mpir/doc/mpir.info-2

This is mpir.info, produced by makeinfo version 4.13 from mpir.texi.

This manual describes how to install and use MPIR, the Multiple
Precision Integers and Rationals library, version 2.5.0.

   Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
Software Foundation, Inc.

   Copyright 2008, 2009, 2010 William Hart

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

INFO-DIR-SECTION GNU libraries
START-INFO-DIR-ENTRY
* mpir: (mpir).                   MPIR Multiple Precision Integers and Rationals Library.
END-INFO-DIR-ENTRY


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

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

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

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


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

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

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

* Menu:

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


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

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

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

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

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

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

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

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

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

   `mpf_get_str' doesn't currently use the algorithm described here, it
multiplies or divides by a power of b to move the radix point to the
just above the highest non-zero digit (or at worst one above that
location), then multiplies by b^n to bring out digits.  This is O(N^2)
and is certainly not optimal.

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

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


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

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

This section is out-of-date.

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

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

* Menu:

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


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

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

This section is somewhat out-of-date.

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

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

This section is out-of-date.

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

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

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

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

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

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


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

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

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

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

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

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

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

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

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

   Using a table avoids lots of calculations on small numbers, and
makes small n go fast.  A bigger table would make more small n go fast,
it's just a question of balancing size against desired speed.  For MPIR
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: mpir.info,  Node: Lucas Numbers Algorithm,  Next: Random Number Algorithms,  Prev: Fibonacci Numbers Algorithm,  Up: Other Algorithms

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

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

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

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

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

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

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


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

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

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

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

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

   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 MPIR 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: mpir.info,  Node: Assembler Coding,  Prev: Other Algorithms,  Up: Algorithms

15.8 Assembler Coding
=====================

The assembler subroutines in MPIR 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 MPIR
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: mpir.info,  Node: Assembler Code Organisation,  Next: Assembler Basics,  Prev: Assembler Coding,  Up: Assembler Coding

15.8.1 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: mpir.info,  Node: Assembler Basics,  Next: Assembler Carry Propagation,  Prev: Assembler Code Organisation,  Up: Assembler Coding

15.8.2 Assembler Basics
-----------------------

`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
overall MPIR 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: mpir.info,  Node: Assembler Carry Propagation,  Next: Assembler Cache Handling,  Prev: Assembler Basics,  Up: Assembler Coding

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

The problem that presents most challenges in MPIR 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: mpir.info,  Node: Assembler Cache Handling,  Next: Assembler Functional Units,  Prev: Assembler Carry Propagation,  Up: Assembler Coding

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

MPIR 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: mpir.info,  Node: Assembler Functional Units,  Next: Assembler Floating Point,  Prev: Assembler Cache Handling,  Up: Assembler Coding

15.8.5 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: mpir.info,  Node: Assembler Floating Point,  Next: Assembler SIMD Instructions,  Prev: Assembler Functional Units,  Up: Assembler Coding

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

Floating point arithmetic is used in MPIR 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: mpir.info,  Node: Assembler SIMD Instructions,  Next: Assembler Software Pipelining,  Prev: Assembler Floating Point,  Up: Assembler Coding

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

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

   SIMD multiplications of say four 16x16 bit multiplies only do as much
work as one 32x32 from MPIR'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: mpir.info,  Node: Assembler Software Pipelining,  Next: Assembler Loop Unrolling,  Prev: Assembler SIMD Instructions,  Up: Assembler Coding

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

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

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

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

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


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

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

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

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

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

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

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

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

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

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


File: mpir.info,  Node: Assembler Writing Guide,  Prev: Assembler Loop Unrolling,  Up: Assembler Coding

15.8.10 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: mpir.info,  Node: Internals,  Next: Contributors,  Prev: Algorithms,  Up: Top

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

*This chapter is provided only for informational purposes and the
various internals described here may change in future MPIR 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: mpir.info,  Node: Integer Internals,  Next: Rational Internals,  Prev: Internals,  Up: Internals

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

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

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

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

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

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

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

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

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


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

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

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

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

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

   This general approach to common factors is badly sub-optimal in the
presence of simple factorizations or little prospect for cancellation,
but MPIR 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: mpir.info,  Node: Float Internals,  Next: Raw Output Internals,  Prev: Rational Internals,  Up: Internals

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

Efficient calculation is the primary aim of MPIR 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: mpir.info,  Node: Raw Output Internals,  Next: C++ Interface Internals,  Prev: Float Internals,  Up: Internals

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

`mpz_out_raw' uses the following format.

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

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

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

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

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


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

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

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

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

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

     struct __gmp_binary_plus
     {
       static void eval(mpf_t f, 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, mp_bitcnt_t precision)
     {
       Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
     }

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

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

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

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

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

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

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


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

Appendix A 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/MPIR but are not listed above, please tell
`http://groups.google.com/group/mpir-devel' about the omission!)

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

   In 2008 GMP was forked and gave rise to the MPIR (Multiple Precision
Integers and Rationals) project. In 2010 version 2.0.0 of MPIR switched
to LGPL v3+ and much code from GMP was again incorporated into MPIR.

   The MPIR project has largely been a collaboration of William Hart,
Brian Gladman and Jason Moxham. MPIR code not obtained from GMP and not
specifically mentioned elsewhere below is likely written by one of
these three.

   William Hart did much of the early MPIR coding including build
system fixes.  His contributions also include Toom 4 and 7 code and
variants, extended GCD based on Niels Mollers ngcd work, asymptotically
fast division code. He does much of the release management work.

   Brian Gladman wrote and maintains MSVC project files. He has also
done much of the conversion of assembly code to yasm format. He rewrote
the benchmark program and developed MSVC ports of tune, speed, try and
the benchmark code.  He helped with many aspects of the merging of GMP
code into MPIR after the switch to LGPL v3+.

   Jason Moxham has contributed a great deal of x86 assembly code. He
has also contributed improved root code and mulhi and mullo routines
and implemented Peter Montgomery's single limb remainder algorithm. He
has also contributed a command line build system for Windows and
numerous build system fixes.

   The following people have either contributed directly to the MPIR
project, made code available on their websites or contributed code to
the official GNU project which has been used in MPIR.

   Pierrick Gaudry wrote some fast assembly support for AMD 64.

   Jason Martin wrote some fast assembly patches for Core 2 and
converted them to intel format. He also did the initial merge of Niels
Moller's fast GCD patches.  He wrote fast addmul functions for Itanium.

   Gonzalo Tornaria helped patch config.guess and associated files to
distinguish modern processors. He also patched mpirbench.

   Michael Abshoff helped resolve some build issues on various
platforms. He served for a while as release manager for the MPIR
project.

   Mariah Lennox contributed patches to mpirbench and various build
failure reports. She has also reported gcc bugs found during MPIR
development.

   Niels Moller wrote the fast ngcd code for computing integer GCD, the
quadratic Hensel division code and precomputed inverse code for
Euclidean division.  He also made contributions to the Toom multiply
code, especially helper functions to simplify Toom evaluations.

   Pierrick Gaudry provided initial AMD 64 assembly support and revised
the FFT code.

   Paul Zimmermann provided an mpz implementation of Toom 4, wrote much
of the FFT code, wrote some of the rootrem code and contributed
invert.c for computing precomputed inverses.

   Alexander Kruppa revised the FFT code.

   Torbjorn Granlund revised the FFT code and wrote a lot of division
code, including the quadratic Euclidean division code, many parts of
the divide and conquer division code, both Hensel and Euclidean, and
his code was also reused for parts of the asymptotically fast division
code. He also helped write the root code and wrote much of the Itanium
assembly code and a couple of Core 2 assembly functions and part of the
basecase middle product assembly code for x86 64 bit. He also wrote the
improved string input and output code and made improvements to the GCD
and extended GCD code. Torbjorn is also responsible for numerous other
bits and pieces that have been used from the GNU project.

   Marco Bodrato and Alberto Zanoni suggested the unbalanced multiply
strategy and found optimal Toom multiplication sequences.

   Marco Bodrato wrote an mpz implementation of the Toom 7 code and
wrote most of the Toom 8.5 multiply and squaring code. He also helped
write the divide and conquer Euclidean division code.

   Robert Gerbicz contributed fast factorial code.

   David Harvey wrote fast middle product code and divide and conquer
approximate quotient code for both Euclidean and Hensel division and
contributed to the quadratic Hensel code.

   T. R. Nicely wrote primality tests used in the benchmark code.

   Jeff Gilchrist assisted with the porting of T. R. Nicely's primality
code to MPIR.

   Peter Shrimpton wrote the BPSW primality test used up to
GMP_LIMB_BITS.

   Thanks to Microsoft for supporting Jason Moxham to work on a command
line build system for Windows and some assembly improvements for
Windows.

   Thanks to the Free Software Foundation France for giving us access
to their build farm.

   Thanks to William Stein for giving us access to his sage.math
machines for testing and for hosting the MPIR website, and for
supporting us in inumerably many other ways.

   Minh Van Nguyen served as release manager for MPIR 2.1.0.


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

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

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

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

   * 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/'

   * Richard Crandall, Carl Pomerance, "Prime Numbers: A Computational
     Perspective" 2nd edition, Springer, 2005.

