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 This is gmp.info, produced by makeinfo version 4.0 from gmp.texi.  This is gmp.info, produced by makeinfo version 4.2 from gmp.texi.
   
   This manual describes how to install and use the GNU multiple precision
   arithmetic library, version 4.1.2.
   
      Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
   2001, 2002 Free Software Foundation, Inc.
   
      Permission is granted to copy, distribute and/or modify this
   document under the terms of the GNU Free Documentation License, Version
   1.1 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  INFO-DIR-SECTION GNU libraries
 START-INFO-DIR-ENTRY  START-INFO-DIR-ENTRY
 * gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.  * gmp: (gmp).                   GNU Multiple Precision Arithmetic Library.
 END-INFO-DIR-ENTRY  END-INFO-DIR-ENTRY
   
    This file documents GNU MP, a library for arbitrary-precision  
 arithmetic.  File: gmp.info,  Node: Number Theoretic Functions,  Next: Integer Comparisons,  Prev: Integer Roots,  Up: Integer Functions
   
    Copyright (C) 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000  Number Theoretic Functions
 Free Software Foundation, Inc.  ==========================
   
    Permission is granted to make and distribute verbatim copies of this   - Function: int mpz_probab_prime_p (mpz_t N, int REPS)
 manual provided the copyright notice and this permission notice are       Determine whether N is prime.  Return 2 if N is definitely prime,
 preserved on all copies.       return 1 if N is probably prime (without being certain), or return
        0 if N is definitely composite.
   
    Permission is granted to copy and distribute modified versions of       This function does some trial divisions, then some Miller-Rabin
 this manual under the conditions for verbatim copying, provided that       probabilistic primality tests.  REPS controls how many such tests
 the entire resulting derived work is distributed under the terms of a       are done, 5 to 10 is a reasonable number, more will reduce the
 permission notice identical to this one.       chances of a composite being returned as "probably prime".
   
    Permission is granted to copy and distribute translations of this       Miller-Rabin and similar tests can be more properly called
 manual into another language, under the above conditions for modified       compositeness tests.  Numbers which fail are known to be composite
 versions, except that this permission notice may be stated in a       but those which pass might be prime or might be composite.  Only a
 translation approved by the Foundation.       few composites pass, hence those which pass are considered
        probably prime.
   
    - Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
 File: gmp.info,  Node: Low-level Functions,  Next: Random Number Functions,  Prev: Floating-point Functions,  Up: Top       Set ROP to the next prime greater than OP.
   
 Low-level Functions       This function uses a probabilistic algorithm to identify primes.
 *******************       For practical purposes it's adequate, the chance of a composite
        passing will be extremely small.
   
    This chapter describes low-level GMP functions, used to implement   - Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 the high-level GMP functions, but also intended for time-critical user       Set ROP to the greatest common divisor of OP1 and OP2.  The result
 code.       is always positive even if one or both input operands are negative.
   
    These functions start with the prefix `mpn_'.   - Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
             unsigned long int OP2)
        Compute the greatest common divisor of OP1 and OP2.  If ROP is not
        `NULL', store the result there.
   
    The `mpn' functions are designed to be as fast as possible, *not* to       If the result is small enough to fit in an `unsigned long int', it
 provide a coherent calling interface.  The different functions have       is returned.  If the result does not fit, 0 is returned, and the
 somewhat similar interfaces, but there are variations that make them       result is equal to the argument OP1.  Note that the result will
 hard to use.  These functions do as little as possible apart from the       always fit if OP2 is non-zero.
 real multiple precision computation, so that no time is spent on things  
 that not all callers need.  
   
    A source operand is specified by a pointer to the least significant   - Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t
 limb and a limb count.  A destination operand is specified by just a            B)
 pointer.  It is the responsibility of the caller to ensure that the       Set G to the greatest common divisor of A and B, and in addition
 destination has enough space for storing the result.       set S and T to coefficients satisfying A*S + B*T = G.  G is always
        positive, even if one or both of A and B are negative.
   
    With this way of specifying operands, it is possible to perform       If T is `NULL' then that value is not computed.
 computations on subranges of an argument, and store the result into a  
 subrange of a destination.  
   
    A common requirement for all functions is that each source area   - Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 needs at least one limb.  No size argument may be zero.  Unless   - Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2)
 otherwise stated, in-place operations are allowed where source and       Set ROP to the least common multiple of OP1 and OP2.  ROP is
 destination are the same, but not where they only partly overlap.       always positive, irrespective of the signs of OP1 and OP2.  ROP
        will be zero if either OP1 or OP2 is zero.
   
    The `mpn' functions are the base for the implementation of the   - Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 `mpz_', `mpf_', and `mpq_' functions.       Compute the inverse of OP1 modulo OP2 and put the result in ROP.
        If the inverse exists, the return value is non-zero and ROP will
        satisfy 0 <= ROP < OP2.  If an inverse doesn't exist the return
        value is zero and ROP is undefined.
   
    This example adds the number beginning at S1P and the number   - Function: int mpz_jacobi (mpz_t A, mpz_t B)
 beginning at S2P and writes the sum at DESTP.  All areas have SIZE       Calculate the Jacobi symbol (A/B).  This is defined only for B odd.
 limbs.  
   
      cy = mpn_add_n (destp, s1p, s2p, size)   - Function: int mpz_legendre (mpz_t A, mpz_t P)
        Calculate the Legendre symbol (A/P).  This is defined only for P
        an odd positive prime, and for such P it's identical to the Jacobi
        symbol.
   
 In the notation used here, a source operand is identified by the   - Function: int mpz_kronecker (mpz_t A, mpz_t B)
 pointer to the least significant limb, and the limb count in braces.   - Function: int mpz_kronecker_si (mpz_t A, long B)
 For example, {s1p, s1size}.   - Function: int mpz_kronecker_ui (mpz_t A, unsigned long B)
    - Function: int mpz_si_kronecker (long A, mpz_t B)
    - Function: int mpz_ui_kronecker (unsigned long A, mpz_t B)
        Calculate the Jacobi symbol (A/B) with the Kronecker extension
        (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
   
  - Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,       When B is odd the Jacobi symbol and Kronecker symbol are
           const mp_limb_t *S2P, mp_size_t SIZE)       identical, so `mpz_kronecker_ui' etc can be used for mixed
      Add {S1P, SIZE} and {S2P, SIZE}, and write the SIZE least       precision Jacobi symbols too.
      significant limbs of the result to RP.  Return carry, either 0 or  
      1.  
   
      This is the lowest-level function for addition.  It is the       For more information see Henri Cohen section 1.4.2 (*note
      preferred function for addition, since it is written in assembly       References::), or any number theory textbook.  See also the
      for most targets.  For addition of a variable to itself (i.e., S1P       example program `demos/qcn.c' which uses `mpz_kronecker_ui'.
      equals S2P, use `mpn_lshift' with a count of 1 for optimal speed.  
   
  - Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,   - Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)
           mp_size_t SIZE, mp_limb_t S2LIMB)       Remove all occurrences of the factor F from OP and store the
      Add {S1P, SIZE} and S2LIMB, and write the SIZE least significant       result in ROP.  The return value is how many such occurrences were
      limbs of the result to RP.  Return carry, either 0 or 1.       removed.
   
  - Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,   - Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
           mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)       Set ROP to OP!, the factorial of OP.
      Add {S1P, S1SIZE} and {S2P, S2SIZE}, and write the S1SIZE least  
      significant limbs of the result to RP.  Return carry, either 0 or  
      1.  
   
      This function requires that S1SIZE is greater than or equal to   - Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
      S2SIZE.   - Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
             unsigned long int K)
        Compute the binomial coefficient N over K and store the result in
        ROP.  Negative values of N are supported by `mpz_bin_ui', using
        the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
        section 1.2.6 part G.
   
  - Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,   - Function: void mpz_fib_ui (mpz_t FN, unsigned long int N)
           const mp_limb_t *S2P, mp_size_t SIZE)   - Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
      Subtract {S2P, S2SIZE} from {S1P, SIZE}, and write the SIZE least            int N)
      significant limbs of the result to RP.  Return borrow, either 0 or       `mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
      1.       `mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
   
      This is the lowest-level function for subtraction.  It is the       These functions are designed for calculating isolated Fibonacci
      preferred function for subtraction, since it is written in       numbers.  When a sequence of values is wanted it's best to start
      assembly for most targets.       with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
        similar.
   
  - Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,   - Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
           mp_size_t SIZE, mp_limb_t S2LIMB)   - Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned long
      Subtract S2LIMB from {S1P, SIZE}, and write the SIZE least            int N)
      significant limbs of the result to RP.  Return borrow, either 0 or       `mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
      1.       `mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
   
  - Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,       These functions are designed for calculating isolated Lucas
           mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)       numbers.  When a sequence of values is wanted it's best to start
      Subtract {S2P, S2SIZE} from {S1P, S1SIZE}, and write the S1SIZE       with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
      least significant limbs of the result to RP.  Return borrow,       or similar.
      either 0 or 1.  
   
      This function requires that S1SIZE is greater than or equal to       The Fibonacci numbers and Lucas numbers are related sequences, so
      S2SIZE.       it's never necessary to call both `mpz_fib2_ui' and
        `mpz_lucnum2_ui'.  The formulas for going from Fibonacci to Lucas
        can be found in *Note Lucas Numbers Algorithm::, the reverse is
        straightforward too.
   
  - Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const  
           mp_limb_t *S2P, mp_size_t SIZE)  File: gmp.info,  Node: Integer Comparisons,  Next: Integer Logic and Bit Fiddling,  Prev: Number Theoretic Functions,  Up: Integer Functions
      Multiply {S1P, SIZE} and {S2P, SIZE}, and write the *entire*  
      result to RP.  
   
      The destination has to have space for 2*SIZE limbs, even if the  Comparison Functions
      significant result might be one limb smaller.  ====================
   
  - Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,   - Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
           mp_size_t SIZE, mp_limb_t S2LIMB)   - Function: int mpz_cmp_d (mpz_t OP1, double OP2)
      Multiply {S1P, SIZE} and S2LIMB, and write the SIZE least   - Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
      significant limbs of the product to RP.  Return the most   - Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
      significant limb of the product.       Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero
        if OP1 = OP2, or a negative value if OP1 < OP2.
   
