This is /home/tege/prec/gmp42/doc/gmp.info, produced by makeinfo
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This manual describes how to install and use the GNU multiple precision
arithmetic library, version 4.2.1.

   Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
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   Permission is granted to copy, distribute and/or modify this
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with no Invariant Sections, with the Front-Cover Texts being "A GNU
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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: Radix to Binary,  Prev: Binary to Radix,  Up: Radix Conversion Algorithms

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

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_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_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_THRESHOLD' usually ends up quite big, around 5000 digits, and
on some processors much bigger still.

   `SET_STR_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_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 10, or more).


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

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

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

Factorial
---------

Factorials are calculated by a combination of removal of twos,
powering, and binary splitting.  The procedure can be best illustrated
with an example,

     23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23

has factors of two removed,

     23! = 2^19.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23

and the resulting terms collected up according to their multiplicity,

     23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)

   Each sequence such as 13.15.17.19.21.23 is evaluated by splitting
into every second term, as for instance (13.17.21).(15.19.23), and the
same recursively on each half.  This is implemented iteratively using
some bit twiddling.

   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.

   Splitting into subsequences of every second term keeps the resulting
products more nearly equal in size than would the simpler approach of
say taking the first half and second half of the sequence.  Nearly
equal products are more efficient for the current multiply
implementation.


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

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

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

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

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, a 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: Assembler Coding,  Prev: Other Algorithms,  Up: Algorithms

Assembler Coding
================

The assembler 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:

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


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

Code Organisation
-----------------

The various `mpn' subdirectories contain machine-dependent code, written
in C or assembler.  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: Assembler Basics,  Next: Assembler Carry Propagation,  Prev: Assembler Code Organisation,  Up: Assembler Coding

Assembler 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 assembler 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: Assembler Carry Propagation,  Next: Assembler Cache Handling,  Prev: Assembler Basics,  Up: Assembler Coding

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 assembler for best
results.


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

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 assembler
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: Assembler Functional Units,  Next: Assembler Floating Point,  Prev: Assembler Cache Handling,  Up: Assembler Coding

Functional Units
----------------

When choosing an approach for an assembler 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 Assembler 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
Assembler 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: Assembler Floating Point,  Next: Assembler SIMD Instructions,  Prev: Assembler Functional Units,  Up: Assembler Coding

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 transfered 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: Assembler SIMD Instructions,  Next: Assembler Software Pipelining,  Prev: Assembler Floating Point,  Up: Assembler Coding

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: Assembler Software Pipelining,  Next: Assembler Loop Unrolling,  Prev: Assembler SIMD Instructions,  Up: Assembler Coding

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 Assembler Loop Unrolling::).


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

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: Assembler Writing Guide,  Prev: Assembler Loop Unrolling,  Up: Assembler Coding

Writing Guide
-------------

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

   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

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

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

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

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.

   `_mp_size' and `_mp_prec' 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.


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

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

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, mpf_t g, 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, unsigned long int 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, unsigned long int 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

Contributors
************

Torbjorn Granlund wrote the original GMP library and is still
developing and maintaining it.  Several other individuals and
organizations have contributed to GMP in various ways.  Here is a list
in chronological order:

   Gunnar Sjoedin and Hans Riesel helped with mathematical problems in
early versions of the library.

   Richard Stallman contributed to 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 of Inria sparked the development of GMP 2, with his
comparisons between bignum packages.

   Ken Weber (Kent State University, Universidade Federal do Rio Grande
do Sul) contributed `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).

   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).

   GNU MP 2 was finished and released by SWOX AB, SWEDEN, in
cooperation with the IDA Center for Computing Sciences, USA.

   Robert Harley of Inria, France and David Seal of ARM, England,
suggested clever improvements for population count.

   Robert Harley also wrote highly optimized Karatsuba and 3-way Toom
multiplication functions for GMP 3.  He also 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.

