms:             Multiplication Algorithms.
                                                              (line   6)
* Nails:                                 Low-level Functions. (line 683)
* Native compilation:                    Build Options.       (line  52)
* NetBSD:                                Notes for Particular Systems.
                                                              (line 100)
* NeXT:                                  Known Build Problems.
                                                              (line  57)
* Next prime function:                   Number Theoretic Functions.
                                                              (line  19)
* Nomenclature:                          Nomenclature and Types.
                                                              (line   6)
* Non-Unix systems:                      Build Options.       (line  11)
* Nth root algorithm:                    Nth Root Algorithm.  (line   6)
* Number sequences:                      Efficiency.          (line 147)
* Number theoretic functions:            Number Theoretic Functions.
                                                              (line   6)
* Numerator and denominator:             Applying Integer Functions.
                                                              (line   6)
* obstack output:                        Formatted Output Functions.
                                                              (line  81)
* OpenBSD:                               Notes for Particular Systems.
                                                              (line 109)
* Optimizing performance:                Performance optimization.
                                                              (line   6)
* ostream output:                        C++ Formatted Output.
                                                              (line   6)
* Other languages:                       Language Bindings.   (line   6)
* Output functions <1>:                  Formatted Output Functions.
                                                              (line   6)
* Output functions <2>:                  I/O of Rationals.    (line   6)
* Output functions <3>:                  I/O of Integers.     (line   6)
* Output functions:                      I/O of Floats.       (line   6)
* Packaged builds:                       Notes for Package Builds.
                                                              (line   6)
* Parameter conventions:                 Parameter Conventions.
                                                              (line   6)
* Parsing expressions demo:              Demonstration Programs.
                                                              (line  21)
* Particular systems:                    Notes for Particular Systems.
                                                              (line   6)
* Past GMP versions:                     Compatibility with older versions.
                                                              (line   6)
* PDF:                                   Build Options.       (line 342)
* Perfect power algorithm:               Perfect Power Algorithm.
                                                              (line   6)
* Perfect power functions:               Integer Roots.       (line  28)
* Perfect square algorithm:              Perfect Square Algorithm.
                                                              (line   6)
* Perfect square functions:              Integer Roots.       (line  37)
* perl:                                  Demonstration Programs.
                                                              (line  35)
* Perl module:                           Demonstration Programs.
                                                              (line  35)
* Postscript:                            Build Options.       (line 342)
* Power/PowerPC <1>:                     Known Build Problems.
                                                              (line  63)
* Power/PowerPC:                         Notes for Particular Systems.
                                                              (line 115)
* Powering algorithms:                   Powering Algorithms. (line   6)
* Powering functions <1>:                Float Arithmetic.    (line  41)
* Powering functions:                    Integer Exponentiation.
                                                              (line   6)
* PowerPC:                               ABI and ISA.         (line 176)
* Precision of floats:                   Floating-point Functions.
                                                              (line   6)
* Precision of hardware floating point:  Notes for Particular Systems.
                                                              (line  34)
* Prefix:                                Build Options.       (line  32)
* Prime testing algorithms:              Prime Testing Algorithm.
                                                              (line   6)
* Prime testing functions:               Number Theoretic Functions.
                                                              (line   7)
* Primorial functions:                   Number Theoretic Functions.
                                                              (line 117)
* printf formatted output:               Formatted Output.    (line   6)
* Probable prime testing functions:      Number Theoretic Functions.
                                                              (line   7)
* prof:                                  Profiling.           (line  24)
* Profiling:                             Profiling.           (line   6)
* Radix conversion algorithms:           Radix Conversion Algorithms.
                                                              (line   6)
* Random number algorithms:              Random Number Algorithms.
                                                              (line   6)
* Random number functions <1>:           Integer Random Numbers.
                                                              (line   6)
* Random number functions <2>:           Random Number Functions.
                                                              (line   6)
* Random number functions:               Miscellaneous Float Functions.
                                                              (line  27)
* Random number seeding:                 Random State Seeding.
                                                              (line   6)
* Random number state:                   Random State Initialization.
                                                              (line   6)
* Random state:                          Nomenclature and Types.
                                                              (line  46)
* Rational arithmetic:                   Efficiency.          (line 113)
* Rational arithmetic functions:         Rational Arithmetic. (line   6)
* Rational assignment functions:         Initializing Rationals.
                                                              (line   6)
* Rational comparison functions:         Comparing Rationals. (line   6)
* Rational conversion functions:         Rational Conversions.
                                                              (line   6)
* Rational initialization functions:     Initializing Rationals.
                                                              (line   6)
* Rational input and output functions:   I/O of Rationals.    (line   6)
* Rational internals:                    Rational Internals.  (line   6)
* Rational number:                       Nomenclature and Types.
                                                              (line  16)
* Rational number functions:             Rational Number Functions.
                                                              (line   6)
* Rational numerator and denominator:    Applying Integer Functions.
                                                              (line   6)
* Rational sign tests:                   Comparing Rationals. (line  28)
* Raw output internals:                  Raw Output Internals.
                                                              (line   6)
* Reallocations:                         Efficiency.          (line  30)
* Reentrancy:                            Reentrancy.          (line   6)
* References:                            References.          (line   6)
* Remove factor functions:               Number Theoretic Functions.
                                                              (line 104)
* Reporting bugs:                        Reporting Bugs.      (line   6)
* Root extraction algorithm:             Nth Root Algorithm.  (line   6)
* Root extraction algorithms:            Root Extraction Algorithms.
                                                              (line   6)
* Root extraction functions <1>:         Float Arithmetic.    (line  37)
* Root extraction functions:             Integer Roots.       (line   6)
* Root testing functions:                Integer Roots.       (line  28)
* Rounding functions:                    Miscellaneous Float Functions.
                                                              (line   9)
* Sample programs:                       Demonstration Programs.