      This is a low-level function that is a building block for general       Note that `mpz_cmp_ui' and `mpz_cmp_si' are macros and will
      multiplication as well as other operations in GMP.  It is written       evaluate their arguments more than once.
      in assembly for most targets.  
   
      Don't call this function if S2LIMB is a power of 2; use   - Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
      `mpn_lshift' with a count equal to the logarithm of S2LIMB   - Function: int mpz_cmpabs_d (mpz_t OP1, double OP2)
      instead, for optimal speed.   - Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
        Compare the absolute values of OP1 and OP2.  Return a positive
        value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
        negative value if abs(OP1) < abs(OP2).
   
  - Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t       Note that `mpz_cmpabs_si' is a macro and will evaluate its
           *S1P, mp_size_t SIZE, mp_limb_t S2LIMB)       arguments more than once.
      Multiply {S1P, SIZE} and S2LIMB, and add the SIZE least  
      significant limbs of the product to {RP, SIZE} and write the  
      result to RP.  Return the most significant limb of the product,  
      plus carry-out from the addition.  
   
      This is a low-level function that is a building block for general   - Macro: int mpz_sgn (mpz_t OP)
      multiplication as well as other operations in GMP.  It is written       Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
      in assembly for most targets.  
   
  - Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t       This function is actually implemented as a macro.  It evaluates
           *S1P, mp_size_t SIZE, mp_limb_t S2LIMB)       its argument multiple times.
      Multiply {S1P, SIZE} and S2LIMB, and subtract the SIZE least  
      significant limbs of the product from {RP, SIZE} and write the  
      result to RP.  Return the most significant limb of the product,  
      minus borrow-out from the subtraction.  
   
      This is a low-level function that is a building block for general  
      multiplication and division as well as other operations in GMP.  File: gmp.info,  Node: Integer Logic and Bit Fiddling,  Next: I/O of Integers,  Prev: Integer Comparisons,  Up: Integer Functions
      It is written in assembly for most targets.  
   
  - Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,  Logical and Bit Manipulation Functions
           mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE)  ======================================
      Multiply {S1P, S1SIZE} and {S2P, S2SIZE}, and write the result to  
      RP.  Return the most significant limb of the result.  
   
      The destination has to have space for S1SIZE + S2SIZE limbs, even     These functions behave as if twos complement arithmetic were used
      if the result might be one limb smaller.  (although sign-magnitude is the actual implementation).  The least
   significant bit is number 0.
   
      This function requires that S1SIZE is greater than or equal to   - Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
      S2SIZE.  The destination must be distinct from either input       Set ROP to OP1 logical-and OP2.
      operands.  
   
  - Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t   - Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
           QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,       Set ROP to OP1 inclusive-or OP2.
           mp_size_t DN)  
      Divide {NP, NN} by {DP, DN}.  Write the quotient at QP and the  
      remainder at RP.  
   
      The quotient written at QP will be NN - DN + 1 limbs.  The   - Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
      remainder written at RP will be DN limbs.       Set ROP to OP1 exclusive-or OP2.
   
      It is required that NN is greater than or equal to DN.  The QXN   - Function: void mpz_com (mpz_t ROP, mpz_t OP)
      operand must be zero.       Set ROP to the one's complement of OP.
   
      The quotient is rounded towards 0.   - Function: unsigned long int mpz_popcount (mpz_t OP)
        If OP>=0, return the population count of OP, which is the number
        of 1 bits in the binary representation.  If OP<0, the number of 1s
        is infinite, and the return value is MAX_ULONG, the largest
        possible `unsigned long'.
   
      No overlap between arguments is permitted.   - Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)
        If OP1 and OP2 are both >=0 or both <0, return the hamming
        distance between the two operands, which is the number of bit
        positions where OP1 and OP2 have different bit values.  If one
        operand is >=0 and the other <0 then the number of bits different
        is infinite, and the return value is MAX_ULONG, the largest
        possible `unsigned long'.
   
  - Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t XSIZE,   - Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int
           mp_limb_t *RS2P, mp_size_t RS2SIZE, const mp_limb_t *S3P,            STARTING_BIT)
           mp_size_t S3SIZE)   - Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int
      [This function is obsolete.  Please call `mpn_tdiv_qr' instead for            STARTING_BIT)
      best performance.]       Scan OP, starting from bit STARTING_BIT, towards more significant
        bits, until the first 0 or 1 bit (respectively) is found.  Return
        the index of the found bit.
   
      Divide {RS2P, RS2SIZE} by {S3P, S3SIZE}, and write the quotient at       If the bit at STARTING_BIT is already what's sought, then
      R1P, with the exception of the most significant limb, which is       STARTING_BIT is returned.
      returned.  The remainder replaces the dividend at RS2P; it will be  
      S3SIZE limbs long (i.e., as many limbs as the divisor).  
   
      In addition to an integer quotient, XSIZE fraction limbs are       If there's no bit found, then MAX_ULONG is returned.  This will
      developed, and stored after the integral limbs.  For most usages,       happen in `mpz_scan0' past the end of a positive number, or
      XSIZE will be zero.       `mpz_scan1' past the end of a negative.
   
      It is required that RS2SIZE is greater than or equal to S3SIZE.   - Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX)
      It is required that the most significant bit of the divisor is set.       Set bit BIT_INDEX in ROP.
   
      If the quotient is not needed, pass RS2P + S3SIZE as R1P.  Aside   - Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)
      from that special case, no overlap between arguments is permitted.       Clear bit BIT_INDEX in ROP.
   
      Return the most significant limb of the quotient, either 0 or 1.   - Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX)
        Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
   
      The area at R1P needs to be RS2SIZE - S3SIZE + XSIZE limbs large.  
   File: gmp.info,  Node: I/O of Integers,  Next: Integer Random Numbers,  Prev: Integer Logic and Bit Fiddling,  Up: Integer Functions
   
  - Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t XSIZE,  Input and Output Functions
           mp_limb_t *S2P, mp_size_t S2SIZE, mp_limb_t S3LIMB)  ==========================
  - Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,  
           mp_size_t S2SIZE, mp_limb_t S3LIMB)  
      Divide {S2P, S2SIZE} by S3LIMB, and write the quotient at R1P.  
      Return the remainder.  
   
      The integer quotient is written to {R1P+XSIZE, S2SIZE} and in     Functions that perform input from a stdio stream, and functions that
      addition XSIZE fraction limbs are developed and written to {R1P,  output to a stdio stream.  Passing a `NULL' pointer for a STREAM
      XSIZE}.  Either or both S2SIZE and XSIZE can be zero.  For most  argument to any of these functions will make them read from `stdin' and
      usages, XSIZE will be zero.  write to `stdout', respectively.
   
      `mpn_divmod_1' exists for upward source compatibility and is     When using any of these functions, it is a good idea to include
      simply a macro calling `mpn_divrem_1' with an XSIZE of 0.  `stdio.h' before `gmp.h', since that will allow `gmp.h' to define
   prototypes for these functions.
   
      The areas at R1P and S2P have to be identical or completely   - Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
      separate, not partially overlapping.       Output OP on stdio stream STREAM, as a string of digits in base
        BASE.  The base may vary from 2 to 36.
   
  - Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,       Return the number of bytes written, or if an error occurred,
           mp_size_t RS2SIZE, const mp_limb_t *S3P, mp_size_t S3SIZE)       return 0.
      *This interface is obsolete.  It will disappear from future  
      releases.  Use `mpn_divrem' in its stead.*  
   
  - Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,   - Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
           mp_size_t SIZE)       Input a possibly white-space preceded string in base BASE from
  - Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP,       stdio stream STREAM, and put the read integer in ROP.  The base
           mp_size_t SIZE, mp_limb_t CARRY)       may vary from 2 to 36.  If BASE is 0, the actual base is
      Divide {SP, SIZE} by 3, expecting it to divide exactly, and       determined from the leading characters: if the first two
      writing the result to {RP, SIZE}.  If 3 divides exactly, the       characters are `0x' or `0X', hexadecimal is assumed, otherwise if
      return value is zero and the result is the quotient.  If not, the       the first character is `0', octal is assumed, otherwise decimal is
      return value is non-zero and the result won't be anything useful.       assumed.
   
      `mpn_divexact_by3c' takes an initial carry parameter, which can be       Return the number of bytes read, or if an error occurred, return 0.
      the return value from a previous call, so a large calculation can  
      be done piece by piece.  `mpn_divexact_by3' is simply a macro  
      calling `mpn_divexact_by3c' with a 0 carry parameter.  
   
      These routines use a multiply-by-inverse and will be faster than   - Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
      `mpn_divrem_1' on CPUs with fast multiplication but slow division.       Output OP on stdio stream STREAM, in raw binary format.  The
        integer is written in a portable format, with 4 bytes of size
        information, and that many bytes of limbs.  Both the size and the
        limbs are written in decreasing significance order (i.e., in
        big-endian).
   
      The source a, result q, size n, initial carry i, and return value       The output can be read with `mpz_inp_raw'.
      c satisfy c*b^n + a-i = 3*q, where b is the size of a limb (2^32  
      or 2^64).  c is always 0, 1 or 2, and the initial carry must also  
      be 0, 1 or 2 (these are both borrows really).  When c=0, clearly  
      q=(a-i)/3.  When c!=0, the remainder (a-i) mod 3 is given by 3-c,  
      because b == 1 mod 3.  
   
  - Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1SIZE,       Return the number of bytes written, or if an error occurred,
           mp_limb_t S2LIMB)       return 0.
      Divide {S1P, S1SIZE} by S2LIMB, and return the remainder.  S1SIZE  
      can be zero.  
   
  - Function: mp_limb_t mpn_preinv_mod_1 (mp_limb_t *S1P, mp_size_t       The output of this can not be read by `mpz_inp_raw' from GMP 1,
           S1SIZE, mp_limb_t S2LIMB, mp_limb_t S3LIMB)       because of changes necessary for compatibility between 32-bit and
      *This interface is obsolete.  It will disappear from future       64-bit machines.
      releases.  Use `mpn_mod_1' in its stead.*  
   
  - Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,   - Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
           mp_size_t S1SIZE, const mp_limb_t *S2P, mp_size_t S2SIZE,       Input from stdio stream STREAM in the format written by
           unsigned long int D)       `mpz_out_raw', and put the result in ROP.  Return the number of
      The function puts the low [D/BITS_PER_MP_LIMB] limbs of Q = {S1P,       bytes read, or if an error occurred, return 0.
      S1SIZE}/{S2P, S2SIZE} mod 2^D at RP, and returns the high D mod  
      BITS_PER_MP_LIMB bits of Q.  
   