   Paul Zimmermann wrote the Divide and Conquer division code, the REDC
code, the REDC-based mpz_powm code, the FFT multiply code, and the
Karatsuba square root code.  He also rewrote the Toom3 code for GMP
4.2.  The ECMNET project Paul is organizing was a driving force behind
many of the optimizations in GMP 3.

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

   Kent Boortz made the Mac OS 9 port.

   Kevin Ryde worked on a 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.

   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.

   GNU MP 4 was finished and released by Torbjorn Granlund and Kevin
Ryde.  Torbjorn's work was partially funded by the IDA Center for
Computing Sciences, USA.

   Jason Moxham rewrote `mpz_fac_ui'.

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

   (This list is chronological, not ordered after significance.  If you
have contributed to GMP but are not listed above, please tell
<tege@swox.com> about the omission!)

   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

References
**********

Books
=====

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

   * 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, "Using and Porting GCC", Free Software
     Foundation, 1999, available online
     `http://gcc.gnu.org/onlinedocs/', and in the GCC package
     `ftp://ftp.gnu.org/gnu/gcc/'

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 2001, `http://www.inria.fr/rrrt/rr-4475.html'

   * 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'

   * Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant
     Integers using Multiplication", in Proceedings of the SIGPLAN
     PLDI'94 Conference, June 1994.  Also available
     `ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz).

   * 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.keio.ac.jp/~nisimura/random/doc/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.

   * 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://www.inria.fr/rrrt/rr-3805.html'

   * 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

GNU Free Documentation License
******************************

                      Version 1.2, November 2002
     Copyright (C) 2000,2001,2002 Free Software Foundation, Inc.
     51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA
     
     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,
     represented in a format whose specification is available to the
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     otherwise Transparent file format whose markup, or absence of
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     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
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     can be read and edited only by proprietary word processors, SGML or
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     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
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     work's title, preceding the beginning of the body of the text.

     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 for under this License.  Any other
     attempt to copy, modify, sublicense or distribute the Document is
     void, and will automatically terminate your rights under this
     License.  However, parties who have received copies, or rights,
     from you under this License will not have their licenses
     terminated so long as such parties remain in full compliance.

 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
     `http://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.

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 1.2
       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
*************

* Menu:

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


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

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

* Menu:

* __GNU_MP_VERSION:                      Useful Macros and Constants.
* __GNU_MP_VERSION_MINOR:                Useful Macros and Constants.
* __GNU_MP_VERSION_PATCHLEVEL:           Useful Macros and Constants.
* _mpz_realloc:                          Integer Special Functions.
* abs <1>:                               C++ Interface Rationals.
* abs <2>:                               C++ Interface Integers.
* abs:                                   C++ Interface Floats.
* ceil:                                  C++ Interface Floats.
* cmp <1>:                               C++ Interface Integers.
* cmp <2>:                               C++ Interface Rationals.
* cmp <3>:                               C++ Interface Integers.
* cmp:                                   C++ Interface Floats.
* floor:                                 C++ Interface Floats.
* gcd:                                   BSD Compatible Functions.
* gmp_asprintf:                          Formatted Output Functions.
* gmp_errno:                             Random State Initialization.
* GMP_ERROR_INVALID_ARGUMENT:            Random State Initialization.
* GMP_ERROR_UNSUPPORTED_ARGUMENT:        Random State Initialization.
* gmp_fprintf:                           Formatted Output Functions.
* gmp_fscanf:                            Formatted Input Functions.
* GMP_LIMB_BITS:                         Low-level Functions.
* GMP_NAIL_BITS:                         Low-level Functions.
* GMP_NAIL_MASK:                         Low-level Functions.
* GMP_NUMB_BITS:                         Low-level Functions.
* GMP_NUMB_MASK:                         Low-level Functions.
* GMP_NUMB_MAX:                          Low-level Functions.
* gmp_obstack_printf:                    Formatted Output Functions.
* gmp_obstack_vprintf:                   Formatted Output Functions.
* gmp_printf:                            Formatted Output Functions.
* GMP_RAND_ALG_DEFAULT:                  Random State Initialization.
* GMP_RAND_ALG_LC:                       Random State Initialization.
* gmp_randclass:                         C++ Interface Random Numbers.
* gmp_randclass::get_f:                  C++ Interface Random Numbers.
* gmp_randclass::get_z_bits:             C++ Interface Random Numbers.
* gmp_randclass::get_z_range:            C++ Interface Random Numbers.
* gmp_randclass::gmp_randclass:          C++ Interface Random Numbers.
* gmp_randclass::seed:                   C++ Interface Random Numbers.
* gmp_randclear:                         Random State Initialization.
* gmp_randinit:                          Random State Initialization.
* gmp_randinit_default:                  Random State Initialization.
* gmp_randinit_lc_2exp:                  Random State Initialization.
* gmp_randinit_lc_2exp_size:             Random State Initialization.
* gmp_randinit_mt:                       Random State Initialization.
* gmp_randinit_set:                      Random State Initialization.
* gmp_randseed:                          Random State Seeding.
* gmp_randseed_ui:                       Random State Seeding.
* gmp_randstate_t:                       Nomenclature and Types.
* gmp_scanf:                             Formatted Input Functions.
* gmp_snprintf:                          Formatted Output Functions.
* gmp_sprintf:                           Formatted Output Functions.
* gmp_sscanf:                            Formatted Input Functions.
* gmp_urandomb_ui:                       Random State Miscellaneous.
* gmp_urandomm_ui:                       Random State Miscellaneous.
* gmp_vasprintf:                         Formatted Output Functions.
* gmp_version:                           Useful Macros and Constants.
* gmp_vfprintf:                          Formatted Output Functions.
* gmp_vfscanf:                           Formatted Input Functions.
* gmp_vprintf:                           Formatted Output Functions.
* gmp_vscanf:                            Formatted Input Functions.
* gmp_vsnprintf:                         Formatted Output Functions.
* gmp_vsprintf:                          Formatted Output Functions.
* gmp_vsscanf:                           Formatted Input Functions.
* hypot:                                 C++ Interface Floats.
* itom:                                  BSD Compatible Functions.
* madd:                                  BSD Compatible Functions.
* mcmp:                                  BSD Compatible Functions.
* mdiv:                                  BSD Compatible Functions.
* mfree:                                 BSD Compatible Functions.