                                                              (line   6)
* Scan bit functions:                    Integer Logic and Bit Fiddling.
                                                              (line  40)
* scanf formatted input:                 Formatted Input.     (line   6)
* SCO:                                   Known Build Problems.
                                                              (line  38)
* Seeding random numbers:                Random State Seeding.
                                                              (line   6)
* Segmentation violation:                Debugging.           (line   7)
* Sequent Symmetry:                      Known Build Problems.
                                                              (line  68)
* Services for Unix:                     Notes for Particular Systems.
                                                              (line  65)
* Shared library versioning:             Notes for Package Builds.
                                                              (line   9)
* Sign tests <1>:                        Comparing Rationals. (line  28)
* Sign tests <2>:                        Float Comparison.    (line  34)
* Sign tests:                            Integer Comparisons. (line  28)
* Size in digits:                        Miscellaneous Integer Functions.
                                                              (line  23)
* Small operands:                        Efficiency.          (line   7)
* Solaris <1>:                           ABI and ISA.         (line 208)
* Solaris:                               Known Build Problems.
                                                              (line  72)
* Sparc:                                 Notes for Particular Systems.
                                                              (line 127)
* Sparc V9:                              ABI and ISA.         (line 208)
* Special integer functions:             Integer Special Functions.
                                                              (line   6)
* Square root algorithm:                 Square Root Algorithm.
                                                              (line   6)
* SSE2:                                  Notes for Particular Systems.
                                                              (line 156)
* Stack backtrace:                       Debugging.           (line  50)
* Stack overflow <1>:                    Debugging.           (line   7)
* Stack overflow:                        Build Options.       (line 274)
* Static linking:                        Efficiency.          (line  14)
* stdarg.h:                              Headers and Libraries.
                                                              (line  17)
* stdio.h:                               Headers and Libraries.
                                                              (line  11)
* Stripped libraries:                    Known Build Problems.
                                                              (line  28)
* Sun:                                   ABI and ISA.         (line 208)
* SunOS:                                 Notes for Particular Systems.
                                                              (line 144)
* Systems:                               Notes for Particular Systems.
                                                              (line   6)
* Temporary memory:                      Build Options.       (line 274)
* Texinfo:                               Build Options.       (line 339)
* Text input/output:                     Efficiency.          (line 153)
* Thread safety:                         Reentrancy.          (line   6)
* Toom multiplication <1>:               Higher degree Toom'n'half.
                                                              (line   6)
* Toom multiplication <2>:               Other Multiplication.
                                                              (line   6)
* Toom multiplication <3>:               Toom 4-Way Multiplication.
                                                              (line   6)
* Toom multiplication:                   Toom 3-Way Multiplication.
                                                              (line   6)
* Types:                                 Nomenclature and Types.
                                                              (line   6)
* ui and si functions:                   Efficiency.          (line  50)
* Unbalanced multiplication:             Unbalanced Multiplication.
                                                              (line   6)
* Upward compatibility:                  Compatibility with older versions.
                                                              (line   6)
* Useful macros and constants:           Useful Macros and Constants.
                                                              (line   6)
* User-defined precision:                Floating-point Functions.
                                                              (line   6)
* Valgrind:                              Debugging.           (line 130)
* Variable conventions:                  Variable Conventions.
                                                              (line   6)
* Version number:                        Useful Macros and Constants.
                                                              (line  12)
* Web page:                              Introduction to GMP. (line  33)
* Windows:                               Notes for Particular Systems.
                                                              (line  70)
* x86:                                   Notes for Particular Systems.
                                                              (line 150)
* x87:                                   Notes for Particular Systems.
                                                              (line  34)
* XML:                                   Build Options.       (line 346)


File: gmp.info,  Node: Function Index,  Prev: Concept Index,  Up: Top

Function and Type Index
***********************

 [index ]
* Menu:

* __GMP_CC:                              Useful Macros and Constants.
                                                              (line  23)
* __GMP_CFLAGS:                          Useful Macros and Constants.
                                                              (line  24)
* __GNU_MP_VERSION:                      Useful Macros and Constants.
                                                              (line  10)
* __GNU_MP_VERSION_MINOR:                Useful Macros and Constants.
                                                              (line  11)
* __GNU_MP_VERSION_PATCHLEVEL:           Useful Macros and Constants.
                                                              (line  12)
* _mpz_realloc:                          Integer Special Functions.
                                                              (line  14)
* abs <1>:                               C++ Interface Rationals.
                                                              (line  49)
* abs <2>:                               C++ Interface Integers.
                                                              (line  47)
* abs:                                   C++ Interface Floats.
                                                              (line  83)
* ceil:                                  C++ Interface Floats.
                                                              (line  84)
* cmp <1>:                               C++ Interface Floats.
                                                              (line  85)
* cmp <2>:                               C++ Interface Integers.
                                                              (line  48)
* cmp <3>:                               C++ Interface Rationals.
                                                              (line  51)
* cmp:                                   C++ Interface Floats.
                                                              (line  86)
* floor:                                 C++ Interface Floats.
                                                              (line  93)
* gcd:                                   C++ Interface Integers.
                                                              (line  64)
* gmp_asprintf:                          Formatted Output Functions.
                                                              (line  65)
* gmp_errno:                             Random State Initialization.
                                                              (line  55)
* GMP_ERROR_INVALID_ARGUMENT:            Random State Initialization.
                                                              (line  55)
* GMP_ERROR_UNSUPPORTED_ARGUMENT:        Random State Initialization.
                                                              (line  55)
* gmp_fprintf:                           Formatted Output Functions.
                                                              (line  29)
* gmp_fscanf:                            Formatted Input Functions.