   * 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/'

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

   * Dan Bernstein, "Detecting perfect powers in essentially linear
     time", Math. Comp. (67) pp. 1253-1283, 1998.

   * 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'

   * Marco Bodrato, Alberto Zanoni, "Integer and Polynomial
     Multiplication: Towards optimal Toom-Cook Matrices", ISAAC 2007
     Proceedings, Ontario, Canada, July 29 - August 1, 2007, ACM Press.
     Available online at `http://ln.bodrato.it/issac2007_pdf'

   * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic",
     version 0.4, November 2009,
     `http://www.loria.fr/~zimmerma/mca/mca-0.4.pdf'

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

   * Agner Fog, "Software optimization resources", online at
     `http://www.agner.org/optimize/'

   * Pierrick Gaudry, Alexander Kruppa, Paul Zimmermann, "A GMP-based
     implementation of Schoenhage-Strassen's large integer
     multiplication algorithm", ISAAC 2007 Proceedings, Ontario,
     Canada, July 29 - August 1, 2007, pp. 167-174, ACM Press. Full
     text available at
     `http://hal.inria.fr/docs/00/14/86/20/PDF/fft.final.pdf'

   * 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).

   * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by
     invariant integers", to appear.

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

   * David Harvey, "The Karatsuba middle product for integers",
     (preprint), 2009. Available at
     `http://www.cims.nyu.edu/~harvey/mulmid/mulmid.pdf'

   * 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, Jeremy R. Johnson, "Efficient Multiprecision
     Floating Point Multiplication with Exact Rounding", Technical
     Report, RISC Linz, 1993, available at
     `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-76.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.

   * Niels Mo"ller, "On Schoenhage's algorithm and subquadratic integer
     GCD computation", Math. Comp. 2007. Available online at
     `http://www.lysator.liu.se/~nisse/archive/S0025-5718-07-02017-0.pdf'

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

   * Thom Mulders, "On short multiplications and divisions", Appl.
     Algebra Engrg. Comm. Comput. 11 (2000), no. 1, pp. 69-88. Tech.
     report No. 276, Dept. of Comp. Sci., ETH Zurich, Nov 1997,
     available online at
     `ftp://ftp.inf.ethz.ch/pub/publications/tech-reports/2xx/276.pdf'

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

   * A. Scho"nhage, A. F. W. Grotefeld and E. Vetter, "Fast Algorithms,
     A Multitape Turing Machine Implementation" BI
     Wissenschafts-Verlag, Mannheim, 1994.

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

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

                     Version 1.3, 3 November 2008

     Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
     `http://fsf.org/'

     Everyone is permitted to copy and distribute verbatim copies
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  0. PREAMBLE

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     We have designed this License in order to use it for manuals for
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====================================================

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

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

[index]
* Menu:

* #include:                              Headers and Libraries.
                                                              (line   6)
* --build:                               Build Options.       (line  59)
* --disable-fft:                         Build Options.       (line 315)
* --disable-shared:                      Build Options.       (line  52)
* --disable-static:                      Build Options.       (line  52)
* --enable-alloca:                       Build Options.       (line 276)
* --enable-assert:                       Build Options.       (line 320)
* --enable-cxx:                          Build Options.       (line 228)
* --enable-fat:                          Build Options.       (line 161)
* --enable-gmpcompat:                    Build Options.       (line  45)
* --enable-profiling <1>:                Profiling.           (line   6)
* --enable-profiling:                    Build Options.       (line 324)
* --exec-prefix:                         Build Options.       (line  32)
* --host:                                Build Options.       (line  73)
* --prefix:                              Build Options.       (line  32)
* -finstrument-functions:                Profiling.           (line  66)
* 2exp functions:                        Efficiency.          (line  43)
* 80x86:                                 Notes for Particular Systems.
                                                              (line 139)
* ABI <1>:                               ABI and ISA.         (line   6)
* ABI:                                   Build Options.       (line 169)
* About this manual:                     Introduction to MPIR.
                                                              (line  48)
* AC_CHECK_LIB:                          Autoconf.            (line  11)
* AIX <1>:                               Notes for Particular Systems.
                                                              (line   7)
* AIX:                                   ABI and ISA.         (line  96)
* Algorithms:                            Algorithms.          (line   6)
* alloca:                                Build Options.       (line 276)
* Allocation of memory:                  Custom Allocation.   (line   6)
* AMD64:                                 ABI and ISA.         (line  45)
* Application Binary Interface:          ABI and ISA.         (line   6)
* Arithmetic functions <1>:              Float Arithmetic.    (line   6)
* Arithmetic functions <2>:              Rational Arithmetic. (line   6)
* Arithmetic functions:                  Integer Arithmetic.  (line   6)
* ARM:                                   Notes for Particular Systems.
                                                              (line  20)
* Assembler cache handling:              Assembler Cache Handling.
                                                              (line   6)
* Assembler carry propagation:           Assembler Carry Propagation.
                                                              (line   6)
* Assembler code organisation:           Assembler Code Organisation.
                                                              (line   6)
* Assembler coding:                      Assembler Coding.    (line   6)
* Assembler floating Point:              Assembler Floating Point.
                                                              (line   6)
* Assembler loop unrolling:              Assembler Loop Unrolling.
                                                              (line   6)
* Assembler SIMD:                        Assembler SIMD Instructions.
                                                              (line   6)
* Assembler software pipelining:         Assembler Software Pipelining.
                                                              (line   6)
* Assembler writing guide:               Assembler Writing Guide.
                                                              (line   6)
* Assertion checking <1>:                Debugging.           (line  76)
* Assertion checking:                    Build Options.       (line 320)
* Assignment functions <1>:              Simultaneous Float Init & Assign.
                                                              (line   6)
* Assignment functions <2>:              Assigning Floats.    (line   6)
* Assignment functions <3>:              Initializing Rationals.
                                                              (line   6)
* Assignment functions <4>:              Simultaneous Integer Init & Assign.
                                                              (line   6)
* Assignment functions:                  Assigning Integers.  (line   6)
* Autoconf:                              Autoconf.            (line   6)
* Basics:                                MPIR Basics.         (line   6)
* Binomial coefficient algorithm:        Binomial Coefficients Algorithm.
                                                              (line   6)
* Binomial coefficient functions:        Number Theoretic Functions.
                                                              (line 151)
* Bit manipulation functions:            Integer Logic and Bit Fiddling.
                                                              (line   6)
* Bit scanning functions:                Integer Logic and Bit Fiddling.
                                                              (line  38)
* Bit shift left:                        Integer Arithmetic.  (line  35)
* Bit shift right:                       Integer Division.    (line  53)
* Bits per limb:                         Useful Macros and Constants.
                                                              (line   7)
* Bug reporting:                         Reporting Bugs.      (line   6)
* Build directory:                       Build Options.       (line  19)
* Build notes for binary packaging:      Notes for Package Builds.
                                                              (line   6)
* Build notes for particular systems:    Notes for Particular Systems.
                                                              (line   6)
* Build options:                         Build Options.       (line   6)
* Build problems known:                  Known Build Problems.
                                                              (line   6)
* Build system:                          Build Options.       (line  59)
* Building MPIR:                         Installing MPIR.     (line   6)
* Bus error:                             Debugging.           (line   7)
* C compiler:                            Build Options.       (line 180)
* C++ compiler:                          Build Options.       (line 252)
* C++ interface:                         C++ Class Interface. (line   6)
* C++ interface internals:               C++ Interface Internals.
                                                              (line   6)
* C++ istream input:                     C++ Formatted Input. (line   6)
* C++ ostream output:                    C++ Formatted Output.
                                                              (line   6)
* C++ support:                           Build Options.       (line 228)
* CC:                                    Build Options.       (line 180)
* CC_FOR_BUILD:                          Build Options.       (line 215)
* CFLAGS:                                Build Options.       (line 180)
* Checker:                               Debugging.           (line 112)
* checkergcc:                            Debugging.           (line 119)
* Code organisation:                     Assembler Code Organisation.
                                                              (line   6)
* Comparison functions <1>:              Float Comparison.    (line   6)
* Comparison functions <2>:              Comparing Rationals. (line   6)
* Comparison functions:                  Integer Comparisons. (line   6)
* Compatibility with older versions:     Compatibility with older versions.
                                                              (line   6)
* Conditions for copying MPIR:           Copying.             (line   6)
* Configuring MPIR:                      Installing MPIR.     (line   6)
* Congruence algorithm:                  Exact Remainder.     (line  30)
* Congruence functions:                  Integer Division.    (line 124)
* Constants:                             Useful Macros and Constants.
                                                              (line   6)
* Contributors:                          Contributors.        (line   6)
* Conventions for parameters:            Parameter Conventions.
                                                              (line   6)
* Conventions for variables:             Variable Conventions.
                                                              (line   6)
* Conversion functions <1>:              Converting Floats.   (line   6)
* Conversion functions <2>:              Rational Conversions.
                                                              (line   6)
* Conversion functions:                  Converting Integers. (line   6)
* Copying conditions:                    Copying.             (line   6)
* CPPFLAGS:                              Build Options.       (line 206)
* CPU types <1>:                         Build Options.       (line 114)
* CPU types:                             Introduction to MPIR.
                                                              (line  24)
* Cross compiling:                       Build Options.       (line  73)
* Custom allocation:                     Custom Allocation.   (line   6)
* CXX:                                   Build Options.       (line 252)
* CXXFLAGS:                              Build Options.       (line 252)
* Cygwin:                                Notes for Particular Systems.
                                                              (line  34)
* Darwin:                                Known Build Problems.
                                                              (line  23)
* Debugging:                             Debugging.           (line   6)
* Digits in an integer:                  Miscellaneous Integer Functions.
                                                              (line  23)
* Divisibility algorithm:                Exact Remainder.     (line  30)
* Divisibility functions:                Integer Division.    (line 112)
* Divisibility testing:                  Efficiency.          (line  91)
* Division algorithms:                   Division Algorithms. (line   6)
* Division functions <1>:                Float Arithmetic.    (line  33)
* Division functions <2>:                Rational Arithmetic. (line  22)
* Division functions:                    Integer Division.    (line   6)
* DLLs:                                  Notes for Particular Systems.
                                                              (line  73)
* DocBook:                               Build Options.       (line 347)
* Documentation formats:                 Build Options.       (line 340)
* Documentation license:                 GNU Free Documentation License.
                                                              (line   6)
* DVI:                                   Build Options.       (line 343)
* Efficiency:                            Efficiency.          (line   6)
* Emacs:                                 Emacs.               (line   6)
* Exact division functions:              Integer Division.    (line 102)
* Exact remainder:                       Exact Remainder.     (line   6)
* Exec prefix:                           Build Options.       (line  32)
* Execution profiling <1>:               Profiling.           (line   6)
* Execution profiling:                   Build Options.       (line 324)
* Exponentiation functions <1>:          Float Arithmetic.    (line  41)
* Exponentiation functions:              Integer Exponentiation.
                                                              (line   6)
* Export:                                Integer Import and Export.
                                                              (line  45)
* Extended GCD:                          Number Theoretic Functions.
                                                              (line  97)
* Factor removal functions:              Number Theoretic Functions.
                                                              (line 141)
* Factorial algorithm:                   Factorial Algorithm. (line   6)
* Factorial functions:                   Number Theoretic Functions.
                                                              (line 146)
* Fast Fourier Transform:                FFT Multiplication.  (line   6)
* Fat binary:                            Build Options.       (line 161)
* FFT multiplication <1>:                FFT Multiplication.  (line   6)
* FFT multiplication:                    Build Options.       (line 315)
* Fibonacci number algorithm:            Fibonacci Numbers Algorithm.
                                                              (line   6)
* Fibonacci sequence functions:          Number Theoretic Functions.
                                                              (line 159)
* Float arithmetic functions:            Float Arithmetic.    (line   6)
* Float assignment functions <1>:        Simultaneous Float Init & Assign.
                                                              (line   6)
* Float assignment functions:            Assigning Floats.    (line   6)
* Float comparison functions:            Float Comparison.    (line   6)
* Float conversion functions:            Converting Floats.   (line   6)
* Float functions:                       Floating-point Functions.
                                                              (line   6)
* Float initialization functions <1>:    Simultaneous Float Init & Assign.
                                                              (line   6)
* Float initialization functions:        Initializing Floats. (line   6)
