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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: Miscellaneous Float Functions,  Prev: I/O of Floats,  Up: Floating-point Functions
   
    Copyright (C) 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000  Miscellaneous Functions
 Free Software Foundation, Inc.  =======================
   
    Permission is granted to make and distribute verbatim copies of this   - Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
 manual provided the copyright notice and this permission notice are   - Function: void mpf_floor (mpf_t ROP, mpf_t OP)
 preserved on all copies.   - Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
        Set ROP to OP rounded to an integer.  `mpf_ceil' rounds to the
        next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
        to the integer towards zero.
   
    Permission is granted to copy and distribute modified versions of   - Function: int mpf_integer_p (mpf_t OP)
 this manual under the conditions for verbatim copying, provided that       Return non-zero if OP is an integer.
 the entire resulting derived work is distributed under the terms of a  
 permission notice identical to this one.  
   
    Permission is granted to copy and distribute translations of this   - Function: int mpf_fits_ulong_p (mpf_t OP)
 manual into another language, under the above conditions for modified   - Function: int mpf_fits_slong_p (mpf_t OP)
 versions, except that this permission notice may be stated in a   - Function: int mpf_fits_uint_p (mpf_t OP)
 translation approved by the Foundation.   - Function: int mpf_fits_sint_p (mpf_t OP)
    - Function: int mpf_fits_ushort_p (mpf_t OP)
    - Function: int mpf_fits_sshort_p (mpf_t OP)
        Return non-zero if OP would fit in the respective C data type, when
        truncated to an integer.
   
    - Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
             unsigned long int NBITS)
        Generate a uniformly distributed random float in ROP, such that 0
        <= ROP < 1, with NBITS significant bits in the mantissa.
   
        The variable STATE must be initialized by calling one of the
        `gmp_randinit' functions (*Note Random State Initialization::)
        before invoking this function.
   
    - Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
             EXP)
        Generate a random float of at most MAX_SIZE limbs, with long
        strings of zeros and ones in the binary representation.  The
        exponent of the number is in the interval -EXP to EXP.  This
        function is useful 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.
   
   
 File: gmp.info,  Node: Function Index,  Prev: Concept Index,  Up: Top  File: gmp.info,  Node: Low-level Functions,  Next: Random Number Functions,  Prev: Floating-point Functions,  Up: Top
   
 Function and Type Index  Low-level Functions
   *******************
   
      This chapter describes low-level GMP functions, used to implement the
   high-level GMP functions, but also intended for time-critical user code.
   
      These functions start with the prefix `mpn_'.
   
      The `mpn' functions are designed to be as fast as possible, *not* to
   provide a coherent calling interface.  The different functions have
   somewhat similar interfaces, but there are variations that make them
   hard to use.  These functions do as little as possible apart from the
   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
   limb and a limb count.  A destination operand is specified by just a
   pointer.  It is the responsibility of the caller to ensure that the
   destination has enough space for storing the result.
   
      With this way of specifying operands, it is possible to perform
   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
   needs at least one limb.  No size argument may be zero.  Unless
   otherwise stated, in-place operations are allowed where source and
   destination are the same, but not where they only partly overlap.
   
      The `mpn' functions are the base for the implementation of the
   `mpz_', `mpf_', and `mpq_' functions.
   
      This example adds the number beginning at S1P and the number
   beginning at S2P and writes the sum at DESTP.  All areas have N limbs.
   
        cy = mpn_add_n (destp, s1p, s2p, n)
   
   In the notation used here, a source operand is identified by the
   pointer to the least significant limb, and the limb count in braces.
   For example, {S1P, S1N}.
   
    - Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
             const mp_limb_t *S2P, mp_size_t N)
        Add {S1P, N} and {S2P, N}, and write the N least significant limbs
        of the result to RP.  Return carry, either 0 or 1.
   
        This is the lowest-level function for addition.  It is the
        preferred function for addition, since it is written in assembly
        for most CPUs.  For addition of a variable to itself (i.e., S1P
        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,
             mp_size_t N, mp_limb_t S2LIMB)
        Add {S1P, N} and S2LIMB, and write the N least significant limbs
        of the result to RP.  Return carry, either 0 or 1.
   
    - Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
             mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
        Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
        limbs of the result to RP.  Return carry, either 0 or 1.
   
        This function requires that S1N is greater than or equal to S2N.
   
    - Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
             const mp_limb_t *S2P, mp_size_t N)
        Subtract {S2P, N} from {S1P, N}, and write the N least significant
        limbs of the result to RP.  Return borrow, either 0 or 1.
   
        This is the lowest-level function for subtraction.  It is the
        preferred function for subtraction, since it is written in
        assembly for most CPUs.
   
    - Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
             mp_size_t N, mp_limb_t S2LIMB)
        Subtract S2LIMB from {S1P, N}, and write the N least significant
        limbs of the result to RP.  Return borrow, either 0 or 1.
   
    - Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
             mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
        Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
        significant limbs of the result to RP.  Return borrow, either 0 or
        1.
   
        This function requires that S1N is greater than or equal to S2N.
   
    - Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const
             mp_limb_t *S2P, mp_size_t N)
        Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
        RP.
   
        The destination has to have space for 2*N limbs, even if the
        product's most significant limb is zero.
   
    - Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
             mp_size_t N, mp_limb_t S2LIMB)
        Multiply {S1P, N} by S2LIMB, and write the N least significant
        limbs of the product to RP.  Return the most significant limb of
        the product.  {S1P, N} and {RP, N} are allowed to overlap provided
        RP <= S1P.
   
        This is a low-level function that is a building block for general
        multiplication as well as other operations in GMP.  It is written
        in assembly for most CPUs.
   
        Don't call this function if S2LIMB is a power of 2; use
        `mpn_lshift' with a count equal to the logarithm of S2LIMB
        instead, for optimal speed.
   
    - Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
             *S1P, mp_size_t N, mp_limb_t S2LIMB)
        Multiply {S1P, N} and S2LIMB, and add the N least significant
        limbs of the product to {RP, N} 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
        multiplication as well as other operations in GMP.  It is written
        in assembly for most CPUs.
   
    - Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
             *S1P, mp_size_t N, mp_limb_t S2LIMB)
        Multiply {S1P, N} and S2LIMB, and subtract the N least significant
        limbs of the product from {RP, N} 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.
        It is written in assembly for most CPUs.
   
    - Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
             mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
        Multiply {S1P, S1N} and {S2P, S2N}, and write the result to RP.
        Return the most significant limb of the result.
   
        The destination has to have space for S1N + S2N limbs, even if the
        result might be one limb smaller.
   
        This function requires that S1N is greater than or equal to S2N.
        The destination must be distinct from both input operands.
   
    - Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
             QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
             mp_size_t DN)
        Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
        and the remainder at {RP, DN}.  The quotient is rounded towards 0.
   
        No overlap is permitted between arguments.  NN must be greater
        than or equal to DN.  The most significant limb of DP must be
        non-zero.  The QXN operand must be zero.
   
    - Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
             mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
             mp_size_t S3N)
        [This function is obsolete.  Please call `mpn_tdiv_qr' instead for
        best performance.]
   
        Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
        with the exception of the most significant limb, which is
        returned.  The remainder replaces the dividend at RS2P; it will be
        S3N limbs long (i.e., as many limbs as the divisor).
   
        In addition to an integer quotient, QXN fraction limbs are
        developed, and stored after the integral limbs.  For most usages,
        QXN will be zero.
   
        It is required that RS2N is greater than or equal to S3N.  It is
        required that the most significant bit of the divisor is set.
   
        If the quotient is not needed, pass RS2P + S3N as R1P.  Aside from
        that special case, no overlap between arguments is permitted.
   
        Return the most significant limb of the quotient, either 0 or 1.
   
        The area at R1P needs to be RS2N - S3N + QXN limbs large.
   
    - Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
             mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
    - Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
             mp_size_t S2N, mp_limb_t S3LIMB)
        Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
        Return the remainder.
   
        The integer quotient is written to {R1P+QXN, S2N} and in addition
        QXN fraction limbs are developed and written to {R1P, QXN}.
        Either or both S2N and QXN can be zero.  For most usages, QXN will
        be zero.
   