                                                              (line  25)
* GMP_LIMB_BITS:                         Low-level Functions. (line 713)
* GMP_NAIL_BITS:                         Low-level Functions. (line 711)
* GMP_NAIL_MASK:                         Low-level Functions. (line 721)
* GMP_NUMB_BITS:                         Low-level Functions. (line 712)
* GMP_NUMB_MASK:                         Low-level Functions. (line 722)
* GMP_NUMB_MAX:                          Low-level Functions. (line 730)
* gmp_obstack_printf:                    Formatted Output Functions.
                                                              (line  79)
* gmp_obstack_vprintf:                   Formatted Output Functions.
                                                              (line  81)
* gmp_printf:                            Formatted Output Functions.
                                                              (line  24)
* GMP_RAND_ALG_DEFAULT:                  Random State Initialization.
                                                              (line  49)
* GMP_RAND_ALG_LC:                       Random State Initialization.
                                                              (line  49)
* gmp_randclass:                         C++ Interface Random Numbers.
                                                              (line   7)
* gmp_randclass::get_f:                  C++ Interface Random Numbers.
                                                              (line  46)
* gmp_randclass::get_z_bits:             C++ Interface Random Numbers.
                                                              (line  39)
* gmp_randclass::get_z_range:            C++ Interface Random Numbers.
                                                              (line  42)
* gmp_randclass::gmp_randclass:          C++ Interface Random Numbers.
                                                              (line  27)
* gmp_randclass::seed:                   C++ Interface Random Numbers.
                                                              (line  34)
* gmp_randclear:                         Random State Initialization.
                                                              (line  62)
* gmp_randinit:                          Random State Initialization.
                                                              (line  47)
* gmp_randinit_default:                  Random State Initialization.
                                                              (line   7)
* gmp_randinit_lc_2exp:                  Random State Initialization.
                                                              (line  18)
* gmp_randinit_lc_2exp_size:             Random State Initialization.
                                                              (line  32)
* gmp_randinit_mt:                       Random State Initialization.
                                                              (line  13)
* gmp_randinit_set:                      Random State Initialization.
                                                              (line  43)
* gmp_randseed:                          Random State Seeding.
                                                              (line   8)
* gmp_randseed_ui:                       Random State Seeding.
                                                              (line  10)
* gmp_randstate_t:                       Nomenclature and Types.
                                                              (line  46)
* gmp_scanf:                             Formatted Input Functions.
                                                              (line  21)
* gmp_snprintf:                          Formatted Output Functions.
                                                              (line  46)
* gmp_sprintf:                           Formatted Output Functions.
                                                              (line  34)
* gmp_sscanf:                            Formatted Input Functions.
                                                              (line  29)
* gmp_urandomb_ui:                       Random State Miscellaneous.
                                                              (line   8)
* gmp_urandomm_ui:                       Random State Miscellaneous.
                                                              (line  14)
* gmp_vasprintf:                         Formatted Output Functions.
                                                              (line  66)
* gmp_version:                           Useful Macros and Constants.
                                                              (line  18)
* gmp_vfprintf:                          Formatted Output Functions.
                                                              (line  30)
* gmp_vfscanf:                           Formatted Input Functions.
                                                              (line  26)
* gmp_vprintf:                           Formatted Output Functions.
                                                              (line  25)
* gmp_vscanf:                            Formatted Input Functions.
                                                              (line  22)
* gmp_vsnprintf:                         Formatted Output Functions.
                                                              (line  48)
* gmp_vsprintf:                          Formatted Output Functions.
                                                              (line  35)
* gmp_vsscanf:                           Formatted Input Functions.
                                                              (line  31)
* hypot:                                 C++ Interface Floats.
                                                              (line  94)
* lcm:                                   C++ Interface Integers.
                                                              (line  65)
* mp_bitcnt_t:                           Nomenclature and Types.
                                                              (line  42)
* mp_bits_per_limb:                      Useful Macros and Constants.
                                                              (line   7)
* mp_exp_t:                              Nomenclature and Types.
                                                              (line  27)
* mp_get_memory_functions:               Custom Allocation.   (line  90)
* mp_limb_t:                             Nomenclature and Types.
                                                              (line  31)
* mp_set_memory_functions:               Custom Allocation.   (line  18)
* mp_size_t:                             Nomenclature and Types.
                                                              (line  37)
* mpf_abs:                               Float Arithmetic.    (line  47)
* mpf_add:                               Float Arithmetic.    (line   7)
* mpf_add_ui:                            Float Arithmetic.    (line   9)
* mpf_ceil:                              Miscellaneous Float Functions.
                                                              (line   7)
* mpf_class:                             C++ Interface General.
                                                              (line  20)
* mpf_class::fits_sint_p:                C++ Interface Floats.
                                                              (line  87)
* mpf_class::fits_slong_p:               C++ Interface Floats.
                                                              (line  88)
* mpf_class::fits_sshort_p:              C++ Interface Floats.
                                                              (line  89)
* mpf_class::fits_uint_p:                C++ Interface Floats.
                                                              (line  90)
* mpf_class::fits_ulong_p:               C++ Interface Floats.
                                                              (line  91)
* mpf_class::fits_ushort_p:              C++ Interface Floats.
                                                              (line  92)
* mpf_class::get_d:                      C++ Interface Floats.
                                                              (line  95)
* mpf_class::get_mpf_t:                  C++ Interface General.
                                                              (line  66)
* mpf_class::get_prec:                   C++ Interface Floats.
                                                              (line 115)
* mpf_class::get_si:                     C++ Interface Floats.
                                                              (line  96)
* mpf_class::get_str:                    C++ Interface Floats.
                                                              (line  98)
* mpf_class::get_ui:                     C++ Interface Floats.
                                                              (line  99)
* mpf_class::mpf_class:                  C++ Interface Floats.
                                                              (line  47)
* mpf_class::operator=:                  C++ Interface Floats.
                                                              (line  60)
* mpf_class::set_prec:                   C++ Interface Floats.