* Float input and output functions:      I/O of Floats.       (line   6)
* Float internals:                       Float Internals.     (line   6)
* Float miscellaneous functions:         Miscellaneous Float Functions.
                                                              (line   6)
* Float random number functions:         Miscellaneous Float Functions.
                                                              (line  27)
* Float rounding functions:              Miscellaneous Float Functions.
                                                              (line   9)
* Float sign tests:                      Float Comparison.    (line  28)
* Floating point mode:                   Notes for Particular Systems.
                                                              (line  25)
* Floating-point functions:              Floating-point Functions.
                                                              (line   6)
* Floating-point number:                 Nomenclature and Types.
                                                              (line  21)
* fnccheck:                              Profiling.           (line  77)
* Formatted input:                       Formatted Input.     (line   6)
* Formatted output:                      Formatted Output.    (line   6)
* Free Documentation License:            GNU Free Documentation License.
                                                              (line   6)
* frexp <1>:                             Converting Floats.   (line  23)
* frexp:                                 Converting Integers. (line  58)
* Function classes:                      Function Classes.    (line   6)
* FunctionCheck:                         Profiling.           (line  77)
* GCC Checker:                           Debugging.           (line 112)
* GCD algorithms:                        Greatest Common Divisor Algorithms.
                                                              (line   6)
* GCD extended:                          Number Theoretic Functions.
                                                              (line  97)
* GCD functions:                         Number Theoretic Functions.
                                                              (line  82)
* GDB:                                   Debugging.           (line  55)
* Generic C:                             Build Options.       (line 150)
* GNU Debugger:                          Debugging.           (line  55)
* GNU Free Documentation License:        GNU Free Documentation License.
                                                              (line   6)
* gprof:                                 Profiling.           (line  41)
* Greatest common divisor algorithms:    Greatest Common Divisor Algorithms.
                                                              (line   6)
* Greatest common divisor functions:     Number Theoretic Functions.
                                                              (line  82)
* Hardware floating point mode:          Notes for Particular Systems.
                                                              (line  25)
* Headers:                               Headers and Libraries.
                                                              (line   6)
* Heap problems:                         Debugging.           (line  24)
* Home page:                             Introduction to MPIR.
                                                              (line  30)
* Host system:                           Build Options.       (line  73)
* HP-UX:                                 ABI and ISA.         (line  69)
* I/O functions <1>:                     I/O of Floats.       (line   6)
* I/O functions <2>:                     I/O of Rationals.    (line   6)
* I/O functions:                         I/O of Integers.     (line   6)
* i386:                                  Notes for Particular Systems.
                                                              (line 139)
* IA-64:                                 ABI and ISA.         (line  69)
* Import:                                Integer Import and Export.
                                                              (line  11)
* In-place operations:                   Efficiency.          (line  57)
* Include files:                         Headers and Libraries.
                                                              (line   6)
* info-lookup-symbol:                    Emacs.               (line   6)
* Initialization functions <1>:          Random State Initialization.
                                                              (line   6)
* Initialization functions <2>:          Simultaneous Float Init & Assign.
                                                              (line   6)
* Initialization functions <3>:          Initializing Floats. (line   6)
* Initialization functions <4>:          Initializing Rationals.
                                                              (line   6)
* Initialization functions <5>:          Simultaneous Integer Init & Assign.
                                                              (line   6)
* Initialization functions:              Initializing Integers.
                                                              (line   6)
* Initializing and clearing:             Efficiency.          (line  21)
* Input functions <1>:                   Formatted Input Functions.
                                                              (line   6)
* Input functions <2>:                   I/O of Floats.       (line   6)
* Input functions <3>:                   I/O of Rationals.    (line   6)
* Input functions:                       I/O of Integers.     (line   6)
* Install prefix:                        Build Options.       (line  32)
* Installing MPIR:                       Installing MPIR.     (line   6)
* Instruction Set Architecture:          ABI and ISA.         (line   6)
* instrument-functions:                  Profiling.           (line  66)
* Integer:                               Nomenclature and Types.
                                                              (line   6)
* Integer arithmetic functions:          Integer Arithmetic.  (line   6)
* Integer assignment functions <1>:      Simultaneous Integer Init & Assign.
                                                              (line   6)
* Integer assignment functions:          Assigning Integers.  (line   6)
* Integer bit manipulation functions:    Integer Logic and Bit Fiddling.
                                                              (line   6)
* Integer comparison functions:          Integer Comparisons. (line   6)
* Integer conversion functions:          Converting Integers. (line   6)
* Integer division functions:            Integer Division.    (line   6)
* Integer exponentiation functions:      Integer Exponentiation.
                                                              (line   6)
* Integer export:                        Integer Import and Export.
                                                              (line  45)
* Integer functions:                     Integer Functions.   (line   6)
* Integer import:                        Integer Import and Export.
                                                              (line  11)
* Integer initialization functions <1>:  Simultaneous Integer Init & Assign.
                                                              (line   6)
* Integer initialization functions:      Initializing Integers.
                                                              (line   6)
* Integer input and output functions:    I/O of Integers.     (line   6)
* Integer internals:                     Integer Internals.   (line   6)
* Integer logical functions:             Integer Logic and Bit Fiddling.
                                                              (line   6)
* Integer miscellaneous functions:       Miscellaneous Integer Functions.
                                                              (line   6)
* Integer random number functions:       Integer Random Numbers.
                                                              (line   6)
* Integer root functions:                Integer Roots.       (line   6)
* Integer sign tests:                    Integer Comparisons. (line  28)
* Integer special functions:             Integer Special Functions.
                                                              (line   6)
* Internals:                             Internals.           (line   6)
* Introduction:                          Introduction to MPIR.
                                                              (line   6)
* Inverse modulo functions:              Number Theoretic Functions.
                                                              (line 110)
* ISA:                                   ABI and ISA.         (line   6)
* istream input:                         C++ Formatted Input. (line   6)
* Jacobi symbol algorithm:               Jacobi Symbol.       (line   6)
* Jacobi symbol functions:               Number Theoretic Functions.
                                                              (line 116)
* Karatsuba multiplication:              Karatsuba Multiplication.
                                                              (line   6)
* Karatsuba square root algorithm:       Square Root Algorithm.
                                                              (line   6)
* Kronecker symbol functions:            Number Theoretic Functions.
                                                              (line 128)
* Language bindings:                     Language Bindings.   (line   6)
* LCM functions:                         Number Theoretic Functions.
                                                              (line 105)
* Least common multiple functions:       Number Theoretic Functions.
                                                              (line 105)
* Legendre symbol functions:             Number Theoretic Functions.
                                                              (line 119)
* libmpir:                               Headers and Libraries.
                                                              (line  22)
* libmpirxx:                             Headers and Libraries.
                                                              (line  28)
* Libraries:                             Headers and Libraries.
                                                              (line  22)
* Libtool:                               Headers and Libraries.
                                                              (line  34)
* Libtool versioning:                    Notes for Package Builds.
                                                              (line   9)
* License conditions:                    Copying.             (line   6)
* Limb:                                  Nomenclature and Types.
                                                              (line  31)
* Limb size:                             Useful Macros and Constants.
                                                              (line   7)
* Linear congruential algorithm:         Random Number Algorithms.
                                                              (line  25)
* Linear congruential random numbers:    Random State Initialization.
                                                              (line  18)
* Linking:                               Headers and Libraries.
                                                              (line  22)
* Logical functions:                     Integer Logic and Bit Fiddling.
                                                              (line   6)
* Low-level functions:                   Low-level Functions. (line   6)
* Lucas number algorithm:                Lucas Numbers Algorithm.
                                                              (line   6)
* Lucas number functions:                Number Theoretic Functions.
                                                              (line 170)
* MacOS X:                               Known Build Problems.
                                                              (line  23)
* Mailing lists:                         Introduction to MPIR.
                                                              (line  35)
* Malloc debugger:                       Debugging.           (line  30)
* Malloc problems:                       Debugging.           (line  24)
* Memory allocation:                     Custom Allocation.   (line   6)
* Memory management:                     Memory Management.   (line   6)
* Mersenne twister algorithm:            Random Number Algorithms.
                                                              (line  17)
* Mersenne twister random numbers:       Random State Initialization.
                                                              (line  13)
* MINGW:                                 Notes for Particular Systems.
                                                              (line  34)
* Miscellaneous float functions:         Miscellaneous Float Functions.
                                                              (line   6)
* Miscellaneous integer functions:       Miscellaneous Integer Functions.
                                                              (line   6)
* MMX:                                   Notes for Particular Systems.
                                                              (line 145)
* Modular inverse functions:             Number Theoretic Functions.
                                                              (line 110)
* Most significant bit:                  Miscellaneous Integer Functions.
                                                              (line  34)
* MPIR version number:                   Useful Macros and Constants.
                                                              (line  12)
* mpir.h:                                Headers and Libraries.
                                                              (line   6)
* mpirxx.h:                              C++ Interface General.