* min:                                   BSD Compatible Functions.
* MINT:                                  BSD Compatible Functions.
* mout:                                  BSD Compatible Functions.
* move:                                  BSD Compatible Functions.
* mp_bits_per_limb:                      Useful Macros and Constants.
* mp_exp_t:                              Nomenclature and Types.
* mp_get_memory_functions:               Custom Allocation.
* mp_limb_t:                             Nomenclature and Types.
* mp_set_memory_functions:               Custom Allocation.
* mp_size_t:                             Nomenclature and Types.
* mpf_abs:                               Float Arithmetic.
* mpf_add:                               Float Arithmetic.
* mpf_add_ui:                            Float Arithmetic.
* mpf_ceil:                              Miscellaneous Float Functions.
* mpf_class:                             C++ Interface General.
* mpf_class::fits_sint_p:                C++ Interface Floats.
* mpf_class::fits_slong_p:               C++ Interface Floats.
* mpf_class::fits_sshort_p:              C++ Interface Floats.
* mpf_class::fits_uint_p:                C++ Interface Floats.
* mpf_class::fits_ulong_p:               C++ Interface Floats.
* mpf_class::fits_ushort_p:              C++ Interface Floats.
* mpf_class::get_d:                      C++ Interface Floats.
* mpf_class::get_mpf_t:                  C++ Interface General.
* mpf_class::get_prec:                   C++ Interface Floats.
* mpf_class::get_si:                     C++ Interface Floats.
* mpf_class::get_str:                    C++ Interface Floats.
* mpf_class::get_ui:                     C++ Interface Floats.
* mpf_class::mpf_class:                  C++ Interface Floats.
* mpf_class::operator=:                  C++ Interface Floats.
* mpf_class::set_prec:                   C++ Interface Floats.
* mpf_class::set_prec_raw:               C++ Interface Floats.
* mpf_class::set_str:                    C++ Interface Floats.
* mpf_clear:                             Initializing Floats.
* mpf_cmp:                               Float Comparison.
* mpf_cmp_d:                             Float Comparison.
* mpf_cmp_si:                            Float Comparison.
* mpf_cmp_ui:                            Float Comparison.
* mpf_div:                               Float Arithmetic.
* mpf_div_2exp:                          Float Arithmetic.
* mpf_div_ui:                            Float Arithmetic.
* mpf_eq:                                Float Comparison.
* mpf_fits_sint_p:                       Miscellaneous Float Functions.
* mpf_fits_slong_p:                      Miscellaneous Float Functions.
* mpf_fits_sshort_p:                     Miscellaneous Float Functions.
* mpf_fits_uint_p:                       Miscellaneous Float Functions.
* mpf_fits_ulong_p:                      Miscellaneous Float Functions.
* mpf_fits_ushort_p:                     Miscellaneous Float Functions.
* mpf_floor:                             Miscellaneous Float Functions.
* mpf_get_d:                             Converting Floats.
* mpf_get_d_2exp:                        Converting Floats.
* mpf_get_default_prec:                  Initializing Floats.
* mpf_get_prec:                          Initializing Floats.
* mpf_get_si:                            Converting Floats.
* mpf_get_str:                           Converting Floats.
* mpf_get_ui:                            Converting Floats.
* mpf_init:                              Initializing Floats.
* mpf_init2:                             Initializing Floats.
* mpf_init_set:                          Simultaneous Float Init & Assign.
* mpf_init_set_d:                        Simultaneous Float Init & Assign.
* mpf_init_set_si:                       Simultaneous Float Init & Assign.
* mpf_init_set_str:                      Simultaneous Float Init & Assign.
* mpf_init_set_ui:                       Simultaneous Float Init & Assign.
* mpf_inp_str:                           I/O of Floats.
* mpf_integer_p:                         Miscellaneous Float Functions.
* mpf_mul:                               Float Arithmetic.
* mpf_mul_2exp:                          Float Arithmetic.
* mpf_mul_ui:                            Float Arithmetic.
* mpf_neg:                               Float Arithmetic.
* mpf_out_str:                           I/O of Floats.
* mpf_pow_ui:                            Float Arithmetic.
* mpf_random2:                           Miscellaneous Float Functions.
* mpf_reldiff:                           Float Comparison.