                                                              (line 116)
* mpf_class::set_prec_raw:               C++ Interface Floats.
                                                              (line 117)
* mpf_class::set_str:                    C++ Interface Floats.
                                                              (line 101)
* mpf_class::swap:                       C++ Interface Floats.
                                                              (line 104)
* mpf_clear:                             Initializing Floats. (line  37)
* mpf_clears:                            Initializing Floats. (line  41)
* mpf_cmp:                               Float Comparison.    (line   7)
* mpf_cmp_d:                             Float Comparison.    (line   9)
* mpf_cmp_si:                            Float Comparison.    (line  11)
* mpf_cmp_ui:                            Float Comparison.    (line  10)
* mpf_cmp_z:                             Float Comparison.    (line   8)
* mpf_div:                               Float Arithmetic.    (line  29)
* mpf_div_2exp:                          Float Arithmetic.    (line  55)
* mpf_div_ui:                            Float Arithmetic.    (line  33)
* mpf_eq:                                Float Comparison.    (line  19)
* mpf_fits_sint_p:                       Miscellaneous Float Functions.
                                                              (line  20)
* mpf_fits_slong_p:                      Miscellaneous Float Functions.
                                                              (line  18)
* mpf_fits_sshort_p:                     Miscellaneous Float Functions.
                                                              (line  22)
* mpf_fits_uint_p:                       Miscellaneous Float Functions.
                                                              (line  19)
* mpf_fits_ulong_p:                      Miscellaneous Float Functions.
                                                              (line  17)
* mpf_fits_ushort_p:                     Miscellaneous Float Functions.
                                                              (line  21)
* mpf_floor:                             Miscellaneous Float Functions.
                                                              (line   8)
* mpf_get_d:                             Converting Floats.   (line   7)
* mpf_get_d_2exp:                        Converting Floats.   (line  17)
* mpf_get_default_prec:                  Initializing Floats. (line  12)
* mpf_get_prec:                          Initializing Floats. (line  62)
* mpf_get_si:                            Converting Floats.   (line  28)
* mpf_get_str:                           Converting Floats.   (line  38)
* mpf_get_ui:                            Converting Floats.   (line  29)
* mpf_init:                              Initializing Floats. (line  19)
* mpf_init2:                             Initializing Floats. (line  26)
* mpf_init_set:                          Simultaneous Float Init & Assign.
                                                              (line  16)
* mpf_init_set_d:                        Simultaneous Float Init & Assign.
                                                              (line  19)
* mpf_init_set_si:                       Simultaneous Float Init & Assign.
                                                              (line  18)
* mpf_init_set_str:                      Simultaneous Float Init & Assign.
                                                              (line  26)
* mpf_init_set_ui:                       Simultaneous Float Init & Assign.
                                                              (line  17)
* mpf_inits:                             Initializing Floats. (line  31)
* mpf_inp_str:                           I/O of Floats.       (line  39)
* mpf_integer_p:                         Miscellaneous Float Functions.
                                                              (line  14)
* mpf_mul:                               Float Arithmetic.    (line  19)
* mpf_mul_2exp:                          Float Arithmetic.    (line  51)
* mpf_mul_ui:                            Float Arithmetic.    (line  21)
* mpf_neg:                               Float Arithmetic.    (line  44)
* mpf_out_str:                           I/O of Floats.       (line  19)
* mpf_pow_ui:                            Float Arithmetic.    (line  41)
* mpf_random2:                           Miscellaneous Float Functions.
                                                              (line  37)
* mpf_reldiff:                           Float Comparison.    (line  30)
* mpf_set:                               Assigning Floats.    (line  10)
* mpf_set_d:                             Assigning Floats.    (line  13)
* mpf_set_default_prec:                  Initializing Floats. (line   7)
* mpf_set_prec:                          Initializing Floats. (line  65)
* mpf_set_prec_raw:                      Initializing Floats. (line  72)
* mpf_set_q:                             Assigning Floats.    (line  15)
* mpf_set_si:                            Assigning Floats.    (line  12)
* mpf_set_str:                           Assigning Floats.    (line  18)
* mpf_set_ui:                            Assigning Floats.    (line  11)
* mpf_set_z:                             Assigning Floats.    (line  14)
* mpf_sgn:                               Float Comparison.    (line  34)
* mpf_sqrt:                              Float Arithmetic.    (line  36)
* mpf_sqrt_ui:                           Float Arithmetic.    (line  37)
* mpf_sub:                               Float Arithmetic.    (line  12)
* mpf_sub_ui:                            Float Arithmetic.    (line  16)
* mpf_swap:                              Assigning Floats.    (line  52)
* mpf_t:                                 Nomenclature and Types.
                                                              (line  21)
* mpf_trunc:                             Miscellaneous Float Functions.
                                                              (line   9)
* mpf_ui_div:                            Float Arithmetic.    (line  31)
* mpf_ui_sub:                            Float Arithmetic.    (line  14)
* mpf_urandomb:                          Miscellaneous Float Functions.