      {S1P, S1SIZE} - Q * {S2P, S2SIZE} mod 2^(S1SIZE*BITS_PER_MP_LIMB)       This routine can read the output from `mpz_out_raw' also from GMP
      is placed at S1P.  Since the low [D/BITS_PER_MP_LIMB] limbs of       1, in spite of changes necessary for compatibility between 32-bit
      this difference are zero, it is possible to overwrite the low       and 64-bit machines.
      limbs at S1P with this difference, provided RP <= S1P.  
   
      This function requires that S1SIZE * BITS_PER_MP_LIMB >= D, and  
      that {S2P, S2SIZE} is odd.  File: gmp.info,  Node: Integer Random Numbers,  Next: Integer Import and Export,  Prev: I/O of Integers,  Up: Integer Functions
   
      *This interface is preliminary.  It might change incompatibly in  Random Number Functions
      future revisions.*  =======================
   
  - Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t     The random number functions of GMP come in two groups; older function
           *SRC_PTR, mp_size_t SRC_SIZE, unsigned long int COUNT)  that rely on a global state, and newer functions that accept a state
      Shift {SRC_PTR, SRC_SIZE} COUNT bits to the left, and write the  parameter that is read and modified.  Please see the *Note Random
      SRC_SIZE least significant limbs of the result to RP.  COUNT might  Number Functions:: for more information on how to use and not to use
      be in the range 1 to n - 1, on an n-bit machine. The bits shifted  random number functions.
      out to the left are returned.  
   
      Overlapping of the destination space and the source space is   - Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
      allowed in this function, provided RP >= SRC_PTR.            unsigned long int N)
        Generate a uniformly distributed random integer in the range 0 to
        2^N-1, inclusive.
   
      This function is written in assembly for most targets.       The variable STATE must be initialized by calling one of the
        `gmp_randinit' functions (*Note Random State Initialization::)
        before invoking this function.
   
  - Function: mp_limp_t mpn_rshift (mp_limb_t *RP, const mp_limb_t   - Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE, mpz_t
           *SRC_PTR, mp_size_t SRC_SIZE, unsigned long int COUNT)            N)
      Shift {SRC_PTR, SRC_SIZE} COUNT bits to the right, and write the       Generate a uniform random integer in the range 0 to N-1, inclusive.
      SRC_SIZE most significant limbs of the result to RP.  COUNT might  
      be in the range 1 to n - 1, on an n-bit machine.  The bits shifted  
      out to the right are returned.  
   
      Overlapping of the destination space and the source space is       The variable STATE must be initialized by calling one of the
      allowed in this function, provided RP <= SRC_PTR.       `gmp_randinit' functions (*Note Random State Initialization::)
        before invoking this function.
   
      This function is written in assembly for most targets.   - Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
             unsigned long int N)
        Generate a random integer with long strings of zeros and ones in
        the binary representation.  Useful for testing functions and
        algorithms, since this kind of random numbers have proven to be
        more likely to trigger corner-case bugs.  The random number will
        be in the range 0 to 2^N-1, inclusive.
   
  - Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,       The variable STATE must be initialized by calling one of the
           mp_size_t SIZE)       `gmp_randinit' functions (*Note Random State Initialization::)
      Compare {S1P, SIZE} and {S2P, SIZE} and return a positive value if       before invoking this function.
      s1 > src2, 0 of they are equal, and a negative value if s1 < src2.  
   
  - Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,   - Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
           mp_size_t S1SIZE, mp_limb_t *S2P, mp_size_t S2SIZE)       Generate a random integer of at most MAX_SIZE limbs.  The generated
      Puts at RP the greatest common divisor of {S1P, S1SIZE} and {S2P,       random number doesn't satisfy any particular requirements of
      S2SIZE}; both source operands are destroyed by the operation.  The       randomness.  Negative random numbers are generated when MAX_SIZE
      size in limbs of the greatest common divisor is returned.       is negative.
   
      {S1P, S1SIZE} must have at least as many bits as {S2P, S2SIZE},       This function is obsolete.  Use `mpz_urandomb' or `mpz_urandomm'
      and {S2P, S2SIZE} must be odd.       instead.
   
  - Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t   - Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
           S1SIZE, mp_limb_t S2LIMB)       Generate a random integer of at most MAX_SIZE limbs, with long
      Return the greatest common divisor of {S1P, S1SIZE} and S2LIMB,       strings of zeros and ones in the binary representation.  Useful
      where S2LIMB (as well as S1SIZE) must be different from 0.       for testing functions and algorithms, since this kind of random
        numbers have proven to be more likely to trigger corner-case bugs.
        Negative random numbers are generated when MAX_SIZE is negative.
   
  - Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P,       This function is obsolete.  Use `mpz_rrandomb' instead.
           mp_size_t *R2SIZE, mp_limb_t *S1P, mp_size_t S1SIZE,  
           mp_limb_t *S2P, mp_size_t S2SIZE)  
      Compute the greatest common divisor of {S1P, S1SIZE} and {S2P,  
      S2SIZE}.  Store the gcd at R1P and return its size in limbs.  
      Write the first cofactor at R2P and store its size in *R2SIZE.  If  
      the cofactor is negative, *R2SIZE is negative and R2P is the  
      absolute value of the cofactor.  
   
      {S1P, S1SIZE} must be greater than or equal to {S2P, S2SIZE}.  
      Neither operand may equal 0.  Both source operands are destroyed,  File: gmp.info,  Node: Integer Import and Export,  Next: Miscellaneous Integer Functions,  Prev: Integer Random Numbers,  Up: Integer Functions
      plus one limb past the end of each, ie. {S1P, S1SIZE+1} and {S2P,  
      S2SIZE+1}.  
   
  - Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,  Integer Import and Export
           const mp_limb_t *SP, mp_size_t SIZE)  =========================
      Compute the square root of {SP, SIZE} and put the result at R1P.  
      Write the remainder at R2P, unless R2P is `NULL'.  
   
      Return the size of the remainder, whether R2P was `NULL' or     `mpz_t' variables can be converted to and from arbitrary words of
      non-`NULL'.  Iff the operand was a perfect square, the return  binary data with the following functions.
      value will be 0.  
   
      The areas at R1P and SP have to be distinct.  The areas at R2P and   - Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER, int
      SP have to be identical or completely separate, not partially            SIZE, int ENDIAN, size_t NAILS, const void *OP)
      overlapping.       Set ROP from an array of word data at OP.
   
      The area at R1P needs to have space for ceil(SIZE/2) limbs.  The       The parameters specify the format of the data.  COUNT many words
      area at R2P needs to be SIZE limbs large.       are read, each SIZE bytes.  ORDER can be 1 for most significant
        word first or -1 for least significant first.  Within each word
        ENDIAN can be 1 for most significant byte first, -1 for least
        significant first, or 0 for the native endianness of the host CPU.
        The most significant NAILS bits of each word are skipped, this
        can be 0 to use the full words.
   
  - Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,       There are no data alignment restrictions on OP, any address is
           mp_limb_t *S1P, mp_size_t S1SIZE)       allowed.
      Convert {S1P, S1SIZE} to a raw unsigned char array in base BASE.  
      The string is not in ASCII; to convert it to printable format, add  
      the ASCII codes for `0' or `A', depending on the base and range.  
      There may be leading zeros in the string.  
   
      The area at S1P is clobbered.       Here's an example converting an array of `unsigned long' data, most
        significant element first and host byte order within each value.
   
      Return the number of characters in STR.            unsigned long  a[20];
             mpz_t          z;
             mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
   
      The area at STR has to have space for the largest possible number       This example assumes the full `sizeof' bytes are used for data in
      represented by a S1SIZE long limb array, plus one extra character.       the given type, which is usually true, and certainly true for
        `unsigned long' everywhere we know of.  However on Cray vector
        systems it may be noted that `short' and `int' are always stored
        in 8 bytes (and with `sizeof' indicating that) but use only 32 or
        46 bits.  The NAILS feature can account for this, by passing for
        instance `8*sizeof(int)-INT_BIT'.
   
  - Function: mp_size_t mpn_set_str (mp_limb_t *R1P, const char *STR,   - Function: void *mpz_export (void *ROP, size_t *COUNT, int ORDER, int
           size_t STRSIZE, int BASE)            SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
      Convert the raw unsigned char array at STR of length STRSIZE to a       Fill ROP with word data from OP.
      limb array {S1P, S1SIZE}.  The base of STR is BASE.  
   
      Return the number of limbs stored in R1P.       The parameters specify the format of the data produced.  Each word
        will be SIZE bytes and ORDER can be 1 for most significant word
        first or -1 for least significant first.  Within each word ENDIAN
        can be 1 for most significant byte first, -1 for least significant
        first, or 0 for the native endianness of the host CPU.  The most
        significant NAILS bits of each word are unused and set to zero,
        this can be 0 to produce full words.
   
  - Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P,       The number of words produced is written to `*COUNT'.  ROP must
           unsigned long int BIT)       have enough space for the data, or if ROP is `NULL' then a result
      Scan S1P from bit position BIT for the next clear bit.       array of the necessary size is allocated using the current GMP
        allocation function (*note Custom Allocation::).  In either case
        the return value is the destination used, ROP or the allocated
        block.
   
      It is required that there be a clear bit within the area at S1P at       If OP is non-zero then the most significant word produced will be
      or beyond bit position BIT, so that the function has something to       non-zero.  If OP is zero then the count returned will be zero and
      return.       nothing written to ROP.  If ROP is `NULL' in this case, no block
        is allocated, just `NULL' is returned.
   
  - Function: unsigned long int mpn_scan1 (const mp_limb_t *S1P,       There are no data alignment restrictions on ROP, any address is
           unsigned long int BIT)       allowed.  The sign of OP is ignored, just the absolute value is
      Scan S1P from bit position BIT for the next set bit.       used.
   