                                                              (line   8)
* MPN_PATH:                              Build Options.       (line 328)
* MS Windows:                            Notes for Particular Systems.
                                                              (line  34)
* MS-DOS:                                Notes for Particular Systems.
                                                              (line  34)
* Multi-threading:                       Reentrancy.          (line   6)
* Multiplication algorithms:             Multiplication Algorithms.
                                                              (line   6)
* Nails:                                 Low-level Functions. (line 531)
* Native compilation:                    Build Options.       (line  59)
* Next likely prime function:            Number Theoretic Functions.
                                                              (line  71)
* Next prime function:                   Number Theoretic Functions.
                                                              (line  60)
* Nomenclature:                          Nomenclature and Types.
                                                              (line   6)
* Non-Unix systems:                      Build Options.       (line  11)
* Nth root algorithm:                    Nth Root Algorithm.  (line   6)
* Number sequences:                      Efficiency.          (line 147)
* Number theoretic functions:            Number Theoretic Functions.
                                                              (line   6)
* Numerator and denominator:             Applying Integer Functions.
                                                              (line   6)
* obstack output:                        Formatted Output Functions.
                                                              (line  81)
* OpenBSD:                               Notes for Particular Systems.
                                                              (line  96)
* Optimizing performance:                Performance optimization.
                                                              (line   6)
* ostream output:                        C++ Formatted Output.
                                                              (line   6)
* Other languages:                       Language Bindings.   (line   6)
* Output functions <1>:                  Formatted Output Functions.
                                                              (line   6)
* Output functions <2>:                  I/O of Floats.       (line   6)
* Output functions <3>:                  I/O of Rationals.    (line   6)
* Output functions:                      I/O of Integers.     (line   6)
* Packaged builds:                       Notes for Package Builds.
                                                              (line   6)
* Parameter conventions:                 Parameter Conventions.
                                                              (line   6)
* Particular systems:                    Notes for Particular Systems.
                                                              (line   6)
* Past GMP/MPIR versions:                Compatibility with older versions.
                                                              (line   6)
* PDF:                                   Build Options.       (line 343)
* Perfect power algorithm:               Perfect Power Algorithm.
                                                              (line   6)
* Perfect power functions:               Integer Roots.       (line  31)
* Perfect square algorithm:              Perfect Square Algorithm.
                                                              (line   6)
* Perfect square functions:              Integer Roots.       (line  40)
* Postscript:                            Build Options.       (line 343)
* Powering algorithms:                   Powering Algorithms. (line   6)
* Powering functions <1>:                Float Arithmetic.    (line  41)
* Powering functions:                    Integer Exponentiation.
                                                              (line   6)
* PowerPC:                               ABI and ISA.         (line  94)
* Precision of floats:                   Floating-point Functions.
                                                              (line   6)
* Precision of hardware floating point:  Notes for Particular Systems.
                                                              (line  25)
* Prefix:                                Build Options.       (line  32)
* Prime testing algorithms:              Prime Testing Algorithm.
                                                              (line   6)
* Prime testing functions:               Number Theoretic Functions.
                                                              (line   8)
* printf formatted output:               Formatted Output.    (line   6)
* Probable prime testing functions:      Number Theoretic Functions.
                                                              (line   8)
* prof:                                  Profiling.           (line  24)
* Profiling:                             Profiling.           (line   6)
* Radix conversion algorithms:           Radix Conversion Algorithms.
                                                              (line   6)
* Random number algorithms:              Random Number Algorithms.
                                                              (line   6)
* Random number functions <1>:           Random Number Functions.
                                                              (line   6)
* Random number functions <2>:           Miscellaneous Float Functions.
                                                              (line  27)
* Random number functions:               Integer Random Numbers.
                                                              (line   6)
* Random number seeding:                 Random State Seeding.
                                                              (line   6)
* Random number state:                   Random State Initialization.
                                                              (line   6)
* Random state:                          Nomenclature and Types.
                                                              (line  45)
* Rational arithmetic:                   Efficiency.          (line 113)
* Rational arithmetic functions:         Rational Arithmetic. (line   6)
* Rational assignment functions:         Initializing Rationals.
                                                              (line   6)
* Rational comparison functions:         Comparing Rationals. (line   6)
* Rational conversion functions:         Rational Conversions.
                                                              (line   6)
* Rational initialization functions:     Initializing Rationals.
                                                              (line   6)
* Rational input and output functions:   I/O of Rationals.    (line   6)
* Rational internals:                    Rational Internals.  (line   6)
* Rational number:                       Nomenclature and Types.
                                                              (line  16)
* Rational number functions:             Rational Number Functions.
                                                              (line   6)
* Rational numerator and denominator:    Applying Integer Functions.
                                                              (line   6)
* Rational sign tests:                   Comparing Rationals. (line  27)
* Raw output internals:                  Raw Output Internals.
                                                              (line   6)
* Reallocations:                         Efficiency.          (line  30)
* Reentrancy:                            Reentrancy.          (line   6)
* References:                            References.          (line   6)
* Remove factor functions:               Number Theoretic Functions.
                                                              (line 141)
* Reporting bugs:                        Reporting Bugs.      (line   6)
* Root extraction algorithm:             Nth Root Algorithm.  (line   6)
* Root extraction algorithms:            Root Extraction Algorithms.
                                                              (line   6)
* Root extraction functions <1>:         Float Arithmetic.    (line  37)
* Root extraction functions:             Integer Roots.       (line   6)
* Root testing functions:                Integer Roots.       (line  31)
* Rounding functions:                    Miscellaneous Float Functions.
                                                              (line   9)
* Scan bit functions:                    Integer Logic and Bit Fiddling.
                                                              (line  38)
* scanf formatted input:                 Formatted Input.     (line   6)
* Seeding random numbers:                Random State Seeding.
                                                              (line   6)
* Segmentation violation:                Debugging.           (line   7)
* Shared library versioning:             Notes for Package Builds.
                                                              (line   9)
* Sign tests <1>:                        Float Comparison.    (line  28)
* Sign tests <2>:                        Comparing Rationals. (line  27)
* Sign tests:                            Integer Comparisons. (line  28)
* Size in digits:                        Miscellaneous Integer Functions.
                                                              (line  23)
* Small operands:                        Efficiency.          (line   7)
* Solaris <1>:                           Known Build Problems.
                                                              (line  29)
* Solaris <2>:                           Notes for Particular Systems.
                                                              (line 135)
* Solaris:                               ABI and ISA.         (line 121)
* Sparc:                                 Notes for Particular Systems.
                                                              (line 102)
* Sparc V9:                              ABI and ISA.         (line 121)
* Special integer functions:             Integer Special Functions.
                                                              (line   6)
* Square root algorithm:                 Square Root Algorithm.
                                                              (line   6)
* SSE2:                                  Notes for Particular Systems.
                                                              (line 145)
* Stack backtrace:                       Debugging.           (line  47)
* Stack overflow <1>:                    Debugging.           (line   7)
* Stack overflow:                        Build Options.       (line 276)
* Static linking:                        Efficiency.          (line  14)
* stdarg.h:                              Headers and Libraries.
                                                              (line  17)
* stdio.h:                               Headers and Libraries.
                                                              (line  11)
* Sun:                                   ABI and ISA.         (line 121)
* Systems:                               Notes for Particular Systems.
                                                              (line   6)
* Temporary memory:                      Build Options.       (line 276)
* Texinfo:                               Build Options.       (line 340)
* Text input/output:                     Efficiency.          (line 153)
* Thread safety:                         Reentrancy.          (line   6)
* Toom multiplication <1>:               Other Multiplication.
                                                              (line   6)
* Toom multiplication <2>:               Toom 4-Way Multiplication.
                                                              (line   6)
* Toom multiplication:                   Toom 3-Way Multiplication.
                                                              (line   6)
* Types:                                 Nomenclature and Types.
                                                              (line   6)
* ui and si functions:                   Efficiency.          (line  50)
* Unbalanced multiplication:             Unbalanced Multiplication.
                                                              (line   6)
* Upward compatibility:                  Compatibility with older versions.
                                                              (line   6)
* Useful macros and constants:           Useful Macros and Constants.
                                                              (line   6)
* User-defined precision:                Floating-point Functions.
                                                              (line   6)
* Valgrind:                              Debugging.           (line 127)
* Variable conventions:                  Variable Conventions.
                                                              (line   6)
* Version number:                        Useful Macros and Constants.
                                                              (line  12)
* Web page:                              Introduction to MPIR.
                                                              (line  30)
* Windows:                               Notes for Particular Systems.
                                                              (line  34)
* x86:                                   Notes for Particular Systems.
                                                              (line 139)
* x87:                                   Notes for Particular Systems.
                                                              (line  25)
* XML:                                   Build Options.       (line 347)