                                                              (line  27)
* mpn_add:                               Low-level Functions. (line  69)
* mpn_add_1:                             Low-level Functions. (line  64)
* mpn_add_n:                             Low-level Functions. (line  54)
* mpn_addmul_1:                          Low-level Functions. (line 150)
* mpn_and_n:                             Low-level Functions. (line 449)
* mpn_andn_n:                            Low-level Functions. (line 464)
* mpn_cmp:                               Low-level Functions. (line 295)
* mpn_cnd_add_n:                         Low-level Functions. (line 542)
* mpn_cnd_sub_n:                         Low-level Functions. (line 544)
* mpn_cnd_swap:                          Low-level Functions. (line 569)
* mpn_com:                               Low-level Functions. (line 489)
* mpn_copyd:                             Low-level Functions. (line 498)
* mpn_copyi:                             Low-level Functions. (line 494)
* mpn_divexact_1:                        Low-level Functions. (line 233)
* mpn_divexact_by3:                      Low-level Functions. (line 240)
* mpn_divexact_by3c:                     Low-level Functions. (line 242)
* mpn_divmod:                            Low-level Functions. (line 228)
* mpn_divmod_1:                          Low-level Functions. (line 212)
* mpn_divrem:                            Low-level Functions. (line 186)
* mpn_divrem_1:                          Low-level Functions. (line 210)
* mpn_gcd:                               Low-level Functions. (line 303)
* mpn_gcd_1:                             Low-level Functions. (line 313)
* mpn_gcdext:                            Low-level Functions. (line 319)
* mpn_get_str:                           Low-level Functions. (line 373)
* mpn_hamdist:                           Low-level Functions. (line 438)
* mpn_ior_n:                             Low-level Functions. (line 454)
* mpn_iorn_n:                            Low-level Functions. (line 469)
* mpn_lshift:                            Low-level Functions. (line 271)
* mpn_mod_1:                             Low-level Functions. (line 266)
* mpn_mul:                               Low-level Functions. (line 116)
* mpn_mul_1:                             Low-level Functions. (line 135)
* mpn_mul_n:                             Low-level Functions. (line 105)
* mpn_nand_n:                            Low-level Functions. (line 474)
* mpn_neg:                               Low-level Functions. (line  98)
* mpn_nior_n:                            Low-level Functions. (line 479)
* mpn_perfect_square_p:                  Low-level Functions. (line 444)
* mpn_popcount:                          Low-level Functions. (line 434)
* mpn_random:                            Low-level Functions. (line 423)
* mpn_random2:                           Low-level Functions. (line 424)
* mpn_rshift:                            Low-level Functions. (line 283)
* mpn_scan0:                             Low-level Functions. (line 408)
* mpn_scan1:                             Low-level Functions. (line 416)
* mpn_sec_add_1:                         Low-level Functions. (line 555)
* mpn_sec_div_qr:                        Low-level Functions. (line 632)
* mpn_sec_div_qr_itch:                   Low-level Functions. (line 633)
* mpn_sec_div_r:                         Low-level Functions. (line 649)
* mpn_sec_div_r_itch:                    Low-level Functions. (line 650)
* mpn_sec_invert:                        Low-level Functions. (line 664)
* mpn_sec_invert_itch:                   Low-level Functions. (line 665)
* mpn_sec_mul:                           Low-level Functions. (line 577)
* mpn_sec_mul_itch:                      Low-level Functions. (line 578)
* mpn_sec_powm:                          Low-level Functions. (line 607)
* mpn_sec_powm_itch:                     Low-level Functions. (line 609)
* mpn_sec_sqr:                           Low-level Functions. (line 592)
* mpn_sec_sqr_itch:                      Low-level Functions. (line 593)
* mpn_sec_sub_1:                         Low-level Functions. (line 557)
* mpn_sec_tabselect:                     Low-level Functions. (line 623)
* mpn_set_str:                           Low-level Functions. (line 388)
* mpn_sizeinbase:                        Low-level Functions. (line 366)
* mpn_sqr:                               Low-level Functions. (line 127)
* mpn_sqrtrem:                           Low-level Functions. (line 348)
* mpn_sub:                               Low-level Functions. (line  90)
* mpn_sub_1:                             Low-level Functions. (line  85)
* mpn_sub_n:                             Low-level Functions. (line  76)
* mpn_submul_1:                          Low-level Functions. (line 162)
* mpn_tdiv_qr:                           Low-level Functions. (line 175)
* mpn_xnor_n:                            Low-level Functions. (line 484)
* mpn_xor_n:                             Low-level Functions. (line 459)
* mpn_zero:                              Low-level Functions. (line 501)
* mpn_zero_p:                            Low-level Functions. (line 299)
* mpq_abs:                               Rational Arithmetic. (line  34)
* mpq_add:                               Rational Arithmetic. (line   8)
* mpq_canonicalize:                      Rational Number Functions.
                                                              (line  22)
* mpq_class:                             C++ Interface General.
                                                              (line  19)
* mpq_class::canonicalize:               C++ Interface Rationals.
                                                              (line  43)
* mpq_class::get_d:                      C++ Interface Rationals.
                                                              (line  52)
* mpq_class::get_den:                    C++ Interface Rationals.
                                                              (line  66)
* mpq_class::get_den_mpz_t:              C++ Interface Rationals.
                                                              (line  76)
* mpq_class::get_mpq_t:                  C++ Interface General.
                                                              (line  65)
* mpq_class::get_num:                    C++ Interface Rationals.
                                                              (line  65)
* mpq_class::get_num_mpz_t:              C++ Interface Rationals.
                                                              (line  75)
* mpq_class::get_str:                    C++ Interface Rationals.
                                                              (line  53)
* mpq_class::mpq_class:                  C++ Interface Rationals.
                                                              (line  12)
* mpq_class::set_str:                    C++ Interface Rationals.
                                                              (line  55)
* mpq_class::swap:                       C++ Interface Rationals.
                                                              (line  57)
* mpq_clear:                             Initializing Rationals.
                                                              (line  16)
* mpq_clears:                            Initializing Rationals.
                                                              (line  20)
* mpq_cmp:                               Comparing Rationals. (line   7)
* mpq_cmp_si:                            Comparing Rationals. (line  18)
* mpq_cmp_ui:                            Comparing Rationals. (line  16)
* mpq_cmp_z:                             Comparing Rationals. (line   8)
* mpq_denref:                            Applying Integer Functions.
                                                              (line  18)
* mpq_div:                               Rational Arithmetic. (line  24)
* mpq_div_2exp:                          Rational Arithmetic. (line  28)
* mpq_equal:                             Comparing Rationals. (line  34)
* mpq_get_d:                             Rational Conversions.