      It is required that there be a set bit within the area at S1P at or       When an application is allocating space itself the required size
      beyond bit position BIT, so that the function has something to       can be determined with a calculation like the following.  Since
      return.       `mpz_sizeinbase' always returns at least 1, `count' here will be
        at least one, which avoids any portability problems with
        `malloc(0)', though if `z' is zero no space at all is actually
        needed.
   
  - Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1SIZE)            numb = 8*size - nail;
  - Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1SIZE)            count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
      Generate a random number of length R1SIZE and store it at R1P.            p = malloc (count * size);
      The most significant limb is always non-zero.  `mpn_random'  
      generates uniformly distributed limb data, `mpn_random2' generates  
      long strings of zeros and ones in the binary representation.  
   
      `mpn_random2' is intended for testing the correctness of the `mpn'  
      routines.  File: gmp.info,  Node: Miscellaneous Integer Functions,  Prev: Integer Import and Export,  Up: Integer Functions
   
  - Function: unsigned long int mpn_popcount (const mp_limb_t *S1P,  Miscellaneous Functions
           unsigned long int SIZE)  =======================
      Count the number of set bits in {S1P, SIZE}.  
   
  - Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P, const   - Function: int mpz_fits_ulong_p (mpz_t OP)
           mp_limb_t *S2P, unsigned long int SIZE)   - Function: int mpz_fits_slong_p (mpz_t OP)
      Compute the hamming distance between {S1P, SIZE} and {S2P, SIZE}.   - Function: int mpz_fits_uint_p (mpz_t OP)
    - Function: int mpz_fits_sint_p (mpz_t OP)
    - Function: int mpz_fits_ushort_p (mpz_t OP)
    - Function: int mpz_fits_sshort_p (mpz_t OP)
        Return non-zero iff the value of OP fits in an `unsigned long int',
        `signed long int', `unsigned int', `signed int', `unsigned short
        int', or `signed short int', respectively.  Otherwise, return zero.
   
  - Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t   - Macro: int mpz_odd_p (mpz_t OP)
           SIZE)   - Macro: int mpz_even_p (mpz_t OP)
      Return non-zero iff {S1P, SIZE} is a perfect square.       Determine whether OP is odd or even, respectively.  Return
        non-zero if yes, zero if no.  These macros evaluate their argument
        more than once.
   
    - Function: size_t mpz_size (mpz_t OP)
 File: gmp.info,  Node: Random Number Functions,  Next: BSD Compatible Functions,  Prev: Low-level Functions,  Up: Top       Return the size of OP measured in number of limbs.  If OP is zero,
        the returned value will be zero.
   
 Random Number Functions   - Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
 ***********************       Return the size of OP measured in number of digits in base BASE.
        The base may vary from 2 to 36.  The sign of OP is ignored, just
        the absolute value is used.  The result will be exact or 1 too
        big.  If BASE is a power of 2, the result will always be exact.
        If OP is zero the return value is always 1.
   
    There are two groups of random number functions in GNU MP; older       This function is useful in order to allocate the right amount of
 functions that call C library random number generators, rely on a global       space before converting OP to a string.  The right amount of
 state, and aren't very random; and newer functions that don't have these       allocation is normally two more than the value returned by
 problems.  The newer functions are self-contained, they accept a random       `mpz_sizeinbase' (one extra for a minus sign and one for the
 state parameter that supplants global state, and generate good random       null-terminator).
 numbers.  
   
    The random state parameter is of the type `gmp_randstate_t'.  It  
 must be initialized by a call to one of the `gmp_randinit' functions  File: gmp.info,  Node: Rational Number Functions,  Next: Floating-point Functions,  Prev: Integer Functions,  Up: Top
 (*Note Random State Initialization::).  The initial seed is set using  
 one of the `gmp_randseed' functions (*Note Random State  
 Initialization::).  
   
    The size of the seed determines the number of different sequences of  Rational Number Functions
 random numbers that is possible to generate.  The "quality" of the seed  *************************
 is the randomness of a given seed compared to the previous seed used  
 and affects the randomness of separate number sequences.  
   
    The algorithm for assigning seed is critical if the generated random     This chapter describes the GMP functions for performing arithmetic
 numbers are to be used for important applications, such as generating  on rational numbers.  These functions start with the prefix `mpq_'.
 cryptographic keys.  
   
    The traditional method is to use the current system time for     Rational numbers are stored in objects of type `mpq_t'.
 seeding.  One has to be careful when using the current time though.  If  
 the application seeds the random functions very often, say several  
 times per second, and the resolution of the system clock is  
 comparatively low, like one second, the same sequence of numbers will  
 be generated until the system clock ticks.  Furthermore, the current  
 system time is quite easy to guess, so a system depending on any  
 unpredictability of the random number sequence should absolutely not  
 use that as its only source for a seed value.  
   
    On some systems there is a special device, often called     All rational arithmetic functions assume operands have a canonical
 `/dev/random', which provides a source of somewhat random numbers more  form, and canonicalize their result.  The canonical from means that the
 usable as seed.  denominator and the numerator have no common factors, and that the
   denominator is positive.  Zero has the unique representation 0/1.
   
    The functions actually generating random functions are documented     Pure assignment functions do not canonicalize the assigned variable.
 under "Miscellaneous Functions" in their respective function class:  It is the responsibility of the user to canonicalize the assigned
 *Note Miscellaneous Integer Functions::, *Note Miscellaneous Float  variable before any arithmetic operations are performed on that
 Functions::.  variable.
   
    - Function: void mpq_canonicalize (mpq_t OP)
        Remove any factors that are common to the numerator and
        denominator of OP, and make the denominator positive.
   
 * Menu:  * Menu:
   
 * Random State Initialization::  How to initialize a random state.  * Initializing Rationals::
   * Rational Conversions::
   * Rational Arithmetic::
   * Comparing Rationals::
   * Applying Integer Functions::
   * I/O of Rationals::
   
   
 File: gmp.info,  Node: Random State Initialization,  Prev: Random Number Functions,  Up: Random Number Functions  File: gmp.info,  Node: Initializing Rationals,  Next: Rational Conversions,  Prev: Rational Number Functions,  Up: Rational Number Functions
   
 Random State Initialization  Initialization and Assignment Functions
 ===========================  =======================================
   
    See *Note Random Number Functions:: for a discussion on how to   - Function: void mpq_init (mpq_t DEST_RATIONAL)
 choose the initial seed value passed to these functions.       Initialize DEST_RATIONAL and set it to 0/1.  Each variable should
        normally only be initialized once, or at least cleared out (using
        the function `mpq_clear') between each initialization.
   
  - Function: void gmp_randinit (gmp_randstate_t STATE, gmp_randalg_t   - Function: void mpq_clear (mpq_t RATIONAL_NUMBER)
           ALG, ...)       Free the space occupied by RATIONAL_NUMBER.  Make sure to call this
      Initialize random state variable STATE.       function for all `mpq_t' variables when you are done with them.
   
      ALG denotes what algorithm to use for random number generation.   - Function: void mpq_set (mpq_t ROP, mpq_t OP)
      Use one of   - Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
         - GMP_RAND_ALG_LC -- Linear congruential.       Assign ROP from OP.
   
           A fast generator defined by X = (aX + c) mod m.   - Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
             unsigned long int OP2)
    - Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
             long int OP2)
        Set the value of ROP to OP1/OP2.  Note that if OP1 and OP2 have
        common factors, ROP has to be passed to `mpq_canonicalize' before
        any operations are performed on ROP.
   
           A third argument SIZE of type unsigned long int is required.   - Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE)
           SIZE is the size of the largest good quality random number to       Set ROP from a null-terminated string STR in the given BASE.
           be generated, expressed in number of bits.  If the random  
           generation functions are asked for a bigger random number  
           than indicated by this parameter, two or more numbers of SIZE  
           bits will be generated and concatenated, resulting in a "bad"  
           random number.  This can be used to generate big random  
           numbers relatively cheap if the quality of randomness isn't  
           of great importance.  
   
           a, c, and m are picked from a table where the modulus (m) is       The string can be an integer like "41" or a fraction like
           a power of 2 and the multiplier is congruent to 5 (mod 8).       "41/152".  The fraction must be in canonical form (*note Rational
           The choice is based on the SIZE parameter.  The maximum SIZE       Number Functions::), or if not then `mpq_canonicalize' must be
           supported by this algorithm is 128.  If you need bigger       called.
           random numbers, use your own scheme and call one of the other  
           `gmp_randinit' functions.  
   
        The numerator and optional denominator are parsed the same as in
        `mpz_set_str' (*note Assigning Integers::).  White space is
        allowed in the string, and is simply ignored.  The BASE can vary
        from 2 to 36, or if BASE is 0 then the leading characters are
        used: `0x' for hex, `0' for octal, or decimal otherwise.  Note
        that this is done separately for the numerator and denominator, so
        for instance `0xEF/100' is 239/100, whereas `0xEF/0x100' is
        239/256.
   
      If ALG is 0 or GMP_RAND_ALG_DEFAULT, the default algorithm is       The return value is 0 if the entire string is a valid number, or
      used.  The default algorithm is typically a fast algorithm like       -1 if not.
      the linear congruential and requires a third SIZE argument (see  
      GMP_RAND_ALG_LC).  
   
      When you're done with a STATE variable, call `gmp_randclear' to   - Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
      deallocate any memory allocated by this function.       Swap the values ROP1 and ROP2 efficiently.
   
      `gmp_randinit' may set the following bits in GMP_ERRNO:  
         * GMP_ERROR_UNSUPPORTED_ARGUMENT -- ALG is unsupported  File: gmp.info,  Node: Rational Conversions,  Next: Rational Arithmetic,  Prev: Initializing Rationals,  Up: Rational Number Functions
   
         * GMP_ERROR_INVALID_ARGUMENT -- SIZE is too big  Conversion Functions
   ====================
   
  - Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t A,   - Function: double mpq_get_d (mpq_t OP)
      unsigned long int C, unsigned long int M2EXP)       Convert OP to a `double'.
   