* mpf_set:                               Assigning Floats.
* mpf_set_d:                             Assigning Floats.
* mpf_set_default_prec:                  Initializing Floats.
* mpf_set_prec:                          Initializing Floats.
* mpf_set_prec_raw:                      Initializing Floats.
* mpf_set_q:                             Assigning Floats.
* mpf_set_si:                            Assigning Floats.
* mpf_set_str:                           Assigning Floats.
* mpf_set_ui:                            Assigning Floats.
* mpf_set_z:                             Assigning Floats.
* mpf_sgn:                               Float Comparison.
* mpf_sqrt:                              Float Arithmetic.
* mpf_sqrt_ui:                           Float Arithmetic.
* mpf_sub:                               Float Arithmetic.
* mpf_sub_ui:                            Float Arithmetic.
* mpf_swap:                              Assigning Floats.
* mpf_t:                                 Nomenclature and Types.
* mpf_trunc:                             Miscellaneous Float Functions.
* mpf_ui_div:                            Float Arithmetic.
* mpf_ui_sub:                            Float Arithmetic.
* mpf_urandomb:                          Miscellaneous Float Functions.
* mpn_add:                               Low-level Functions.
* mpn_add_1:                             Low-level Functions.
* mpn_add_n:                             Low-level Functions.
* mpn_addmul_1:                          Low-level Functions.
* mpn_bdivmod:                           Low-level Functions.
* mpn_cmp:                               Low-level Functions.
* mpn_divexact_by3:                      Low-level Functions.
* mpn_divexact_by3c:                     Low-level Functions.
* mpn_divmod:                            Low-level Functions.
* mpn_divmod_1:                          Low-level Functions.
* mpn_divrem:                            Low-level Functions.
* mpn_divrem_1:                          Low-level Functions.
* mpn_gcd:                               Low-level Functions.
* mpn_gcd_1:                             Low-level Functions.
* mpn_gcdext:                            Low-level Functions.
* mpn_get_str:                           Low-level Functions.
* mpn_hamdist:                           Low-level Functions.
* mpn_lshift:                            Low-level Functions.
* mpn_mod_1:                             Low-level Functions.
* mpn_mul:                               Low-level Functions.
* mpn_mul_1:                             Low-level Functions.
* mpn_mul_n:                             Low-level Functions.
* mpn_perfect_square_p:                  Low-level Functions.
* mpn_popcount:                          Low-level Functions.
* mpn_random:                            Low-level Functions.
* mpn_random2:                           Low-level Functions.
* mpn_rshift:                            Low-level Functions.
* mpn_scan0:                             Low-level Functions.
* mpn_scan1:                             Low-level Functions.
* mpn_set_str:                           Low-level Functions.
* mpn_sqrtrem:                           Low-level Functions.
* mpn_sub:                               Low-level Functions.
* mpn_sub_1:                             Low-level Functions.
* mpn_sub_n:                             Low-level Functions.
* mpn_submul_1:                          Low-level Functions.
* mpn_tdiv_qr:                           Low-level Functions.
* mpq_abs:                               Rational Arithmetic.
* mpq_add:                               Rational Arithmetic.
* mpq_canonicalize:                      Rational Number Functions.
* mpq_class:                             C++ Interface General.
* mpq_class::canonicalize:               C++ Interface Rationals.
* mpq_class::get_d:                      C++ Interface Rationals.
* mpq_class::get_den:                    C++ Interface Rationals.
* mpq_class::get_den_mpz_t:              C++ Interface Rationals.
* mpq_class::get_mpq_t:                  C++ Interface General.
* mpq_class::get_num:                    C++ Interface Rationals.
* mpq_class::get_num_mpz_t:              C++ Interface Rationals.
* mpq_class::get_str:                    C++ Interface Rationals.
* mpq_class::mpq_class:                  C++ Interface Rationals.
* mpq_class::set_str:                    C++ Interface Rationals.
* mpq_clear:                             Initializing Rationals.
* mpq_cmp:                               Comparing Rationals.
* mpq_cmp_si:                            Comparing Rationals.