                                                              (line   7)
* mpq_get_den:                           Applying Integer Functions.
                                                              (line  24)
* mpq_get_num:                           Applying Integer Functions.
                                                              (line  23)
* mpq_get_str:                           Rational Conversions.
                                                              (line  22)
* mpq_init:                              Initializing Rationals.
                                                              (line   7)
* mpq_inits:                             Initializing Rationals.
                                                              (line  12)
* mpq_inp_str:                           I/O of Rationals.    (line  27)
* mpq_inv:                               Rational Arithmetic. (line  37)
* mpq_mul:                               Rational Arithmetic. (line  16)
* mpq_mul_2exp:                          Rational Arithmetic. (line  20)
* mpq_neg:                               Rational Arithmetic. (line  31)
* mpq_numref:                            Applying Integer Functions.
                                                              (line  17)
* mpq_out_str:                           I/O of Rationals.    (line  19)
* mpq_set:                               Initializing Rationals.
                                                              (line  24)
* mpq_set_d:                             Rational Conversions.
                                                              (line  17)
* mpq_set_den:                           Applying Integer Functions.
                                                              (line  26)
* mpq_set_f:                             Rational Conversions.
                                                              (line  18)
* mpq_set_num:                           Applying Integer Functions.
                                                              (line  25)
* mpq_set_si:                            Initializing Rationals.
                                                              (line  31)
* mpq_set_str:                           Initializing Rationals.
                                                              (line  36)
* mpq_set_ui:                            Initializing Rationals.
                                                              (line  29)
* mpq_set_z:                             Initializing Rationals.
                                                              (line  25)
* mpq_sgn:                               Comparing Rationals. (line  28)
* mpq_sub:                               Rational Arithmetic. (line  12)
* mpq_swap:                              Initializing Rationals.
                                                              (line  56)
* mpq_t:                                 Nomenclature and Types.
                                                              (line  16)
* mpz_2fac_ui:                           Number Theoretic Functions.
                                                              (line 110)
* mpz_abs:                               Integer Arithmetic.  (line  45)
* mpz_add:                               Integer Arithmetic.  (line   7)
* mpz_add_ui:                            Integer Arithmetic.  (line   9)
* mpz_addmul:                            Integer Arithmetic.  (line  26)
* mpz_addmul_ui:                         Integer Arithmetic.  (line  28)
* mpz_and:                               Integer Logic and Bit Fiddling.
                                                              (line  11)
* mpz_array_init:                        Integer Special Functions.
                                                              (line  11)
* mpz_bin_ui:                            Number Theoretic Functions.
                                                              (line 122)
* mpz_bin_uiui:                          Number Theoretic Functions.
                                                              (line 124)
* mpz_cdiv_q:                            Integer Division.    (line  13)
* mpz_cdiv_q_2exp:                       Integer Division.    (line  26)
* mpz_cdiv_q_ui:                         Integer Division.    (line  18)
* mpz_cdiv_qr:                           Integer Division.    (line  16)
* mpz_cdiv_qr_ui:                        Integer Division.    (line  22)
* mpz_cdiv_r:                            Integer Division.    (line  14)
* mpz_cdiv_r_2exp:                       Integer Division.    (line  28)
* mpz_cdiv_r_ui:                         Integer Division.    (line  20)
* mpz_cdiv_ui:                           Integer Division.    (line  24)
* mpz_class:                             C++ Interface General.
                                                              (line  18)
* mpz_class::fits_sint_p:                C++ Interface Integers.
                                                              (line  50)
* mpz_class::fits_slong_p:               C++ Interface Integers.
                                                              (line  51)
* mpz_class::fits_sshort_p:              C++ Interface Integers.
                                                              (line  52)
* mpz_class::fits_uint_p:                C++ Interface Integers.
                                                              (line  53)
* mpz_class::fits_ulong_p:               C++ Interface Integers.
                                                              (line  54)
* mpz_class::fits_ushort_p:              C++ Interface Integers.
                                                              (line  55)
* mpz_class::get_d:                      C++ Interface Integers.
                                                              (line  56)
* mpz_class::get_mpz_t:                  C++ Interface General.
                                                              (line  64)
* mpz_class::get_si:                     C++ Interface Integers.
                                                              (line  57)
* mpz_class::get_str:                    C++ Interface Integers.
                                                              (line  58)
* mpz_class::get_ui:                     C++ Interface Integers.
                                                              (line  59)
* mpz_class::mpz_class:                  C++ Interface Integers.
                                                              (line   7)
* mpz_class::set_str:                    C++ Interface Integers.
                                                              (line  61)
* mpz_class::swap:                       C++ Interface Integers.
                                                              (line  66)
* mpz_clear:                             Initializing Integers.
                                                              (line  49)
* mpz_clears:                            Initializing Integers.
                                                              (line  53)
* mpz_clrbit:                            Integer Logic and Bit Fiddling.
                                                              (line  56)
* mpz_cmp:                               Integer Comparisons. (line   7)
* mpz_cmp_d:                             Integer Comparisons. (line   8)
* mpz_cmp_si:                            Integer Comparisons. (line   9)
* mpz_cmp_ui:                            Integer Comparisons. (line  10)
* mpz_cmpabs:                            Integer Comparisons. (line  18)
* mpz_cmpabs_d:                          Integer Comparisons. (line  19)
* mpz_cmpabs_ui:                         Integer Comparisons. (line  20)
* mpz_com:                               Integer Logic and Bit Fiddling.
                                                              (line  20)
* mpz_combit:                            Integer Logic and Bit Fiddling.