      Initialize random state variable STATE with given linear   - Function: void mpq_set_d (mpq_t ROP, double OP)
      congruential scheme.   - Function: void mpq_set_f (mpq_t ROP, mpf_t OP)
        Set ROP to the value of OP, without rounding.
   
      Parameters A, C, and M2EXP are the multiplier, adder, and modulus   - Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP)
      for the linear congruential scheme to use, respectively.  The       Convert OP to a string of digits in base BASE.  The base may vary
      modulus is expressed as a power of 2, so that M = 2^M2EXP.       from 2 to 36.  The string will be of the form `num/den', or if the
        denominator is 1 then just `num'.
   
      The least significant bits of a random number generated by the       If STR is `NULL', the result string is allocated using the current
      linear congruential algorithm where the modulus is a power of two       allocation function (*note Custom Allocation::).  The block will be
      are not very random.  Therefore, the lower half of a random number       `strlen(str)+1' bytes, that being exactly enough for the string and
      generated by an LC scheme initialized with this function is       null-terminator.
      discarded.  Thus, the size of a random number is M2EXP / 2  
      (rounded upwards) bits when this function has been used for  
      initializing the random state.  
   
      When you're done with a STATE variable, call `gmp_randclear' to       If STR is not `NULL', it should point to a block of storage large
      deallocate any memory allocated by this function.       enough for the result, that being
   
  - Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)            mpz_sizeinbase (mpq_numref(OP), BASE)
  - Function: void gmp_randseed_ui (gmp_randstate_t STATE, unsigned long            + mpz_sizeinbase (mpq_denref(OP), BASE) + 3
           int SEED)  
      Set the initial seed value.  
   
      Parameter SEED is the initial random seed.  The function       The three extra bytes are for a possible minus sign, possible
      `gmp_randseed_ui' takes the SEED as an unsigned long int rather       slash, and the null-terminator.
      than as an mpz_t.  
   
  - Function: void gmp_randclear (gmp_randstate_t STATE)       A pointer to the result string is returned, being either the
      Free all memory occupied by STATE.  Make sure to call this       allocated block, or the given STR.
      function for all `gmp_randstate_t' variables when you are done with  
      them.  
   
   
 File: gmp.info,  Node: BSD Compatible Functions,  Next: Custom Allocation,  Prev: Random Number Functions,  Up: Top  File: gmp.info,  Node: Rational Arithmetic,  Next: Comparing Rationals,  Prev: Rational Conversions,  Up: Rational Number Functions
   
 Berkeley MP Compatible Functions  Arithmetic Functions
 ********************************  ====================
   
    These functions are intended to be fully compatible with the   - Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
 Berkeley MP library which is available on many BSD derived U*ix       Set SUM to ADDEND1 + ADDEND2.
 systems.  The `--enable-mpbsd' option must be used when building GNU MP  
 to make these available (*note Installing GMP::).  
   
    The original Berkeley MP library has a usage restriction: you cannot   - Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
 use the same variable as both source and destination in a single            SUBTRAHEND)
 function call.  The compatible functions in GNU MP do not share this       Set DIFFERENCE to MINUEND - SUBTRAHEND.
 restriction--inputs and outputs may overlap.  
   
    It is not recommended that new programs are written using these   - Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
 functions.  Apart from the incomplete set of functions, the interface            MULTIPLICAND)
 for initializing `MINT' objects is more error prone, and the `pow'       Set PRODUCT to MULTIPLIER times MULTIPLICAND.
 function collides with `pow' in `libm.a'.  
   
    Include the header `mp.h' to get the definition of the necessary   - Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, unsigned long int
 types and functions.  If you are on a BSD derived system, make sure to            OP2)
 include GNU `mp.h' if you are going to link the GNU `libmp.a' to your       Set ROP to OP1 times 2 raised to OP2.
 program.  This means that you probably need to give the -I<dir> option  
 to the compiler, where <dir> is the directory where you have GNU `mp.h'.  
   
  - Function: MINT * itom (signed short int INITIAL_VALUE)   - Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
      Allocate an integer consisting of a `MINT' object and dynamic limb            DIVISOR)
      space.  Initialize the integer to INITIAL_VALUE.  Return a pointer       Set QUOTIENT to DIVIDEND/DIVISOR.
      to the `MINT' object.  
   
  - Function: MINT * xtom (char *INITIAL_VALUE)   - Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, unsigned long int
      Allocate an integer consisting of a `MINT' object and dynamic limb            OP2)
      space.  Initialize the integer from INITIAL_VALUE, a hexadecimal,       Set ROP to OP1 divided by 2 raised to OP2.
      '\0'-terminate C string.  Return a pointer to the `MINT' object.  
   
  - Function: void move (MINT *SRC, MINT *DEST)   - Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
      Set DEST to SRC by copying.  Both variables must be previously       Set NEGATED_OPERAND to -OPERAND.
      initialized.  
   
  - Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)   - Function: void mpq_abs (mpq_t ROP, mpq_t OP)
      Add SRC_1 and SRC_2 and put the sum in DESTINATION.       Set ROP to the absolute value of OP.
   
  - Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)   - Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
      Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.       Set INVERTED_NUMBER to 1/NUMBER.  If the new denominator is zero,
        this routine will divide by zero.
   
  - Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)  
      Multiply SRC_1 and SRC_2 and put the product in DESTINATION.  File: gmp.info,  Node: Comparing Rationals,  Next: Applying Integer Functions,  Prev: Rational Arithmetic,  Up: Rational Number Functions
   
  - Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,  Comparison Functions
           MINT *REMAINDER)  ====================
  - Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT  
           *QUOTIENT, signed short int *REMAINDER)  
      Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod  
      DIVISOR.  The quotient is rounded towards zero; the remainder has  
      the same sign as the dividend unless it is zero.  
   
      Some implementations of these functions work differently--or not   - Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
      at all--for negative arguments.       Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero
        if OP1 = OP2, and a negative value if OP1 < OP2.
   
  - Function: void msqrt (MINT *OPERAND, MINT *ROOT, MINT *REMAINDER)       To determine if two rationals are equal, `mpq_equal' is faster than
      Set ROOT to the truncated integer part of the square root of       `mpq_cmp'.
      OPERAND.  Set REMAINDER to OPERAND-ROOT*ROOT, (i.e., zero if  
      OPERAND is a perfect square).  
   
      If ROOT and REMAINDER are the same variable, the results are   - Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
      undefined.            long int DEN2)
    - Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int
             DEN2)
        Compare OP1 and NUM2/DEN2.  Return a positive value if OP1 >
        NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
        NUM2/DEN2.
   
  - Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)       NUM2 and DEN2 are allowed to have common factors.
      Set DEST to (BASE raised to EXP) modulo MOD.  
   
  - Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)       These functions are implemented as a macros and evaluate their
      Set DEST to BASE raised to EXP.       arguments multiple times.
   
  - Function: void gcd (MINT *OPERAND1, MINT *OPERAND2, MINT *RES)   - Macro: int mpq_sgn (mpq_t OP)
      Set RES to the greatest common divisor of OPERAND1 and OPERAND2.       Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
   
  - Function: int mcmp (MINT *OPERAND1, MINT *OPERAND2)       This function is actually implemented as a macro.  It evaluates its
      Compare OPERAND1 and OPERAND2.  Return a positive value if       arguments multiple times.
      OPERAND1 > OPERAND2, zero if OPERAND1 = OPERAND2, and a negative  
      value if OPERAND1 < OPERAND2.  
   
  - Function: void min (MINT *DEST)   - Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
      Input a decimal string from `stdin', and put the read integer in       Return non-zero if OP1 and OP2 are equal, zero if they are
      DEST.  SPC and TAB are allowed in the number string, and are       non-equal.  Although `mpq_cmp' can be used for the same purpose,
      ignored.       this function is much faster.
   
  - Function: void mout (MINT *SRC)  
      Output SRC to `stdout', as a decimal string.  Also output a  File: gmp.info,  Node: Applying Integer Functions,  Next: I/O of Rationals,  Prev: Comparing Rationals,  Up: Rational Number Functions
      newline.  
   
  - Function: char * mtox (MINT *OPERAND)  Applying Integer Functions to Rationals
      Convert OPERAND to a hexadecimal string, and return a pointer to  =======================================
      the string.  The returned string is allocated using the default  
      memory allocation function, `malloc' by default.  
   
  - Function: void mfree (MINT *OPERAND)     The set of `mpq' functions is quite small.  In particular, there are
      De-allocate, the space used by OPERAND.  *This function should  few functions for either input or output.  The following functions give
      only be passed a value returned by `itom' or `xtom'.*  direct access to the numerator and denominator of an `mpq_t'.
   
      Note that if an assignment to the numerator and/or denominator could
   take an `mpq_t' out of the canonical form described at the start of
   this chapter (*note Rational Number Functions::) then
   `mpq_canonicalize' must be called before any other `mpq' functions are
   applied to that `mpq_t'.
   
    - Macro: mpz_t mpq_numref (mpq_t OP)
    - Macro: mpz_t mpq_denref (mpq_t OP)
        Return a reference to the numerator and denominator of OP,
        respectively.  The `mpz' functions can be used on the result of
        these macros.
   
    - Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
    - Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
    - Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
    - Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
        Get or set the numerator or denominator of a rational.  These
        functions are equivalent to calling `mpz_set' with an appropriate
        `mpq_numref' or `mpq_denref'.  Direct use of `mpq_numref' or
        `mpq_denref' is recommended instead of these functions.
   
   
 File: gmp.info,  Node: Custom Allocation,  Next: Contributors,  Prev: BSD Compatible Functions,  Up: Top  File: gmp.info,  Node: I/O of Rationals,  Prev: Applying Integer Functions,  Up: Rational Number Functions
   
 Custom Allocation  Input and Output Functions
 *****************  ==========================
   
    By default, GMP uses `malloc', `realloc' and `free' for memory     When using any of these functions, it's a good idea to include
 allocation.  If `malloc' or `realloc' fails, GMP prints a message to  `stdio.h' before `gmp.h', since that will allow `gmp.h' to define
 the standard error output and terminates execution.  prototypes for these functions.
   