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

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

[index]
* Menu:

* __GNU_MP_VERSION:                      Useful Macros and Constants.
                                                              (line  10)
* __GNU_MP_VERSION_MINOR:                Useful Macros and Constants.
                                                              (line  11)
* __GNU_MP_VERSION_PATCHLEVEL:           Useful Macros and Constants.
                                                              (line  12)
* __MPIR_VERSION:                        Useful Macros and Constants.
                                                              (line  18)
* __MPIR_VERSION_MINOR:                  Useful Macros and Constants.
                                                              (line  19)
* __MPIR_VERSION_PATCHLEVEL:             Useful Macros and Constants.
                                                              (line  20)
* _mpz_realloc:                          Integer Special Functions.
                                                              (line  54)
* abs <1>:                               C++ Interface Floats.
                                                              (line  70)
* abs <2>:                               C++ Interface Rationals.
                                                              (line  43)
* abs:                                   C++ Interface Integers.
                                                              (line  42)
* ceil:                                  C++ Interface Floats.
                                                              (line  71)
* cmp <1>:                               C++ Interface Floats.
                                                              (line  72)
* cmp <2>:                               C++ Interface Rationals.
                                                              (line  44)
* cmp:                                   C++ Interface Integers.
                                                              (line  43)
* floor:                                 C++ Interface Floats.
                                                              (line  80)
* gmp_asprintf:                          Formatted Output Functions.
                                                              (line  65)
* gmp_fprintf:                           Formatted Output Functions.
                                                              (line  29)
* gmp_fscanf:                            Formatted Input Functions.
                                                              (line  25)
* GMP_LIMB_BITS:                         Low-level Functions. (line 564)
* GMP_NAIL_BITS:                         Low-level Functions. (line 562)
* GMP_NAIL_MASK:                         Low-level Functions. (line 572)
* GMP_NUMB_BITS:                         Low-level Functions. (line 563)
* GMP_NUMB_MASK:                         Low-level Functions. (line 573)
* GMP_NUMB_MAX:                          Low-level Functions. (line 581)
* gmp_obstack_printf:                    Formatted Output Functions.
                                                              (line  79)
* gmp_obstack_vprintf:                   Formatted Output Functions.
                                                              (line  81)
* gmp_printf:                            Formatted Output Functions.
                                                              (line  24)
* gmp_randclass:                         C++ Interface Random Numbers.
                                                              (line   7)
* gmp_randclass::get_f:                  C++ Interface Random Numbers.
                                                              (line  39)
* gmp_randclass::get_z_bits:             C++ Interface Random Numbers.
                                                              (line  32)
* gmp_randclass::get_z_range:            C++ Interface Random Numbers.
                                                              (line  36)
* gmp_randclass::gmp_randclass:          C++ Interface Random Numbers.
                                                              (line  13)
* gmp_randclass::seed:                   C++ Interface Random Numbers.
                                                              (line  27)
* gmp_randclear:                         Random State Initialization.
                                                              (line  46)
* gmp_randinit_default:                  Random State Initialization.
                                                              (line   7)
* gmp_randinit_lc_2exp:                  Random State Initialization.
                                                              (line  18)
* gmp_randinit_lc_2exp_size:             Random State Initialization.
                                                              (line  32)
* gmp_randinit_mt:                       Random State Initialization.
                                                              (line  13)
* gmp_randinit_set:                      Random State Initialization.
                                                              (line  43)
* gmp_randseed:                          Random State Seeding.
                                                              (line   7)
* gmp_randseed_ui:                       Random State Seeding.
                                                              (line   9)
* gmp_randstate_t:                       Nomenclature and Types.
                                                              (line  45)
* gmp_scanf:                             Formatted Input Functions.
                                                              (line  21)
* gmp_snprintf:                          Formatted Output Functions.
                                                              (line  46)
* gmp_sprintf:                           Formatted Output Functions.
                                                              (line  34)
* gmp_sscanf:                            Formatted Input Functions.
                                                              (line  29)
* gmp_urandomb_ui:                       Random State Miscellaneous.
                                                              (line   8)
* gmp_urandomm_ui:                       Random State Miscellaneous.
                                                              (line  14)
* gmp_vasprintf:                         Formatted Output Functions.
                                                              (line  66)
* gmp_version:                           Useful Macros and Constants.
                                                              (line  25)
* gmp_vfprintf:                          Formatted Output Functions.
                                                              (line  30)
* gmp_vfscanf:                           Formatted Input Functions.
                                                              (line  26)
* gmp_vprintf:                           Formatted Output Functions.
                                                              (line  25)
* gmp_vscanf:                            Formatted Input Functions.
                                                              (line  22)
* gmp_vsnprintf:                         Formatted Output Functions.
                                                              (line  48)
* gmp_vsprintf:                          Formatted Output Functions.
                                                              (line  35)
* gmp_vsscanf:                           Formatted Input Functions.
                                                              (line  31)
* hypot:                                 C++ Interface Floats.
                                                              (line  81)
* mp_bitcnt_t:                           Nomenclature and Types.
                                                              (line  41)
* mp_bits_per_limb:                      Useful Macros and Constants.
                                                              (line   7)
* mp_exp_t:                              Nomenclature and Types.
                                                              (line  27)
* mp_get_memory_functions:               Custom Allocation.   (line  98)
* mp_limb_t:                             Nomenclature and Types.
                                                              (line  31)
* mp_set_memory_functions:               Custom Allocation.   (line  18)
* mp_size_t:                             Nomenclature and Types.
                                                              (line  37)
* mpf_abs:                               Float Arithmetic.    (line  47)
* mpf_add:                               Float Arithmetic.    (line   7)
* mpf_add_ui:                            Float Arithmetic.    (line   9)
* mpf_ceil:                              Miscellaneous Float Functions.
                                                              (line   7)
* mpf_class:                             C++ Interface General.
                                                              (line  20)
* mpf_class::fits_sint_p:                C++ Interface Floats.
                                                              (line  74)
* mpf_class::fits_slong_p:               C++ Interface Floats.
                                                              (line  75)
* mpf_class::fits_sshort_p:              C++ Interface Floats.
                                                              (line  76)
* mpf_class::fits_uint_p:                C++ Interface Floats.
                                                              (line  77)
* mpf_class::fits_ulong_p:               C++ Interface Floats.
                                                              (line  78)
* mpf_class::fits_ushort_p:              C++ Interface Floats.
                                                              (line  79)
* mpf_class::get_d:                      C++ Interface Floats.
                                                              (line  82)
* mpf_class::get_mpf_t:                  C++ Interface General.
                                                              (line  66)
* mpf_class::get_prec:                   C++ Interface Floats.
                                                              (line 100)
* mpf_class::get_si:                     C++ Interface Floats.
                                                              (line  83)
* mpf_class::get_str:                    C++ Interface Floats.
                                                              (line  85)
* mpf_class::get_ui:                     C++ Interface Floats.
                                                              (line  86)
* mpf_class::mpf_class:                  C++ Interface Floats.
                                                              (line  12)
* mpf_class::operator=:                  C++ Interface Floats.
                                                              (line  47)
* mpf_class::set_prec:                   C++ Interface Floats.
                                                              (line 101)
* mpf_class::set_prec_raw:               C++ Interface Floats.
                                                              (line 102)
* mpf_class::set_str:                    C++ Interface Floats.
                                                              (line  87)
* mpf_clear:                             Initializing Floats. (line  37)
* mpf_clears:                            Initializing Floats. (line  41)
* mpf_cmp:                               Float Comparison.    (line   7)
* mpf_cmp_d:                             Float Comparison.    (line   8)
* mpf_cmp_si:                            Float Comparison.    (line  10)
* mpf_cmp_ui:                            Float Comparison.    (line   9)
* mpf_div:                               Float Arithmetic.    (line  29)
* mpf_div_2exp:                          Float Arithmetic.    (line  53)
* mpf_div_ui:                            Float Arithmetic.    (line  33)
* mpf_eq:                                Float Comparison.    (line  17)
* mpf_fits_sint_p:                       Miscellaneous Float Functions.
                                                              (line  20)
* mpf_fits_slong_p:                      Miscellaneous Float Functions.
                                                              (line  18)
* mpf_fits_sshort_p:                     Miscellaneous Float Functions.
                                                              (line  22)
* mpf_fits_uint_p:                       Miscellaneous Float Functions.
                                                              (line  19)
* mpf_fits_ulong_p:                      Miscellaneous Float Functions.
                                                              (line  17)
* mpf_fits_ushort_p:                     Miscellaneous Float Functions.
                                                              (line  21)
* mpf_floor:                             Miscellaneous Float Functions.
                                                              (line   8)
* mpf_get_d:                             Converting Floats.   (line   7)
* mpf_get_d_2exp:                        Converting Floats.   (line  16)
* mpf_get_default_prec:                  Initializing Floats. (line  12)
* mpf_get_prec:                          Initializing Floats. (line  62)
* mpf_get_si:                            Converting Floats.   (line  27)
* mpf_get_str:                           Converting Floats.   (line  37)
* mpf_get_ui:                            Converting Floats.   (line  28)
* mpf_init:                              Initializing Floats. (line  19)
* mpf_init2:                             Initializing Floats. (line  26)
* mpf_init_set:                          Simultaneous Float Init & Assign.
                                                              (line  16)
* mpf_init_set_d:                        Simultaneous Float Init & Assign.
                                                              (line  19)
* mpf_init_set_si:                       Simultaneous Float Init & Assign.
                                                              (line  18)
* mpf_init_set_str:                      Simultaneous Float Init & Assign.
                                                              (line  25)
* mpf_init_set_ui:                       Simultaneous Float Init & Assign.
                                                              (line  17)
* mpf_inits:                             Initializing Floats. (line  31)
* mpf_inp_str:                           I/O of Floats.       (line  36)
* mpf_integer_p:                         Miscellaneous Float Functions.
                                                              (line  14)
* mpf_mul:                               Float Arithmetic.    (line  19)
* mpf_mul_2exp:                          Float Arithmetic.    (line  50)
* mpf_mul_ui:                            Float Arithmetic.    (line  21)
* mpf_neg:                               Float Arithmetic.    (line  44)
* mpf_out_str:                           I/O of Floats.       (line  17)
* mpf_pow_ui:                            Float Arithmetic.    (line  41)
* mpf_random2:                           Miscellaneous Float Functions.
                                                              (line  49)
* mpf_reldiff:                           Float Comparison.    (line  24)
* mpf_rrandomb:                          Miscellaneous Float Functions.
                                                              (line  36)
* mpf_set:                               Assigning Floats.    (line  10)