                                                              (line  59)
* mpz_congruent_2exp_p:                  Integer Division.    (line 137)
* mpz_congruent_p:                       Integer Division.    (line 133)
* mpz_congruent_ui_p:                    Integer Division.    (line 135)
* mpz_divexact:                          Integer Division.    (line 110)
* mpz_divexact_ui:                       Integer Division.    (line 112)
* mpz_divisible_2exp_p:                  Integer Division.    (line 123)
* mpz_divisible_p:                       Integer Division.    (line 120)
* mpz_divisible_ui_p:                    Integer Division.    (line 122)
* mpz_even_p:                            Miscellaneous Integer Functions.
                                                              (line  18)
* mpz_export:                            Integer Import and Export.
                                                              (line  45)
* mpz_fac_ui:                            Number Theoretic Functions.
                                                              (line 109)
* mpz_fdiv_q:                            Integer Division.    (line  30)
* mpz_fdiv_q_2exp:                       Integer Division.    (line  43)
* mpz_fdiv_q_ui:                         Integer Division.    (line  35)
* mpz_fdiv_qr:                           Integer Division.    (line  33)
* mpz_fdiv_qr_ui:                        Integer Division.    (line  39)
* mpz_fdiv_r:                            Integer Division.    (line  31)
* mpz_fdiv_r_2exp:                       Integer Division.    (line  45)
* mpz_fdiv_r_ui:                         Integer Division.    (line  37)
* mpz_fdiv_ui:                           Integer Division.    (line  41)
* mpz_fib2_ui:                           Number Theoretic Functions.
                                                              (line 132)
* mpz_fib_ui:                            Number Theoretic Functions.
                                                              (line 130)
* mpz_fits_sint_p:                       Miscellaneous Integer Functions.
                                                              (line  10)
* mpz_fits_slong_p:                      Miscellaneous Integer Functions.
                                                              (line   8)
* mpz_fits_sshort_p:                     Miscellaneous Integer Functions.
                                                              (line  12)
* mpz_fits_uint_p:                       Miscellaneous Integer Functions.
                                                              (line   9)
* mpz_fits_ulong_p:                      Miscellaneous Integer Functions.
                                                              (line   7)
* mpz_fits_ushort_p:                     Miscellaneous Integer Functions.
                                                              (line  11)
* mpz_gcd:                               Number Theoretic Functions.
                                                              (line  26)
* mpz_gcd_ui:                            Number Theoretic Functions.
                                                              (line  33)
* mpz_gcdext:                            Number Theoretic Functions.
                                                              (line  43)
* mpz_get_d:                             Converting Integers. (line  27)
* mpz_get_d_2exp:                        Converting Integers. (line  36)
* mpz_get_si:                            Converting Integers. (line  18)
* mpz_get_str:                           Converting Integers. (line  47)
* mpz_get_ui:                            Converting Integers. (line  11)
* mpz_getlimbn:                          Integer Special Functions.
                                                              (line  23)
* mpz_hamdist:                           Integer Logic and Bit Fiddling.
                                                              (line  29)
* mpz_import:                            Integer Import and Export.
                                                              (line  11)
* mpz_init:                              Initializing Integers.
                                                              (line  26)
* mpz_init2:                             Initializing Integers.
                                                              (line  33)
* mpz_init_set:                          Simultaneous Integer Init & Assign.
                                                              (line  27)
* mpz_init_set_d:                        Simultaneous Integer Init & Assign.
                                                              (line  30)
* mpz_init_set_si:                       Simultaneous Integer Init & Assign.
                                                              (line  29)
* mpz_init_set_str:                      Simultaneous Integer Init & Assign.
                                                              (line  35)
* mpz_init_set_ui:                       Simultaneous Integer Init & Assign.
                                                              (line  28)
* mpz_inits:                             Initializing Integers.
                                                              (line  29)
* mpz_inp_raw:                           I/O of Integers.     (line  62)
* mpz_inp_str:                           I/O of Integers.     (line  31)
* mpz_invert:                            Number Theoretic Functions.
                                                              (line  70)
* mpz_ior:                               Integer Logic and Bit Fiddling.
                                                              (line  14)
* mpz_jacobi:                            Number Theoretic Functions.
                                                              (line  79)
* mpz_kronecker:                         Number Theoretic Functions.
                                                              (line  87)
* mpz_kronecker_si:                      Number Theoretic Functions.
                                                              (line  88)
* mpz_kronecker_ui:                      Number Theoretic Functions.
                                                              (line  89)
* mpz_lcm:                               Number Theoretic Functions.
                                                              (line  62)
* mpz_lcm_ui:                            Number Theoretic Functions.
                                                              (line  64)
* mpz_legendre:                          Number Theoretic Functions.
                                                              (line  82)
* mpz_limbs_finish:                      Integer Special Functions.
                                                              (line  48)
* mpz_limbs_modify:                      Integer Special Functions.
                                                              (line  41)
* mpz_limbs_read:                        Integer Special Functions.
                                                              (line  35)
* mpz_limbs_write:                       Integer Special Functions.
                                                              (line  40)
* mpz_lucnum2_ui:                        Number Theoretic Functions.
                                                              (line 143)
* mpz_lucnum_ui:                         Number Theoretic Functions.
                                                              (line 141)
* mpz_mfac_uiui:                         Number Theoretic Functions.
                                                              (line 112)
* mpz_mod:                               Integer Division.    (line 100)
* mpz_mod_ui:                            Integer Division.    (line 102)
* mpz_mul:                               Integer Arithmetic.  (line  19)
* mpz_mul_2exp:                          Integer Arithmetic.  (line  38)
* mpz_mul_si:                            Integer Arithmetic.  (line  20)
* mpz_mul_ui:                            Integer Arithmetic.  (line  22)
* mpz_neg:                               Integer Arithmetic.  (line  42)
* mpz_nextprime:                         Number Theoretic Functions.
                                                              (line  19)
* mpz_odd_p:                             Miscellaneous Integer Functions.