    Some applications might want to allocate memory in other ways, or     Passing a `NULL' pointer for a STREAM argument to any of these
 might not want a fatal error when there is no more memory available.  functions will make them read from `stdin' and write to `stdout',
 To accomplish this, you can specify alternative memory allocation  respectively.
 functions.  
   
    This can be done in the Berkeley compatibility library as well as   - Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
 the main GMP library.       Output OP on stdio stream STREAM, as a string of digits in base
        BASE.  The base may vary from 2 to 36.  Output is in the form
        `num/den' or if the denominator is 1 then just `num'.
   
  - Function: void mp_set_memory_functions (       Return the number of bytes written, or if an error occurred,
           void *(*ALLOC_FUNC_PTR) (size_t),       return 0.
           void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),  
           void (*FREE_FUNC_PTR) (void *, size_t))  
      Replace the current allocation functions from the arguments.  If  
      an argument is `NULL', the corresponding default function is  
      retained.  
   
      *Be sure to call this function only when there are no active GMP   - Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
      objects allocated using the previous memory functions!  Usually,       Read a string of digits from STREAM and convert them to a rational
      that means that you have to call this function before any other       in ROP.  Any initial white-space characters are read and
      GMP function.*       discarded.  Return the number of characters read (including white
        space), or 0 if a rational could not be read.
   
    The functions you supply should fit the following declarations:       The input can be a fraction like `17/63' or just an integer like
        `123'.  Reading stops at the first character not in this form, and
        white space is not permitted within the string.  If the input
        might not be in canonical form, then `mpq_canonicalize' must be
        called (*note Rational Number Functions::).
   
  - Function: void * allocate_function (size_t ALLOC_SIZE)       The BASE can be between 2 and 36, or can be 0 in which case the
      This function should return a pointer to newly allocated space       leading characters of the string determine the base, `0x' or `0X'
      with at least ALLOC_SIZE storage units.       for hexadecimal, `0' for octal, or decimal otherwise.  The leading
        characters are examined separately for the numerator and
        denominator of a fraction, so for instance `0x10/11' is 16/11,
        whereas `0x10/0x11' is 16/17.
   
  - Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,  
           size_t NEW_SIZE)  File: gmp.info,  Node: Floating-point Functions,  Next: Low-level Functions,  Prev: Rational Number Functions,  Up: Top
      This function should return a pointer to newly allocated space of  
      at least NEW_SIZE storage units, after copying at least the first  
      OLD_SIZE storage units from PTR.  It should also de-allocate the  
      space at PTR.  
   
      You can assume that the space at PTR was formerly returned from  Floating-point Functions
      `allocate_function' or `reallocate_function', for a request for  ************************
      OLD_SIZE storage units.  
   
  - Function: void deallocate_function (void *PTR, size_t SIZE)     GMP floating point numbers are stored in objects of type `mpf_t' and
      De-allocate the space pointed to by PTR.  functions operating on them have an `mpf_' prefix.
   
      You can assume that the space at PTR was formerly returned from     The mantissa of each float has a user-selectable precision, limited
      `allocate_function' or `reallocate_function', for a request for  only by available memory.  Each variable has its own precision, and
      SIZE storage units.  that can be increased or decreased at any time.
   
    (A "storage unit" is the unit in which the `sizeof' operator returns     The exponent of each float is a fixed precision, one machine word on
 the size of an object, normally an 8 bit byte.)  most systems.  In the current implementation the exponent is a count of
   limbs, so for example on a 32-bit system this means a range of roughly
   2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
   greater.  Note however `mpf_get_str' can only return an exponent which
   fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
   bigger than a `long'.
   
      Each variable keeps a size for the mantissa data actually in use.
   This means that if a float is exactly represented in only a few bits
   then only those bits will be used in a calculation, even if the
   selected precision is high.
   
      All calculations are performed to the precision of the destination
   variable.  Each function is defined to calculate with "infinite
   precision" followed by a truncation to the destination precision, but
   of course the work done is only what's needed to determine a result
   under that definition.
   
      The precision selected for a variable is a minimum value, GMP may
   increase it a little to facilitate efficient calculation.  Currently
   this means rounding up to a whole limb, and then sometimes having a
   further partial limb, depending on the high limb of the mantissa.  But
   applications shouldn't be concerned by such details.
   
      The mantissa in stored in binary, as might be imagined from the fact
   precisions are expressed in bits.  One consequence of this is that
   decimal fractions like 0.1 cannot be represented exactly.  The same is
   true of plain IEEE `double' floats.  This makes both highly unsuitable
   for calculations involving money or other values that should be exact
   decimal fractions.  (Suitably scaled integers, or perhaps rationals,
   are better choices.)
   
      `mpf' functions and variables have no special notion of infinity or
   not-a-number, and applications must take care not to overflow the
   exponent or results will be unpredictable.  This might change in a
   future release.
   
      Note that the `mpf' functions are _not_ intended as a smooth
   extension to IEEE P754 arithmetic.  In particular results obtained on
   one computer often differ from the results on a computer with a
   different word size.
   
   * Menu:
   
   * Initializing Floats::
   * Assigning Floats::
   * Simultaneous Float Init & Assign::
   * Converting Floats::
   * Float Arithmetic::
   * Float Comparison::
   * I/O of Floats::
   * Miscellaneous Float Functions::
   
   
 File: gmp.info,  Node: Contributors,  Next: References,  Prev: Custom Allocation,  Up: Top  File: gmp.info,  Node: Initializing Floats,  Next: Assigning Floats,  Prev: Floating-point Functions,  Up: Floating-point Functions
   
 Contributors  Initialization Functions
 ************  ========================
   
    Torbjorn Granlund wrote the original GMP library and is still   - Function: void mpf_set_default_prec (unsigned long int PREC)
 developing and maintaining it.  Several other individuals and       Set the default precision to be *at least* PREC bits.  All
 organizations have contributed to GMP in various ways.  Here is a list       subsequent calls to `mpf_init' will use this precision, but
 in chronological order:       previously initialized variables are unaffected.
   
    Gunnar Sjoedin and Hans Riesel helped with mathematical problems in   - Function: unsigned long int mpf_get_default_prec (void)
 early versions of the library.       Return the default default precision actually used.
   
    Richard Stallman contributed to the interface design and revised the     An `mpf_t' object must be initialized before storing the first value
 first version of this manual.  in it.  The functions `mpf_init' and `mpf_init2' are used for that
   purpose.
   
    Brian Beuning and Doug Lea helped with testing of early versions of   - Function: void mpf_init (mpf_t X)
 the library and made creative suggestions.       Initialize X to 0.  Normally, a variable should be initialized
        once only or at least be cleared, using `mpf_clear', between
        initializations.  The precision of X is undefined unless a default
        precision has already been established by a call to
        `mpf_set_default_prec'.
   
    John Amanatides of York University in Canada contributed the function   - Function: void mpf_init2 (mpf_t X, unsigned long int PREC)
 `mpz_probab_prime_p'.       Initialize X to 0 and set its precision to be *at least* PREC
        bits.  Normally, a variable should be initialized once only or at
        least be cleared, using `mpf_clear', between initializations.
   
    Paul Zimmermann of Inria sparked the development of GMP 2, with his   - Function: void mpf_clear (mpf_t X)
 comparisons between bignum packages.       Free the space occupied by X.  Make sure to call this function for
        all `mpf_t' variables when you are done with them.
   
    Ken Weber (Kent State University, Universidade Federal do Rio Grande     Here is an example on how to initialize floating-point variables:
 do Sul) contributed `mpz_gcd', `mpz_divexact', `mpn_gcd', and       {
 `mpn_bdivmod', partially supported by CNPq (Brazil) grant 301314194-2.         mpf_t x, y;
          mpf_init (x);           /* use default precision */
          mpf_init2 (y, 256);     /* precision _at least_ 256 bits */
          ...
          /* Unless the program is about to exit, do ... */
          mpf_clear (x);
          mpf_clear (y);
        }
   
    Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'     The following three functions are useful for changing the precision
 configure.  He has also made valuable suggestions and tested numerous  during a calculation.  A typical use would be for adjusting the
 intermediary releases.  precision gradually in iterative algorithms like Newton-Raphson, making
   the computation precision closely match the actual accurate part of the
   numbers.
   
    Joachim Hollman was involved in the design of the `mpf' interface,   - Function: unsigned long int mpf_get_prec (mpf_t OP)
 and in the `mpz' design revisions for version 2.       Return the current precision of OP, in bits.
   
    Bennet Yee contributed the functions `mpz_jacobi' and `mpz_legendre'.   - Function: void mpf_set_prec (mpf_t ROP, unsigned long int PREC)
        Set the precision of ROP to be *at least* PREC bits.  The value in
        ROP will be truncated to the new precision.
   
    Andreas Schwab contributed the files `mpn/m68k/lshift.S' and       This function requires a call to `realloc', and so should not be
 `mpn/m68k/rshift.S'.       used in a tight loop.
   
    The development of floating point functions of GNU MP 2, were   - Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC)
 supported in part by the ESPRIT-BRA (Basic Research Activities) 6846       Set the precision of ROP to be *at least* PREC bits, without
 project POSSO (POlynomial System SOlving).       changing the memory allocated.
   
    GNU MP 2 was finished and released by SWOX AB (formerly known as TMG       PREC must be no more than the allocated precision for ROP, that
 Datakonsult), Swedenborgsgatan 23, SE-118 27 STOCKHOLM, SWEDEN, in       being the precision when ROP was initialized, or in the most recent
 cooperation with the IDA Center for Computing Sciences, USA.       `mpf_set_prec'.
   
    Robert Harley of Inria, France and David Seal of ARM, England,       The value in ROP is unchanged, and in particular if it had a higher
 suggested clever improvements for population count.       precision than PREC it will retain that higher precision.  New
        values written to ROP will use the new PREC.
   
    Robert Harley also wrote highly optimized Karatsuba and 3-way Toom       Before calling `mpf_clear' or the full `mpf_set_prec', another
 multiplication functions for GMP 3.  He also contributed the ARM       `mpf_set_prec_raw' call must be made to restore ROP to its original
 assembly code.       allocated precision.  Failing to do so will have unpredictable
        results.
   