* mpq_cmp_ui:                            Comparing Rationals.
* mpq_denref:                            Applying Integer Functions.
* mpq_div:                               Rational Arithmetic.
* mpq_div_2exp:                          Rational Arithmetic.
* mpq_equal:                             Comparing Rationals.
* mpq_get_d:                             Rational Conversions.
* mpq_get_den:                           Applying Integer Functions.
* mpq_get_num:                           Applying Integer Functions.
* mpq_get_str:                           Rational Conversions.
* mpq_init:                              Initializing Rationals.
* mpq_inp_str:                           I/O of Rationals.
* mpq_inv:                               Rational Arithmetic.
* mpq_mul:                               Rational Arithmetic.
* mpq_mul_2exp:                          Rational Arithmetic.
* mpq_neg:                               Rational Arithmetic.
* mpq_numref:                            Applying Integer Functions.
* mpq_out_str:                           I/O of Rationals.
* mpq_set:                               Initializing Rationals.
* mpq_set_d:                             Rational Conversions.
* mpq_set_den:                           Applying Integer Functions.
* mpq_set_f:                             Rational Conversions.
* mpq_set_num:                           Applying Integer Functions.
* mpq_set_si:                            Initializing Rationals.
* mpq_set_str:                           Initializing Rationals.
* mpq_set_ui:                            Initializing Rationals.
* mpq_set_z:                             Initializing Rationals.
* mpq_sgn:                               Comparing Rationals.
* mpq_sub:                               Rational Arithmetic.
* mpq_swap:                              Initializing Rationals.
* mpq_t:                                 Nomenclature and Types.
* mpz_abs:                               Integer Arithmetic.
* mpz_add:                               Integer Arithmetic.
* mpz_add_ui:                            Integer Arithmetic.
* mpz_addmul:                            Integer Arithmetic.
* mpz_addmul_ui:                         Integer Arithmetic.
* mpz_and:                               Integer Logic and Bit Fiddling.
* mpz_array_init:                        Integer Special Functions.
* mpz_bin_ui:                            Number Theoretic Functions.
* mpz_bin_uiui:                          Number Theoretic Functions.
* mpz_cdiv_q:                            Integer Division.
* mpz_cdiv_q_2exp:                       Integer Division.
* mpz_cdiv_q_ui:                         Integer Division.
* mpz_cdiv_qr:                           Integer Division.
* mpz_cdiv_qr_ui:                        Integer Division.
* mpz_cdiv_r:                            Integer Division.
* mpz_cdiv_r_2exp:                       Integer Division.
* mpz_cdiv_r_ui:                         Integer Division.
* mpz_cdiv_ui:                           Integer Division.
* mpz_class:                             C++ Interface General.
* mpz_class::fits_sint_p:                C++ Interface Integers.
* mpz_class::fits_slong_p:               C++ Interface Integers.
* mpz_class::fits_sshort_p:              C++ Interface Integers.
* mpz_class::fits_uint_p:                C++ Interface Integers.
* mpz_class::fits_ulong_p:               C++ Interface Integers.
* mpz_class::fits_ushort_p:              C++ Interface Integers.
* mpz_class::get_d:                      C++ Interface Integers.
* mpz_class::get_mpz_t:                  C++ Interface General.
* mpz_class::get_si:                     C++ Interface Integers.
* mpz_class::get_str:                    C++ Interface Integers.
* mpz_class::get_ui:                     C++ Interface Integers.
* mpz_class::mpz_class:                  C++ Interface Integers.
* mpz_class::set_str:                    C++ Interface Integers.
* mpz_clear:                             Initializing Integers.
* mpz_clrbit:                            Integer Logic and Bit Fiddling.
* mpz_cmp:                               Integer Comparisons.
* mpz_cmp_d:                             Integer Comparisons.
* mpz_cmp_si:                            Integer Comparisons.
* mpz_cmp_ui:                            Integer Comparisons.
* mpz_cmpabs:                            Integer Comparisons.
* mpz_cmpabs_d:                          Integer Comparisons.
* mpz_cmpabs_ui:                         Integer Comparisons.
* mpz_com:                               Integer Logic and Bit Fiddling.