                                                              (line  17)
* mpz_out_raw:                           I/O of Integers.     (line  46)
* mpz_out_str:                           I/O of Integers.     (line  19)
* mpz_perfect_power_p:                   Integer Roots.       (line  28)
* mpz_perfect_square_p:                  Integer Roots.       (line  37)
* mpz_popcount:                          Integer Logic and Bit Fiddling.
                                                              (line  23)
* mpz_pow_ui:                            Integer Exponentiation.
                                                              (line  31)
* mpz_powm:                              Integer Exponentiation.
                                                              (line   8)
* mpz_powm_sec:                          Integer Exponentiation.
                                                              (line  18)
* mpz_powm_ui:                           Integer Exponentiation.
                                                              (line  10)
* mpz_primorial_ui:                      Number Theoretic Functions.
                                                              (line 117)
* mpz_probab_prime_p:                    Number Theoretic Functions.
                                                              (line   7)
* mpz_random:                            Integer Random Numbers.
                                                              (line  42)
* mpz_random2:                           Integer Random Numbers.
                                                              (line  51)
* mpz_realloc2:                          Initializing Integers.
                                                              (line  57)
* mpz_remove:                            Number Theoretic Functions.
                                                              (line 104)
* mpz_roinit_n:                          Integer Special Functions.
                                                              (line  69)
* MPZ_ROINIT_N:                          Integer Special Functions.
                                                              (line  84)
* mpz_root:                              Integer Roots.       (line   8)
* mpz_rootrem:                           Integer Roots.       (line  14)
* mpz_rrandomb:                          Integer Random Numbers.
                                                              (line  31)
* mpz_scan0:                             Integer Logic and Bit Fiddling.
                                                              (line  38)
* mpz_scan1:                             Integer Logic and Bit Fiddling.
                                                              (line  40)
* mpz_set:                               Assigning Integers.  (line  10)
* mpz_set_d:                             Assigning Integers.  (line  13)
* mpz_set_f:                             Assigning Integers.  (line  15)
* mpz_set_q:                             Assigning Integers.  (line  14)
* mpz_set_si:                            Assigning Integers.  (line  12)
* mpz_set_str:                           Assigning Integers.  (line  21)
* mpz_set_ui:                            Assigning Integers.  (line  11)
* mpz_setbit:                            Integer Logic and Bit Fiddling.
                                                              (line  53)
* mpz_sgn:                               Integer Comparisons. (line  28)
* mpz_si_kronecker:                      Number Theoretic Functions.
                                                              (line  90)
* mpz_size:                              Integer Special Functions.
                                                              (line  31)
* mpz_sizeinbase:                        Miscellaneous Integer Functions.
                                                              (line  23)
* mpz_sqrt:                              Integer Roots.       (line  18)
* mpz_sqrtrem:                           Integer Roots.       (line  21)
* mpz_sub:                               Integer Arithmetic.  (line  12)
* mpz_sub_ui:                            Integer Arithmetic.  (line  14)
* mpz_submul:                            Integer Arithmetic.  (line  32)
* mpz_submul_ui:                         Integer Arithmetic.  (line  34)
* mpz_swap:                              Assigning Integers.  (line  37)
* mpz_t:                                 Nomenclature and Types.
                                                              (line   6)
* mpz_tdiv_q:                            Integer Division.    (line  47)
* mpz_tdiv_q_2exp:                       Integer Division.    (line  60)
* mpz_tdiv_q_ui:                         Integer Division.    (line  52)
* mpz_tdiv_qr:                           Integer Division.    (line  50)
* mpz_tdiv_qr_ui:                        Integer Division.    (line  56)
* mpz_tdiv_r:                            Integer Division.    (line  48)
* mpz_tdiv_r_2exp:                       Integer Division.    (line  62)
* mpz_tdiv_r_ui:                         Integer Division.    (line  54)
* mpz_tdiv_ui:                           Integer Division.    (line  58)
* mpz_tstbit:                            Integer Logic and Bit Fiddling.
                                                              (line  62)
* mpz_ui_kronecker:                      Number Theoretic Functions.
                                                              (line  91)
* mpz_ui_pow_ui:                         Integer Exponentiation.
                                                              (line  33)
* mpz_ui_sub:                            Integer Arithmetic.  (line  16)
* mpz_urandomb:                          Integer Random Numbers.
                                                              (line  14)
* mpz_urandomm:                          Integer Random Numbers.
                                                              (line  23)
* mpz_xor:                               Integer Logic and Bit Fiddling.
                                                              (line  17)
* operator"" <1>:                        C++ Interface Floats.
                                                              (line  56)
* operator"" <2>:                        C++ Interface Rationals.
                                                              (line  38)
* operator"":                            C++ Interface Integers.
                                                              (line  30)
* operator%:                             C++ Interface Integers.
                                                              (line  35)
* operator/:                             C++ Interface Integers.
                                                              (line  34)
* operator<<:                            C++ Formatted Output.
                                                              (line  33)
* operator>> <1>:                        C++ Formatted Input. (line  14)
* operator>> <2>:                        C++ Interface Rationals.
                                                              (line  85)
* operator>>:                            C++ Formatted Input. (line  25)
* sgn <1>:                               C++ Interface Integers.
                                                              (line  62)
* sgn <2>:                               C++ Interface Rationals.
                                                              (line  56)
* sgn:                                   C++ Interface Floats.
                                                              (line 102)
* sqrt <1>:                              C++ Interface Integers.
                                                              (line  63)
* sqrt:                                  C++ Interface Floats.
                                                              (line 103)
* swap <1>:                              C++ Interface Floats.
                                                              (line 105)
* swap <2>:                              C++ Interface Rationals.
                                                              (line  58)
* swap:                                  C++ Interface Integers.
                                                              (line  67)
* trunc:                                 C++ Interface Floats.
                                                              (line 106)