    Torsten Ekedahl of the Mathematical department of Stockholm       `mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
 University provided significant inspiration during several phases of       original allocated precision.  After `mpf_set_prec_raw' it
 the GMP development.  His mathematical expertise helped improve several       reflects the PREC value set.
 algorithms.  
   
    Paul Zimmermann wrote the Burnikel-Ziegler division code, the REDC       `mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
 code, the REDC-based mpz_powm code, and the FFT multiply code.  The       at different precisions during a calculation, perhaps to gradually
 ECMNET project Paul is organizing has been a driving force behind many       increase precision in an iteration, or just to use various
 of the optimization of GMP 3.       different precisions for different purposes during a calculation.
   
    Linus Nordberg wrote the new configure system based on autoconf and  
 implemented the new random functions.  File: gmp.info,  Node: Assigning Floats,  Next: Simultaneous Float Init & Assign,  Prev: Initializing Floats,  Up: Floating-point Functions
   
    Kent Boortz made the Macintosh port.  Assignment Functions
   ====================
   
    Kevin Ryde wrote a lot of very high quality x86 code, optimized for     These functions assign new values to already initialized floats
 most CPU variants.  He also made countless other valuable contributions.  (*note Initializing Floats::).
   
    Steve Root helped write the optimized alpha 21264 assembly code.   - Function: void mpf_set (mpf_t ROP, mpf_t OP)
    - Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
    - Function: void mpf_set_si (mpf_t ROP, signed long int OP)
    - Function: void mpf_set_d (mpf_t ROP, double OP)
    - Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
    - Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
        Set the value of ROP from OP.
   
    GNU MP 3.1 was finished and released by Torbjorn Granlund and Kevin   - Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
 Ryde.  Torbjorn's work was partially funded by the IDA Center for       Set the value of ROP from the string in STR.  The string is of the
 Computing Sciences, USA.       form `M@N' or, if the base is 10 or less, alternatively `MeN'.
        `M' is the mantissa and `N' is the exponent.  The mantissa is
        always in the specified base.  The exponent is either in the
        specified base or, if BASE is negative, in decimal.  The decimal
        point expected is taken from the current locale, on systems
        providing `localeconv'.
   
    (This list is chronological, not ordered after significance.  If you       The argument BASE may be in the ranges 2 to 36, or -36 to -2.
 have contributed to GMP but are not listed above, please tell       Negative values are used to specify that the exponent is in
 <tege@swox.com> about the omission!)       decimal.
   
        Unlike the corresponding `mpz' function, the base will not be
        determined from the leading characters of the string if BASE is 0.
        This is so that numbers like `0.23' are not interpreted as octal.
   
        White space is allowed in the string, and is simply ignored.
        [This is not really true; white-space is ignored in the beginning
        of the string and within the mantissa, but not in other places,
        such as after a minus sign or in the exponent.  We are considering
        changing the definition of this function, making it fail when
        there is any white-space in the input, since that makes a lot of
        sense.  Please tell us your opinion about this change.  Do you
        really want it to accept "3 14" as meaning 314 as it does now?]
   
        This function returns 0 if the entire string is a valid number in
        base BASE.  Otherwise it returns -1.
   
    - Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
        Swap ROP1 and ROP2 efficiently.  Both the values and the
        precisions of the two variables are swapped.
   
   
 File: gmp.info,  Node: References,  Next: Concept Index,  Prev: Contributors,  Up: Top  File: gmp.info,  Node: Simultaneous Float Init & Assign,  Next: Converting Floats,  Prev: Assigning Floats,  Up: Floating-point Functions
   
 References  Combined Initialization and Assignment Functions
 **********  ================================================
   
    * Donald E. Knuth, "The Art of Computer Programming", vol 2,     For convenience, GMP provides a parallel series of
      "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1988.  initialize-and-set functions which initialize the output and then store
   the value there.  These functions' names have the form `mpf_init_set...'
   
    * John D. Lipson, "Elements of Algebra and Algebraic Computing", The     Once the float has been initialized by any of the `mpf_init_set...'
      Benjamin Cummings Publishing Company Inc, 1981.  functions, it can be used as the source or destination operand for the
   ordinary float functions.  Don't use an initialize-and-set function on
   a variable already initialized!
   
    * Richard M. Stallman, "Using and Porting GCC", Free Software   - Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
      Foundation, 1999, available online   - Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
      `http://www.gnu.org/software/gcc/onlinedocs/', and in the GCC   - Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
      package `ftp://ftp.gnu.org/pub/gnu/gcc/'.   - Function: void mpf_init_set_d (mpf_t ROP, double OP)
        Initialize ROP and set its value from OP.
   
    * Peter L. Montgomery, "Modular Multiplication Without Trial       The precision of ROP will be taken from the active default
      Division", in Mathematics of Computation, volume 44, number 170,       precision, as set by `mpf_set_default_prec'.
      April 1985.  
   
    * Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant   - Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
      Integers using Multiplication", in Proceedings of the SIGPLAN       Initialize ROP and set its value from the string in STR.  See
      PLDI'94 Conference, June 1994.  Available online,       `mpf_set_str' above for details on the assignment operation.
      `ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz too).  
   
    * Tudor Jebelean, "An algorithm for exact division", Journal of       Note that ROP is initialized even if an error occurs.  (I.e., you
      Symbolic Computation, v. 15, 1993, pp. 169-180.  Research report       have to call `mpf_clear' for it.)
      version available online  
      `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'  
   
    * Kenneth Weber, "The accelerated integer GCD algorithm", ACM       The precision of ROP will be taken from the active default
      Transactions on Mathematical Software, v. 21 (March), 1995, pp.       precision, as set by `mpf_set_default_prec'.
      111-122.  
   
    * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",  
      Max-Planck-Institut fuer Informatik Research Report  File: gmp.info,  Node: Converting Floats,  Next: Float Arithmetic,  Prev: Simultaneous Float Init & Assign,  Up: Floating-point Functions
      MPI-I-98-1-022,  
      `http://www.mpi-sb.mpg.de/~ziegler/TechRep.ps.gz'.  
   
    * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,  Conversion Functions
      "Handbook of Applied Cryptography",  ====================
      `http://cacr.math.uwaterloo.ca/hac/'.  
   
    * Henri Cohen, "A Course in Computational Algebraic Number Theory",   - Function: double mpf_get_d (mpf_t OP)
      Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.       Convert OP to a `double'.
      Errata available online  
      `http://www.math.u-bordeaux.fr/~cohen'  
   
    - Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP)
        Find D and EXP such that D times 2 raised to EXP, with
        0.5<=abs(D)<1, is a good approximation to OP.  This is similar to
        the standard C function `frexp'.
   
    - Function: long mpf_get_si (mpf_t OP)
    - Function: unsigned long mpf_get_ui (mpf_t OP)
        Convert OP to a `long' or `unsigned long', truncating any fraction
        part.  If OP is too big for the return type, the result is
        undefined.
   
        See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
        Miscellaneous Float Functions::).
   
    - Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE,
             size_t N_DIGITS, mpf_t OP)
        Convert OP to a string of digits in base BASE.  BASE can be 2 to
        36.  Up to N_DIGITS digits will be generated.  Trailing zeros are
        not returned.  No more digits than can be accurately represented
        by OP are ever generated.  If N_DIGITS is 0 then that accurate
        maximum number of digits are generated.
   
        If STR is `NULL', the result string is allocated using the current
        allocation function (*note Custom Allocation::).  The block will be
        `strlen(str)+1' bytes, that being exactly enough for the string and
        null-terminator.
   
        If STR is not `NULL', it should point to a block of N_DIGITS + 2
        bytes, that being enough for the mantissa, a possible minus sign,
        and a null-terminator.  When N_DIGITS is 0 to get all significant
        digits, an application won't be able to know the space required,
        and STR should be `NULL' in that case.
   
        The generated string is a fraction, with an implicit radix point
        immediately to the left of the first digit.  The applicable
        exponent is written through the EXPPTR pointer.  For example, the
        number 3.1416 would be returned as string "31416" and exponent 1.
   
        When OP is zero, an empty string is produced and the exponent
        returned is 0.
   
        A pointer to the result string is returned, being either the
        allocated block or the given STR.
   
   
 File: gmp.info,  Node: Concept Index,  Next: Function Index,  Prev: References,  Up: Top  File: gmp.info,  Node: Float Arithmetic,  Next: Float Comparison,  Prev: Converting Floats,  Up: Floating-point Functions
   
 Concept Index  Arithmetic Functions
 *************  ====================
   
 * Menu:   - Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
    - Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int
             OP2)
        Set ROP to OP1 + OP2.
   