* mpz_combit:                            Integer Logic and Bit Fiddling.
* mpz_congruent_2exp_p:                  Integer Division.
* mpz_congruent_p:                       Integer Division.
* mpz_congruent_ui_p:                    Integer Division.
* mpz_divexact:                          Integer Division.
* mpz_divexact_ui:                       Integer Division.
* mpz_divisible_2exp_p:                  Integer Division.
* mpz_divisible_p:                       Integer Division.
* mpz_divisible_ui_p:                    Integer Division.
* mpz_even_p:                            Miscellaneous Integer Functions.
* mpz_export:                            Integer Import and Export.
* mpz_fac_ui:                            Number Theoretic Functions.
* mpz_fdiv_q:                            Integer Division.
* mpz_fdiv_q_2exp:                       Integer Division.
* mpz_fdiv_q_ui:                         Integer Division.
* mpz_fdiv_qr:                           Integer Division.
* mpz_fdiv_qr_ui:                        Integer Division.
* mpz_fdiv_r:                            Integer Division.
* mpz_fdiv_r_2exp:                       Integer Division.
* mpz_fdiv_r_ui:                         Integer Division.
* mpz_fdiv_ui:                           Integer Division.
* mpz_fib2_ui:                           Number Theoretic Functions.
* mpz_fib_ui:                            Number Theoretic Functions.
* mpz_fits_sint_p:                       Miscellaneous Integer Functions.
* mpz_fits_slong_p:                      Miscellaneous Integer Functions.
* mpz_fits_sshort_p:                     Miscellaneous Integer Functions.
* mpz_fits_uint_p:                       Miscellaneous Integer Functions.
* mpz_fits_ulong_p:                      Miscellaneous Integer Functions.
* mpz_fits_ushort_p:                     Miscellaneous Integer Functions.
* mpz_gcd:                               Number Theoretic Functions.
* mpz_gcd_ui:                            Number Theoretic Functions.
* mpz_gcdext:                            Number Theoretic Functions.
* mpz_get_d:                             Converting Integers.
* mpz_get_d_2exp:                        Converting Integers.
* mpz_get_si:                            Converting Integers.
* mpz_get_str:                           Converting Integers.
* mpz_get_ui:                            Converting Integers.
* mpz_getlimbn:                          Integer Special Functions.
* mpz_hamdist:                           Integer Logic and Bit Fiddling.
* mpz_import:                            Integer Import and Export.
* mpz_init:                              Initializing Integers.
* mpz_init2:                             Initializing Integers.
* mpz_init_set:                          Simultaneous Integer Init & Assign.
* mpz_init_set_d:                        Simultaneous Integer Init & Assign.
* mpz_init_set_si:                       Simultaneous Integer Init & Assign.
* mpz_init_set_str:                      Simultaneous Integer Init & Assign.
* mpz_init_set_ui:                       Simultaneous Integer Init & Assign.
* mpz_inp_raw:                           I/O of Integers.
* mpz_inp_str:                           I/O of Integers.
* mpz_invert:                            Number Theoretic Functions.
* mpz_ior:                               Integer Logic and Bit Fiddling.
* mpz_jacobi:                            Number Theoretic Functions.
* mpz_kronecker:                         Number Theoretic Functions.
* mpz_kronecker_si:                      Number Theoretic Functions.
* mpz_kronecker_ui:                      Number Theoretic Functions.
* mpz_lcm:                               Number Theoretic Functions.
* mpz_lcm_ui:                            Number Theoretic Functions.
* mpz_legendre:                          Number Theoretic Functions.
* mpz_lucnum2_ui:                        Number Theoretic Functions.
* mpz_lucnum_ui:                         Number Theoretic Functions.
* mpz_mod:                               Integer Division.
* mpz_mod_ui:                            Integer Division.
* mpz_mul:                               Integer Arithmetic.
* mpz_mul_2exp:                          Integer Arithmetic.
* mpz_mul_si:                            Integer Arithmetic.
* mpz_mul_ui:                            Integer Arithmetic.
* mpz_neg:                               Integer Arithmetic.
* mpz_nextprime:                         Number Theoretic Functions.
* mpz_odd_p:                             Miscellaneous Integer Functions.
* mpz_out_raw:                           I/O of Integers.