* mpf_set_d:                             Assigning Floats.    (line  13)
* mpf_set_default_prec:                  Initializing Floats. (line   7)
* mpf_set_prec:                          Initializing Floats. (line  65)
* mpf_set_prec_raw:                      Initializing Floats. (line  72)
* mpf_set_q:                             Assigning Floats.    (line  15)
* mpf_set_si:                            Assigning Floats.    (line  12)
* mpf_set_str:                           Assigning Floats.    (line  18)
* mpf_set_ui:                            Assigning Floats.    (line  11)
* mpf_set_z:                             Assigning Floats.    (line  14)
* mpf_sgn:                               Float Comparison.    (line  28)
* mpf_sqrt:                              Float Arithmetic.    (line  36)
* mpf_sqrt_ui:                           Float Arithmetic.    (line  37)
* mpf_sub:                               Float Arithmetic.    (line  12)
* mpf_sub_ui:                            Float Arithmetic.    (line  16)
* mpf_swap:                              Assigning Floats.    (line  52)
* mpf_t:                                 Nomenclature and Types.
                                                              (line  21)
* mpf_trunc:                             Miscellaneous Float Functions.
                                                              (line   9)
* mpf_ui_div:                            Float Arithmetic.    (line  31)
* mpf_ui_sub:                            Float Arithmetic.    (line  14)
* mpf_urandomb:                          Miscellaneous Float Functions.
                                                              (line  27)
* mpir_version:                          Useful Macros and Constants.
                                                              (line  29)
* mpn_add:                               Low-level Functions. (line  70)
* mpn_add_1:                             Low-level Functions. (line  65)
* mpn_add_n:                             Low-level Functions. (line  55)
* mpn_addmul_1:                          Low-level Functions. (line 131)
* mpn_and_n:                             Low-level Functions. (line 473)
* mpn_andn_n:                            Low-level Functions. (line 488)
* mpn_bdivmod:                           Low-level Functions. (line 257)
* mpn_cmp:                               Low-level Functions. (line 298)
* mpn_com:                               Low-level Functions. (line 513)
* mpn_copyd:                             Low-level Functions. (line 522)
* mpn_copyi:                             Low-level Functions. (line 518)
* mpn_divexact_by3:                      Low-level Functions. (line 225)
* mpn_divexact_by3c:                     Low-level Functions. (line 227)
* mpn_divmod_1:                          Low-level Functions. (line 209)
* mpn_divrem:                            Low-level Functions. (line 183)
* mpn_divrem_1:                          Low-level Functions. (line 207)
* mpn_gcd:                               Low-level Functions. (line 303)
* mpn_gcd_1:                             Low-level Functions. (line 314)
* mpn_gcdext:                            Low-level Functions. (line 320)
* mpn_get_str:                           Low-level Functions. (line 362)
* mpn_hamdist:                           Low-level Functions. (line 463)
* mpn_ior_n:                             Low-level Functions. (line 478)
* mpn_iorn_n:                            Low-level Functions. (line 493)
* mpn_lshift:                            Low-level Functions. (line 274)
* mpn_mod_1:                             Low-level Functions. (line 251)
* mpn_mul:                               Low-level Functions. (line 153)
* mpn_mul_1:                             Low-level Functions. (line 116)
* mpn_mul_n:                             Low-level Functions. (line 104)
* mpn_nand_n:                            Low-level Functions. (line 498)
* mpn_neg:                               Low-level Functions. (line  99)
* mpn_nior_n:                            Low-level Functions. (line 503)
* mpn_perfect_square_p:                  Low-level Functions. (line 469)
* mpn_popcount:                          Low-level Functions. (line 459)
* mpn_random:                            Low-level Functions. (line 411)
* mpn_random2:                           Low-level Functions. (line 412)
* mpn_randomb:                           Low-level Functions. (line 441)
* mpn_rrandom:                           Low-level Functions. (line 449)
* mpn_rshift:                            Low-level Functions. (line 286)
* mpn_scan0:                             Low-level Functions. (line 396)
* mpn_scan1:                             Low-level Functions. (line 404)
* mpn_set_str:                           Low-level Functions. (line 377)
* mpn_sqr:                               Low-level Functions. (line 164)
* mpn_sqrtrem:                           Low-level Functions. (line 344)
* mpn_sub:                               Low-level Functions. (line  91)
* mpn_sub_1:                             Low-level Functions. (line  86)
* mpn_sub_n:                             Low-level Functions. (line  77)
* mpn_submul_1:                          Low-level Functions. (line 142)
* mpn_tdiv_qr:                           Low-level Functions. (line 173)
* mpn_urandomb:                          Low-level Functions. (line 425)
* mpn_urandomm:                          Low-level Functions. (line 433)
* mpn_xnor_n:                            Low-level Functions. (line 508)
* mpn_xor_n:                             Low-level Functions. (line 483)
* mpn_zero:                              Low-level Functions. (line 525)
* mpq_abs:                               Rational Arithmetic. (line  31)
* mpq_add:                               Rational Arithmetic. (line   7)
* mpq_canonicalize:                      Rational Number Functions.
                                                              (line  22)
* mpq_class:                             C++ Interface General.
                                                              (line  19)
* mpq_class::canonicalize:               C++ Interface Rationals.
                                                              (line  37)
* mpq_class::get_d:                      C++ Interface Rationals.
                                                              (line  46)
* mpq_class::get_den:                    C++ Interface Rationals.
                                                              (line  58)
* mpq_class::get_den_mpz_t:              C++ Interface Rationals.
                                                              (line  68)
* mpq_class::get_mpq_t:                  C++ Interface General.
                                                              (line  65)
* mpq_class::get_num:                    C++ Interface Rationals.
                                                              (line  57)
* mpq_class::get_num_mpz_t:              C++ Interface Rationals.
                                                              (line  67)
* mpq_class::get_str:                    C++ Interface Rationals.
                                                              (line  47)
* mpq_class::mpq_class:                  C++ Interface Rationals.
                                                              (line  11)
* mpq_class::set_str:                    C++ Interface Rationals.
                                                              (line  48)
* mpq_clear:                             Initializing Rationals.
                                                              (line  16)
* mpq_clears:                            Initializing Rationals.
                                                              (line  20)
* mpq_cmp:                               Comparing Rationals. (line   7)
* mpq_cmp_si:                            Comparing Rationals. (line  17)
* mpq_cmp_ui:                            Comparing Rationals. (line  15)
* mpq_denref:                            Applying Integer Functions.
                                                              (line  18)
* mpq_div:                               Rational Arithmetic. (line  22)
* mpq_div_2exp:                          Rational Arithmetic. (line  25)
* mpq_equal:                             Comparing Rationals. (line  33)
* mpq_get_d:                             Rational Conversions.
                                                              (line   7)
* mpq_get_den:                           Applying Integer Functions.
                                                              (line  24)
* mpq_get_num:                           Applying Integer Functions.
                                                              (line  23)
* mpq_get_str:                           Rational Conversions.
                                                              (line  22)
* mpq_init:                              Initializing Rationals.
                                                              (line   7)
* mpq_inits:                             Initializing Rationals.
                                                              (line  12)
* mpq_inp_str:                           I/O of Rationals.    (line  23)
* mpq_inv:                               Rational Arithmetic. (line  34)
* mpq_mul:                               Rational Arithmetic. (line  15)
* mpq_mul_2exp:                          Rational Arithmetic. (line  18)
* mpq_neg:                               Rational Arithmetic. (line  28)
* mpq_numref:                            Applying Integer Functions.
                                                              (line  17)
* mpq_out_str:                           I/O of Rationals.    (line  15)
* mpq_set:                               Initializing Rationals.
                                                              (line  24)
* mpq_set_d:                             Rational Conversions.
                                                              (line  17)
* mpq_set_den:                           Applying Integer Functions.
                                                              (line  26)
* mpq_set_f:                             Rational Conversions.
                                                              (line  18)
* mpq_set_num:                           Applying Integer Functions.
                                                              (line  25)
* mpq_set_si:                            Initializing Rationals.
                                                              (line  31)
* mpq_set_str:                           Initializing Rationals.
                                                              (line  36)
* mpq_set_ui:                            Initializing Rationals.
                                                              (line  29)
* mpq_set_z:                             Initializing Rationals.
                                                              (line  25)
* mpq_sgn:                               Comparing Rationals. (line  27)
* mpq_sub:                               Rational Arithmetic. (line  11)
* mpq_swap:                              Initializing Rationals.
                                                              (line  56)
* mpq_t:                                 Nomenclature and Types.
                                                              (line  16)
* mpz_abs:                               Integer Arithmetic.  (line  42)
* mpz_add:                               Integer Arithmetic.  (line   7)
* mpz_add_ui:                            Integer Arithmetic.  (line   9)
* mpz_addmul:                            Integer Arithmetic.  (line  25)
* mpz_addmul_ui:                         Integer Arithmetic.  (line  27)
* mpz_and:                               Integer Logic and Bit Fiddling.
                                                              (line  11)
* mpz_array_init:                        Integer Special Functions.
                                                              (line  11)
* mpz_bin_ui:                            Number Theoretic Functions.
                                                              (line 149)
* mpz_bin_uiui:                          Number Theoretic Functions.
                                                              (line 151)
* mpz_cdiv_q:                            Integer Division.    (line  13)
* mpz_cdiv_q_2exp:                       Integer Division.    (line  24)
* mpz_cdiv_q_ui:                         Integer Division.    (line  17)
* mpz_cdiv_qr:                           Integer Division.    (line  15)
* mpz_cdiv_qr_ui:                        Integer Division.    (line  21)
* mpz_cdiv_r:                            Integer Division.    (line  14)
* mpz_cdiv_r_2exp:                       Integer Division.    (line  25)
* mpz_cdiv_r_ui:                         Integer Division.    (line  19)
* mpz_cdiv_ui:                           Integer Division.    (line  23)
* mpz_class:                             C++ Interface General.
                                                              (line  18)
* mpz_class::fits_sint_p:                C++ Interface Integers.
                                                              (line  45)
* mpz_class::fits_slong_p:               C++ Interface Integers.
                                                              (line  46)
* mpz_class::fits_sshort_p:              C++ Interface Integers.
                                                              (line  47)
* mpz_class::fits_uint_p:                C++ Interface Integers.
                                                              (line  48)
* mpz_class::fits_ulong_p:               C++ Interface Integers.
                                                              (line  49)
* mpz_class::fits_ushort_p:              C++ Interface Integers.
                                                              (line  50)
* mpz_class::get_d:                      C++ Interface Integers.
                                                              (line  51)
* mpz_class::get_mpz_t:                  C++ Interface General.
                                                              (line  64)
* mpz_class::get_si:                     C++ Interface Integers.
                                                              (line  52)
* mpz_class::get_str:                    C++ Interface Integers.
                                                              (line  53)
* mpz_class::get_ui:                     C++ Interface Integers.
                                                              (line  54)
* mpz_class::mpz_class:                  C++ Interface Integers.
                                                              (line   7)
* mpz_class::set_str:                    C++ Interface Integers.
                                                              (line  55)
* mpz_clear:                             Initializing Integers.
                                                              (line  41)
* mpz_clears:                            Initializing Integers.
                                                              (line  45)
* mpz_clrbit:                            Integer Logic and Bit Fiddling.
                                                              (line  54)
* mpz_cmp:                               Integer Comparisons. (line   7)
* mpz_cmp_d:                             Integer Comparisons. (line   8)
* mpz_cmp_si:                            Integer Comparisons. (line   9)
* mpz_cmp_ui:                            Integer Comparisons. (line  10)
* mpz_cmpabs:                            Integer Comparisons. (line  18)
* mpz_cmpabs_d:                          Integer Comparisons. (line  19)
