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 2.6.0. 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<=R48 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 > operator+(const mpf_class &f, const mpf_class &g) { return __gmp_expr <__gmp_binary_expr >(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 mpf_class & mpf_class::operator=(const __gmp_expr &expr) { expr.eval(this->get_mpf_t(), this->precision()); return *this; } template void __gmp_expr<__gmp_binary_expr >::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 __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > operator+(const __gmp_expr &expr1, const __gmp_expr &expr2) { return __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, __gmp_binary_plus> > (expr1, expr2); } And the corresponding specializations of `__gmp_expr::eval': template void __gmp_expr <__gmp_binary_expr<__gmp_expr, __gmp_expr, 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. Case Vanhorsen helped with release testing. David Cleaver filed a bug report.  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' * Marco Bodrato, "High degree Toom`n'half for balanced and unbalanced multiplication", E. Antelo, D. Hough and P. Ienne, editors, Proceedings of the 20th IEEE Symposium on Computer Arithmetic, IEEE, Tubingen, Germany, July 25-27, 2011, pp. 15-222. See `http://bodrato.it/papers' * 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 of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software. We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference. 1. APPLICABILITY AND DEFINITIONS This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you". You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. A "Modified Version" of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language. A "Secondary Section" is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them. The "Invariant Sections" are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none. The "Cover Texts" are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words. A "Transparent" copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not "Transparent" is called "Opaque". Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only. The "Title Page" means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text. The "publisher" means any person or entity that distributes copies of the Document to the public. A section "Entitled XYZ" means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as "Acknowledgements", "Dedications", "Endorsements", or "History".) To "Preserve the Title" of such a section when you modify the Document means that it remains a section "Entitled XYZ" according to this definition. The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. 2. VERBATIM COPYING You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3. You may also lend copies, under the same conditions stated above, and you may publicly display copies. 3. COPYING IN QUANTITY If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects. If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages. If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public. It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document. 4. MODIFICATIONS You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version: A. Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission. B. List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement. C. State on the Title page the name of the publisher of the Modified Version, as the publisher. D. Preserve all the copyright notices of the Document. E. Add an appropriate copyright notice for your modifications adjacent to the other copyright notices. F. Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below. G. Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice. H. Include an unaltered copy of this License. I. Preserve the section Entitled "History", Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled "History" in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence. J. Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the "History" section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission. K. For any section Entitled "Acknowledgements" or "Dedications", Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein. L. Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles. M. Delete any section Entitled "Endorsements". Such a section may not be included in the Modified Version. N. Do not retitle any existing section to be Entitled "Endorsements" or to conflict in title with any Invariant Section. O. Preserve any Warranty Disclaimers. If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles. You may add a section Entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties--for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard. You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one. The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version. 5. COMBINING DOCUMENTS You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers. The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work. In the combination, you must combine any sections Entitled "History" in the various original documents, forming one section Entitled "History"; likewise combine any sections Entitled "Acknowledgements", and any sections Entitled "Dedications". You must delete all sections Entitled "Endorsements." 6. COLLECTIONS OF DOCUMENTS You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects. You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document. 7. AGGREGATION WITH INDEPENDENT WORKS A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an "aggregate" if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document. If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate. 8. TRANSLATION Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail. If a section in the Document is Entitled "Acknowledgements", "Dedications", or "History", the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title. 9. TERMINATION You may not copy, modify, sublicense, or distribute the Document except as expressly provided under this License. Any attempt otherwise to copy, modify, sublicense, or distribute it is void, and will automatically terminate your rights under this License. However, if you cease all violation of this License, then your license from a particular copyright holder is reinstated (a) provisionally, unless and until the copyright holder explicitly and finally terminates your license, and (b) permanently, if the copyright holder fails to notify you of the violation by some reasonable means prior to 60 days after the cessation. Moreover, your license from a particular copyright holder is reinstated permanently if the copyright holder notifies you of the violation by some reasonable means, this is the first time you have received notice of violation of this License (for any work) from that copyright holder, and you cure the violation prior to 30 days after your receipt of the notice. Termination of your rights under this section does not terminate the licenses of parties who have received copies or rights from you under this License. If your rights have been terminated and not permanently reinstated, receipt of a copy of some or all of the same material does not give you any rights to use it. 10. FUTURE REVISIONS OF THIS LICENSE The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See `http://www.gnu.org/copyleft/'. Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. If the Document specifies that a proxy can decide which future versions of this License can be used, that proxy's public statement of acceptance of a version permanently authorizes you to choose that version for the Document. 11. RELICENSING "Massive Multiauthor Collaboration Site" (or "MMC Site") means any World Wide Web server that publishes copyrightable works and also provides prominent facilities for anybody to edit those works. A public wiki that anybody can edit is an example of such a server. A "Massive Multiauthor Collaboration" (or "MMC") contained in the site means any set of copyrightable works thus published on the MMC site. "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0 license published by Creative Commons Corporation, a not-for-profit corporation with a principal place of business in San Francisco, California, as well as future copyleft versions of that license published by that same organization. "Incorporate" means to publish or republish a Document, in whole or in part, as part of another Document. An MMC is "eligible for relicensing" if it is licensed under this License, and if all works that were first published under this License somewhere other than this MMC, and subsequently incorporated in whole or in part into the MMC, (1) had no cover texts or invariant sections, and (2) were thus incorporated prior to November 1, 2008. The operator of an MMC Site may republish an MMC contained in the site under CC-BY-SA on the same site at any time before August 1, 2009, provided the MMC is eligible for relicensing. ADDENDUM: How to use this License for your documents ==================================================== To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page: Copyright (C) YEAR YOUR NAME. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.  File: 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 60) * --disable-fft: Build Options. (line 316) * --disable-shared: Build Options. (line 53) * --disable-static: Build Options. (line 53) * --enable-alloca: Build Options. (line 277) * --enable-assert: Build Options. (line 321) * --enable-cxx: Build Options. (line 229) * --enable-fat: Build Options. (line 162) * --enable-gmpcompat: Build Options. (line 45) * --enable-profiling <1>: Profiling. (line 6) * --enable-profiling: Build Options. (line 325) * --exec-prefix: Build Options. (line 32) * --host: Build Options. (line 74) * --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 170) * 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 277) * 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 321) * 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 60) * Building MPIR: Installing MPIR. (line 6) * Bus error: Debugging. (line 7) * C compiler: Build Options. (line 181) * C++ compiler: Build Options. (line 253) * 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 229) * CC: Build Options. (line 181) * CC_FOR_BUILD: Build Options. (line 216) * CFLAGS: Build Options. (line 181) * 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 207) * CPU types <1>: Build Options. (line 115) * CPU types: Introduction to MPIR. (line 24) * Cross compiling: Build Options. (line 74) * Custom allocation: Custom Allocation. (line 6) * CXX: Build Options. (line 253) * CXXFLAGS: Build Options. (line 253) * 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 348) * Documentation formats: Build Options. (line 341) * Documentation license: GNU Free Documentation License. (line 6) * DVI: Build Options. (line 344) * 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 325) * 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 162) * FFT multiplication <1>: FFT Multiplication. (line 6) * FFT multiplication: Build Options. (line 316) * 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 151) * 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 74) * 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 329) * 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 60) * 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 344) * 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 344) * 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 277) * 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 277) * Texinfo: Build Options. (line 341) * 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 348)  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 19) * 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 21) * 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)