 * ABI:                                   ABI and ISA.   - Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 * About this manual:                     Introduction to GMP.   - Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t
 * alloca:                                Build Options.            OP2)
 * Allocation of memory:                  Custom Allocation.   - Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
 * Anonymous FTP of latest version:       Getting the Latest Version of GMP.            OP2)
 * Arithmetic functions <1>:              Float Arithmetic.       Set ROP to OP1 - OP2.
 * Arithmetic functions <2>:              Rational Arithmetic.  
 * Arithmetic functions:                  Integer Arithmetic.   - Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 * Assignment functions <1>:              Assigning Floats.   - Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int
 * Assignment functions:                  Assigning Integers.            OP2)
 * Basics:                                GMP Basics.       Set ROP to OP1 times OP2.
 * Berkeley MP compatible functions:      BSD Compatible Functions.  
 * Binomial coefficient functions:        Number Theoretic Functions.     Division is undefined if the divisor is zero, and passing a zero
 * Bit manipulation functions:            Integer Logic and Bit Fiddling.  divisor to the divide functions will make these functions intentionally
 * Bit shift left:                        Integer Arithmetic.  divide by zero.  This lets the user handle arithmetic exceptions in
 * Bit shift right:                       Integer Division.  these functions in the same manner as other arithmetic exceptions.
 * Bits per limb:                         Useful Macros and Constants.  
 * BSD MP compatible functions:           BSD Compatible Functions.   - Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 * Bug reporting:                         Reporting Bugs.   - Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
 * Build notes for binary packaging:      Notes for Package Builds.            OP2)
 * Build notes for particular systems:    Notes for Particular Systems.   - Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
 * Build options:                         Build Options.            OP2)
 * Build problems known:                  Known Build Problems.       Set ROP to OP1/OP2.
 * Comparison functions <1>:              Integer Comparisons.  
 * Comparison functions <2>:              Comparing Rationals.   - Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
 * Comparison functions:                  Float Comparison.   - Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
 * Compatibility with older versions:     Compatibility with older versions.       Set ROP to the square root of OP.
 * Conditions for copying GNU MP:         Copying.  
 * Configuring GMP:                       Installing GMP.   - Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
 * Constants:                             Useful Macros and Constants.            OP2)
 * Contributors:                          Contributors.       Set ROP to OP1 raised to the power OP2.
 * Conventions for variables:             GMP Variable Conventions.  
 * Conversion functions <1>:              Converting Integers.   - Function: void mpf_neg (mpf_t ROP, mpf_t OP)
 * Conversion functions:                  Converting Floats.       Set ROP to -OP.
 * Copying conditions:                    Copying.  
 * CPUs supported:                        Introduction to GMP.   - Function: void mpf_abs (mpf_t ROP, mpf_t OP)
 * Custom allocation:                     Custom Allocation.       Set ROP to the absolute value of OP.
 * Demonstration programs:                Build Options.  
 * Division functions <1>:                Integer Division.   - Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long int
 * Division functions <2>:                Rational Arithmetic.            OP2)
 * Division functions:                    Float Arithmetic.       Set ROP to OP1 times 2 raised to OP2.
 * Exact division functions:              Integer Division.  
 * Example programs:                      Build Options.   - Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long int
 * Exponentiation functions <1>:          Float Arithmetic.            OP2)
 * Exponentiation functions:              Integer Exponentiation.       Set ROP to OP1 divided by 2 raised to OP2.
 * Extended GCD:                          Number Theoretic Functions.  
 * Factorial functions:                   Number Theoretic Functions.  
 * Fibonacci sequence functions:          Number Theoretic Functions.  File: gmp.info,  Node: Float Comparison,  Next: I/O of Floats,  Prev: Float Arithmetic,  Up: Floating-point Functions
 * Float arithmetic functions:            Float Arithmetic.  
 * Float assignment functions:            Assigning Floats.  Comparison Functions
 * Float comparison functions:            Float Comparison.  ====================
 * Float conversion functions:            Converting Floats.  
 * Float functions:                       Floating-point Functions.   - Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
 * Float init and assign functions:       Simultaneous Float Init & Assign.   - Function: int mpf_cmp_d (mpf_t OP1, double OP2)
 * Float initialization functions:        Initializing Floats.   - Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
 * Float input and output functions:      I/O of Floats.   - Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
 * Float miscellaneous functions:         Miscellaneous Float Functions.       Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero
 * Floating-point functions:              Floating-point Functions.       if OP1 = OP2, and a negative value if OP1 < OP2.
 * Floating-point number:                 Nomenclature and Types.  
 * FTP of latest version:                 Getting the Latest Version of GMP.   - Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3)
 * Function classes:                      Function Classes.       Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
 * GMP version number:                    Useful Macros and Constants.       zero otherwise.  I.e., test of OP1 and OP2 are approximately equal.
 * gmp.h:                                 GMP Basics.  
 * Greatest common divisor functions:     Number Theoretic Functions.       Caution: Currently only whole limbs are compared, and only in an
 * Home page:                             Introduction to GMP.       exact fashion.  In the future values like 1000 and 0111 may be
 * I/O functions <1>:                     I/O of Rationals.       considered the same to 3 bits (on the basis that their difference
 * I/O functions <2>:                     I/O of Integers.       is that small).
 * I/O functions:                         I/O of Floats.  
 * Initialization and assignment functions <1>: Initializing Rationals.   - Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
 * Initialization and assignment functions <2>: Simultaneous Float Init & Assign.       Compute the relative difference between OP1 and OP2 and store the
 * Initialization and assignment functions: Simultaneous Integer Init & Assign.       result in ROP.  This is abs(OP1-OP2)/OP1.
 * Initialization functions <1>:          Initializing Integers.  
 * Initialization functions:              Initializing Floats.   - Macro: int mpf_sgn (mpf_t OP)
 * Input functions <1>:                   I/O of Floats.       Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
 * Input functions <2>:                   I/O of Rationals.  
 * Input functions:                       I/O of Integers.       This function is actually implemented as a macro.  It evaluates
 * Installing GMP:                        Installing GMP.       its arguments multiple times.
 * Integer:                               Nomenclature and Types.  
 * Integer arithmetic functions:          Integer Arithmetic.  
 * Integer assignment functions:          Assigning Integers.  File: gmp.info,  Node: I/O of Floats,  Next: Miscellaneous Float Functions,  Prev: Float Comparison,  Up: Floating-point Functions
 * Integer bit manipulation functions:    Integer Logic and Bit Fiddling.  
 * Integer comparison functions:          Integer Comparisons.  Input and Output Functions
 * Integer conversion functions:          Converting Integers.  ==========================
 * Integer division functions:            Integer Division.  
 * Integer exponentiation functions:      Integer Exponentiation.     Functions that perform input from a stdio stream, and functions that
 * Integer functions:                     Integer Functions.  output to a stdio stream.  Passing a `NULL' pointer for a STREAM
 * Integer init and assign:               Simultaneous Integer Init & Assign.  argument to any of these functions will make them read from `stdin' and
 * Integer initialization functions:      Initializing Integers.  write to `stdout', respectively.
 * Integer input and output functions:    I/O of Integers.  
 * Integer miscellaneous functions:       Miscellaneous Integer Functions.     When using any of these functions, it is a good idea to include
 * Integer random number functions:       Integer Random Numbers.  `stdio.h' before `gmp.h', since that will allow `gmp.h' to define
 * Integer root functions:                Integer Roots.  prototypes for these functions.
 * Introduction:                          Introduction to GMP.  
 * ISA:                                   ABI and ISA.   - Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
 * Jabobi symbol functions:               Number Theoretic Functions.            N_DIGITS, mpf_t OP)
 * Kronecker symbol functions:            Number Theoretic Functions.       Print OP to STREAM, as a string of digits.  Return the number of
 * Latest version of GMP:                 Getting the Latest Version of GMP.       bytes written, or if an error occurred, return 0.
 * Least common multiple functions:       Number Theoretic Functions.  
 * Libtool versioning:                    Notes for Package Builds.       The mantissa is prefixed with an `0.' and is in the given BASE,
 * Limb:                                  Nomenclature and Types.       which may vary from 2 to 36.  An exponent then printed, separated
 * Limb size:                             Useful Macros and Constants.       by an `e', or if BASE is greater than 10 then by an `@'.  The
 * Logical functions:                     Integer Logic and Bit Fiddling.       exponent is always in decimal.  The decimal point follows the
 * Low-level functions:                   Low-level Functions.       current locale, on systems providing `localeconv'.
 * Mailing list:                          Introduction to GMP.  
 * Memory allocation:                     Custom Allocation.       Up to N_DIGITS will be printed from the mantissa, except that no
 * Miscellaneous float functions:         Miscellaneous Float Functions.       more digits than are accurately representable by OP will be
 * Miscellaneous integer functions:       Miscellaneous Integer Functions.       printed.  N_DIGITS can be 0 to select that accurate maximum.
 * Miscellaneous rational functions:      Miscellaneous Rational Functions.  
 * Modular inverse functions:             Number Theoretic Functions.   - Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
 * mp.h:                                  BSD Compatible Functions.       Read a string in base BASE from STREAM, and put the read float in
 * Multi-threading:                       GMP and Reentrancy.       ROP.  The string is of the form `M@N' or, if the base is 10 or
 * Nomenclature:                          Nomenclature and Types.       less, alternatively `MeN'.  `M' is the mantissa and `N' is the
 * Number theoretic functions:            Number Theoretic Functions.       exponent.  The mantissa is always in the specified base.  The
 * Numerator and denominator:             Applying Integer Functions.       exponent is either in the specified base or, if BASE is negative,
 * Output functions <1>:                  I/O of Floats.       in decimal.  The decimal point expected is taken from the current
 * Output functions <2>:                  I/O of Integers.       locale, on systems providing `localeconv'.
 * Output functions:                      I/O of Rationals.  
 * Packaged builds:                       Notes for Package Builds.       The argument BASE may be in the ranges 2 to 36, or -36 to -2.
 * Parameter conventions:                 GMP Variable Conventions.       Negative values are used to specify that the exponent is in
 * Precision of floats:                   Floating-point Functions.       decimal.
 * Prime testing functions:               Number Theoretic Functions.  
 * Random number functions <1>:           Integer Random Numbers.       Unlike the corresponding `mpz' function, the base will not be
 * Random number functions:               Random Number Functions.       determined from the leading characters of the string if BASE is 0.
 * Random number state:                   Random State Initialization.       This is so that numbers like `0.23' are not interpreted as octal.
 * Rational arithmetic functions:         Rational Arithmetic.  
 * Rational comparison functions:         Comparing Rationals.       Return the number of bytes read, or if an error occurred, return 0.
 * Rational init and assign:              Initializing Rationals.  
 * Rational input and output functions:   I/O of Rationals.  
 * Rational miscellaneous functions:      Miscellaneous Rational Functions.  
 * Rational number:                       Nomenclature and Types.  
 * Rational number functions:             Rational Number Functions.  
 * Rational numerator and denominator:    Applying Integer Functions.  
 * Reentrancy:                            GMP and Reentrancy.  
 * References:                            References.  
 * Reporting bugs:                        Reporting Bugs.  
 * Root extraction functions <1>:         Float Arithmetic.  
 * Root extraction functions:             Integer Roots.  
 * Stack overflow segfaults:              Build Options.  
 * Stripped libraries:                    Known Build Problems.  
 * Thread safety:                         GMP and Reentrancy.  
 * Types:                                 Nomenclature and Types.  
 * Upward compatibility:                  Compatibility with older versions.  
 * Useful macros and constants:           Useful Macros and Constants.  
 * User-defined precision:                Floating-point Functions.  
 * Variable conventions:                  GMP Variable Conventions.  
 * Version number:                        Useful Macros and Constants.  
 * Web page:                              Introduction to GMP.  
   
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