* mpz_cmpabs_ui:                         Integer Comparisons. (line  20)
* mpz_com:                               Integer Logic and Bit Fiddling.
                                                              (line  20)
* mpz_combit:                            Integer Logic and Bit Fiddling.
                                                              (line  57)
* mpz_congruent_2exp_p:                  Integer Division.    (line 124)
* mpz_congruent_p:                       Integer Division.    (line 121)
* mpz_congruent_ui_p:                    Integer Division.    (line 123)
* mpz_divexact:                          Integer Division.    (line 101)
* mpz_divexact_ui:                       Integer Division.    (line 102)
* mpz_divisible_2exp_p:                  Integer Division.    (line 112)
* mpz_divisible_p:                       Integer Division.    (line 110)
* mpz_divisible_ui_p:                    Integer Division.    (line 111)
* mpz_even_p:                            Miscellaneous Integer Functions.
                                                              (line  18)
* mpz_export:                            Integer Import and Export.
                                                              (line  45)
* mpz_fac_ui:                            Number Theoretic Functions.
                                                              (line 146)
* mpz_fdiv_q:                            Integer Division.    (line  27)
* mpz_fdiv_q_2exp:                       Integer Division.    (line  38)
* mpz_fdiv_q_ui:                         Integer Division.    (line  31)
* mpz_fdiv_qr:                           Integer Division.    (line  29)
* mpz_fdiv_qr_ui:                        Integer Division.    (line  35)
* mpz_fdiv_r:                            Integer Division.    (line  28)
* mpz_fdiv_r_2exp:                       Integer Division.    (line  39)
* mpz_fdiv_r_ui:                         Integer Division.    (line  33)
* mpz_fdiv_ui:                           Integer Division.    (line  37)
* mpz_fib2_ui:                           Number Theoretic Functions.
                                                              (line 159)
* mpz_fib_ui:                            Number Theoretic Functions.
                                                              (line 157)
* mpz_fits_sint_p:                       Miscellaneous Integer Functions.
                                                              (line  10)
* mpz_fits_slong_p:                      Miscellaneous Integer Functions.
                                                              (line   8)
* mpz_fits_sshort_p:                     Miscellaneous Integer Functions.
                                                              (line  12)
* mpz_fits_uint_p:                       Miscellaneous Integer Functions.
                                                              (line   9)
* mpz_fits_ulong_p:                      Miscellaneous Integer Functions.
                                                              (line   7)
* mpz_fits_ushort_p:                     Miscellaneous Integer Functions.
                                                              (line  11)
* mpz_gcd:                               Number Theoretic Functions.
                                                              (line  82)
* mpz_gcd_ui:                            Number Theoretic Functions.
                                                              (line  87)
* mpz_gcdext:                            Number Theoretic Functions.
                                                              (line  97)
* mpz_get_d:                             Converting Integers. (line  43)
* mpz_get_d_2exp:                        Converting Integers. (line  51)
* mpz_get_si:                            Converting Integers. (line  18)
* mpz_get_str:                           Converting Integers. (line  62)
* mpz_get_sx:                            Converting Integers. (line  35)
* mpz_get_ui:                            Converting Integers. (line  11)
* mpz_get_ux:                            Converting Integers. (line  27)
* mpz_getlimbn:                          Integer Special Functions.
                                                              (line  63)
* mpz_hamdist:                           Integer Logic and Bit Fiddling.
                                                              (line  29)
* mpz_import:                            Integer Import and Export.
                                                              (line  11)
* mpz_init:                              Initializing Integers.
                                                              (line  26)
* mpz_init2:                             Initializing Integers.
                                                              (line  33)
* mpz_init_set:                          Simultaneous Integer Init & Assign.
                                                              (line  27)
* mpz_init_set_d:                        Simultaneous Integer Init & Assign.
                                                              (line  32)
* mpz_init_set_si:                       Simultaneous Integer Init & Assign.
                                                              (line  29)
* mpz_init_set_str:                      Simultaneous Integer Init & Assign.
                                                              (line  37)
* mpz_init_set_sx:                       Simultaneous Integer Init & Assign.
                                                              (line  31)
* mpz_init_set_ui:                       Simultaneous Integer Init & Assign.
                                                              (line  28)
* mpz_init_set_ux:                       Simultaneous Integer Init & Assign.
                                                              (line  30)
* mpz_inits:                             Initializing Integers.
                                                              (line  29)
* mpz_inp_raw:                           I/O of Integers.     (line  59)
* mpz_inp_str:                           I/O of Integers.     (line  28)
* mpz_invert:                            Number Theoretic Functions.
                                                              (line 110)
* mpz_ior:                               Integer Logic and Bit Fiddling.
                                                              (line  14)
* mpz_jacobi:                            Number Theoretic Functions.
                                                              (line 116)
* mpz_kronecker:                         Number Theoretic Functions.
                                                              (line 124)
* mpz_kronecker_si:                      Number Theoretic Functions.
                                                              (line 125)
* mpz_kronecker_ui:                      Number Theoretic Functions.
                                                              (line 126)
* mpz_lcm:                               Number Theoretic Functions.
                                                              (line 104)
* mpz_lcm_ui:                            Number Theoretic Functions.
                                                              (line 105)
* mpz_legendre:                          Number Theoretic Functions.
                                                              (line 119)
* mpz_likely_prime_p:                    Number Theoretic Functions.
                                                              (line  26)
* mpz_lucnum2_ui:                        Number Theoretic Functions.
                                                              (line 170)
* mpz_lucnum_ui:                         Number Theoretic Functions.
                                                              (line 168)
* mpz_mod:                               Integer Division.    (line  91)
* mpz_mod_ui:                            Integer Division.    (line  93)
* mpz_mul:                               Integer Arithmetic.  (line  19)
* mpz_mul_2exp:                          Integer Arithmetic.  (line  35)
* mpz_mul_si:                            Integer Arithmetic.  (line  20)
* mpz_mul_ui:                            Integer Arithmetic.  (line  22)
* mpz_neg:                               Integer Arithmetic.  (line  39)
* mpz_next_likely_prime:                 Number Theoretic Functions.
                                                              (line  71)
* mpz_nextprime:                         Number Theoretic Functions.
                                                              (line  60)
* mpz_nthroot:                           Integer Roots.       (line  13)
* mpz_odd_p:                             Miscellaneous Integer Functions.
                                                              (line  17)
* mpz_out_raw:                           I/O of Integers.     (line  43)
* mpz_out_str:                           I/O of Integers.     (line  16)
* mpz_perfect_power_p:                   Integer Roots.       (line  31)
* mpz_perfect_square_p:                  Integer Roots.       (line  40)
* mpz_popcount:                          Integer Logic and Bit Fiddling.
                                                              (line  23)
* mpz_pow_ui:                            Integer Exponentiation.
                                                              (line  18)
* mpz_powm:                              Integer Exponentiation.
                                                              (line   8)
* mpz_powm_ui:                           Integer Exponentiation.
                                                              (line  10)
* mpz_probab_prime_p:                    Number Theoretic Functions.
                                                              (line  41)
* mpz_probable_prime_p:                  Number Theoretic Functions.
                                                              (line   8)
* mpz_realloc2:                          Initializing Integers.
                                                              (line  49)
* mpz_remove:                            Number Theoretic Functions.
                                                              (line 141)
* mpz_root:                              Integer Roots.       (line   7)
* mpz_rootrem:                           Integer Roots.       (line  17)
* mpz_rrandomb:                          Integer Random Numbers.
                                                              (line  31)
* mpz_scan0:                             Integer Logic and Bit Fiddling.
                                                              (line  37)
* mpz_scan1:                             Integer Logic and Bit Fiddling.
                                                              (line  38)
* mpz_set:                               Assigning Integers.  (line  10)
* mpz_set_d:                             Assigning Integers.  (line  15)
* mpz_set_f:                             Assigning Integers.  (line  17)
* mpz_set_q:                             Assigning Integers.  (line  16)
* mpz_set_si:                            Assigning Integers.  (line  12)
* mpz_set_str:                           Assigning Integers.  (line  24)
* mpz_set_sx:                            Assigning Integers.  (line  14)
* mpz_set_ui:                            Assigning Integers.  (line  11)
* mpz_set_ux:                            Assigning Integers.  (line  13)
* mpz_setbit:                            Integer Logic and Bit Fiddling.
                                                              (line  51)
* mpz_sgn:                               Integer Comparisons. (line  28)
* mpz_si_kronecker:                      Number Theoretic Functions.
                                                              (line 127)
* mpz_size:                              Integer Special Functions.
                                                              (line  71)
* mpz_sizeinbase:                        Miscellaneous Integer Functions.
                                                              (line  23)
* mpz_sqrt:                              Integer Roots.       (line  21)
* mpz_sqrtrem:                           Integer Roots.       (line  24)
* mpz_sub:                               Integer Arithmetic.  (line  12)
* mpz_sub_ui:                            Integer Arithmetic.  (line  14)
* mpz_submul:                            Integer Arithmetic.  (line  30)
* mpz_submul_ui:                         Integer Arithmetic.  (line  32)
* mpz_swap:                              Assigning Integers.  (line  40)
* mpz_t:                                 Nomenclature and Types.
                                                              (line   6)
* mpz_tdiv_q:                            Integer Division.    (line  41)
* mpz_tdiv_q_2exp:                       Integer Division.    (line  52)
* mpz_tdiv_q_ui:                         Integer Division.    (line  45)
* mpz_tdiv_qr:                           Integer Division.    (line  43)
* mpz_tdiv_qr_ui:                        Integer Division.    (line  49)
* mpz_tdiv_r:                            Integer Division.    (line  42)
* mpz_tdiv_r_2exp:                       Integer Division.    (line  53)
* mpz_tdiv_r_ui:                         Integer Division.    (line  47)
* mpz_tdiv_ui:                           Integer Division.    (line  51)
* mpz_tstbit:                            Integer Logic and Bit Fiddling.
                                                              (line  60)
* mpz_ui_kronecker:                      Number Theoretic Functions.
                                                              (line 128)
* mpz_ui_pow_ui:                         Integer Exponentiation.
                                                              (line  20)
* mpz_ui_sub:                            Integer Arithmetic.  (line  16)
* mpz_urandomb:                          Integer Random Numbers.
                                                              (line  14)
* mpz_urandomm:                          Integer Random Numbers.
                                                              (line  23)
* mpz_xor:                               Integer Logic and Bit Fiddling.
                                                              (line  17)
* operator%:                             C++ Interface Integers.
                                                              (line  30)
* operator/:                             C++ Interface Integers.
                                                              (line  29)
* operator<<:                            C++ Formatted Output.
                                                              (line  11)
* operator>> <1>:                        C++ Interface Rationals.
                                                              (line  77)
* operator>>:                            C++ Formatted Input. (line  11)
* sgn <1>:                               C++ Interface Floats.
                                                              (line  89)
* sgn <2>:                               C++ Interface Rationals.
                                                              (line  50)
* sgn:                                   C++ Interface Integers.
                                                              (line  57)
* sqrt <1>:                              C++ Interface Floats.
                                                              (line  90)
* sqrt:                                  C++ Interface Integers.
                                                              (line  58)
* trunc:                                 C++ Interface Floats.
                                                              (line  91)