* mpz_out_str:                           I/O of Integers.
* mpz_perfect_power_p:                   Integer Roots.
* mpz_perfect_square_p:                  Integer Roots.
* mpz_popcount:                          Integer Logic and Bit Fiddling.
* mpz_pow_ui:                            Integer Exponentiation.
* mpz_powm:                              Integer Exponentiation.
* mpz_powm_ui:                           Integer Exponentiation.
* mpz_probab_prime_p:                    Number Theoretic Functions.
* mpz_random:                            Integer Random Numbers.
* mpz_random2:                           Integer Random Numbers.
* mpz_realloc2:                          Initializing Integers.
* mpz_remove:                            Number Theoretic Functions.
* mpz_root:                              Integer Roots.
* mpz_rootrem:                           Integer Roots.
* mpz_rrandomb:                          Integer Random Numbers.
* mpz_scan0:                             Integer Logic and Bit Fiddling.
* mpz_scan1:                             Integer Logic and Bit Fiddling.
* mpz_set:                               Assigning Integers.
* mpz_set_d:                             Assigning Integers.
* mpz_set_f:                             Assigning Integers.
* mpz_set_q:                             Assigning Integers.
* mpz_set_si:                            Assigning Integers.
* mpz_set_str:                           Assigning Integers.
* mpz_set_ui:                            Assigning Integers.
* mpz_setbit:                            Integer Logic and Bit Fiddling.
* mpz_sgn:                               Integer Comparisons.
* mpz_si_kronecker:                      Number Theoretic Functions.
* mpz_size:                              Integer Special Functions.
* mpz_sizeinbase:                        Miscellaneous Integer Functions.
* mpz_sqrt:                              Integer Roots.
* mpz_sqrtrem:                           Integer Roots.
* mpz_sub:                               Integer Arithmetic.
* mpz_sub_ui:                            Integer Arithmetic.
* mpz_submul:                            Integer Arithmetic.
* mpz_submul_ui:                         Integer Arithmetic.
* mpz_swap:                              Assigning Integers.
* mpz_t:                                 Nomenclature and Types.
* mpz_tdiv_q:                            Integer Division.
* mpz_tdiv_q_2exp:                       Integer Division.
* mpz_tdiv_q_ui:                         Integer Division.
* mpz_tdiv_qr:                           Integer Division.
* mpz_tdiv_qr_ui:                        Integer Division.
* mpz_tdiv_r:                            Integer Division.
* mpz_tdiv_r_2exp:                       Integer Division.
* mpz_tdiv_r_ui:                         Integer Division.
* mpz_tdiv_ui:                           Integer Division.
* mpz_tstbit:                            Integer Logic and Bit Fiddling.
* mpz_ui_kronecker:                      Number Theoretic Functions.
* mpz_ui_pow_ui:                         Integer Exponentiation.
* mpz_ui_sub:                            Integer Arithmetic.
* mpz_urandomb:                          Integer Random Numbers.
* mpz_urandomm:                          Integer Random Numbers.
* mpz_xor:                               Integer Logic and Bit Fiddling.
* msqrt:                                 BSD Compatible Functions.
* msub:                                  BSD Compatible Functions.
* mtox:                                  BSD Compatible Functions.
* mult:                                  BSD Compatible Functions.
* operator%:                             C++ Interface Integers.
* operator/:                             C++ Interface Integers.
* operator<<:                            C++ Formatted Output.
* operator>> <1>:                        C++ Formatted Input.
* operator>>:                            C++ Interface Rationals.
* pow:                                   BSD Compatible Functions.
* rpow:                                  BSD Compatible Functions.
* sdiv:                                  BSD Compatible Functions.
* sgn <1>:                               C++ Interface Rationals.
* sgn <2>:                               C++ Interface Floats.
* sgn:                                   C++ Interface Integers.
* sqrt <1>:                              C++ Interface Floats.
* sqrt:                                  C++ Interface Integers.
* trunc:                                 C++ Interface Floats.
* xtom:                                  BSD Compatible Functions.