        `mpn_divmod_1' exists for upward source compatibility and is
        simply a macro calling `mpn_divrem_1' with a QXN of 0.
   
        The areas at R1P and S2P have to be identical or completely
        separate, not partially overlapping.
   
    - Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
             mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
        [This function is obsolete.  Please call `mpn_tdiv_qr' instead for
        best performance.]
   
    - Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
             mp_size_t N)
    - Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP,
             mp_size_t N, mp_limb_t CARRY)
        Divide {SP, N} by 3, expecting it to divide exactly, and writing
        the result to {RP, N}.  If 3 divides exactly, the return value is
        zero and the result is the quotient.  If not, the return value is
        non-zero and the result won't be anything useful.
   
        `mpn_divexact_by3c' takes an initial carry parameter, which can be
        the return value from a previous call, so a large calculation can
        be done piece by piece from low to high.  `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
        `mpn_divrem_1' on CPUs with fast multiplication but slow division.
   
        The source a, result q, size n, initial carry i, and return value
        c satisfy c*b^n + a-i = 3*q, where b=2^mp_bits_per_limb.  The
        return c is always 0, 1 or 2, and the initial carry i 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 (when `mp_bits_per_limb' is even, which is
        always so currently).
   
    - Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
             mp_limb_t S2LIMB)
        Divide {S1P, S1N} by S2LIMB, and return the remainder.  S1N can be
        zero.
   
    - Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,
             mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N, unsigned
             long int D)
        This function puts the low floor(D/mp_bits_per_limb) limbs of Q =
        {S1P, S1N}/{S2P, S2N} mod 2^D at RP, and returns the high D mod
        `mp_bits_per_limb' bits of Q.
   
        {S1P, S1N} - Q * {S2P, S2N} mod 2^(S1N*mp_bits_per_limb) is placed
        at S1P.  Since the low floor(D/mp_bits_per_limb) limbs of this
        difference are zero, it is possible to overwrite the low limbs at
        S1P with this difference, provided RP <= S1P.
   
        This function requires that S1N * mp_bits_per_limb >= D, and that
        {S2P, S2N} is odd.
   
        *This interface is preliminary.  It might change incompatibly in
        future revisions.*
   
    - Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
             mp_size_t N, unsigned int COUNT)
        Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
        The bits shifted out at the left are returned in the least
        significant COUNT bits of the return value (the rest of the return
        value is zero).
   
        COUNT must be in the range 1 to mp_bits_per_limb-1.  The regions
        {SP, N} and {RP, N} may overlap, provided RP >= SP.
   
        This function is written in assembly for most CPUs.
   
    - Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
             mp_size_t N, unsigned int COUNT)
        Shift {SP, N} right by COUNT bits, and write the result to {RP,
        N}.  The bits shifted out at the right are returned in the most
        significant COUNT bits of the return value (the rest of the return
        value is zero).
   
        COUNT must be in the range 1 to mp_bits_per_limb-1.  The regions
        {SP, N} and {RP, N} may overlap, provided RP <= SP.
   
        This function is written in assembly for most CPUs.
   
    - Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
             mp_size_t N)
        Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
        S2, 0 if they are equal, or a negative value if S1 < S2.
   
    - Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,
             mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N)
        Set {RP, RETVAL} to the greatest common divisor of {S1P, S1N} and
        {S2P, S2N}.  The result can be up to S2N limbs, the return value
        is the actual number produced.  Both source operands are destroyed.
   
        {S1P, S1N} must have at least as many bits as {S2P, S2N}.  {S2P,
        S2N} must be odd.  Both operands must have non-zero most
        significant limbs.  No overlap is permitted between {S1P, S1N} and
        {S2P, S2N}.
   
    - Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1N,
             mp_limb_t S2LIMB)
        Return the greatest common divisor of {S1P, S1N} and S2LIMB.  Both
        operands must be non-zero.
   
    - Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P,
             mp_size_t *R2N, mp_limb_t *S1P, mp_size_t S1N, mp_limb_t
             *S2P, mp_size_t S2N)
        Calculate the greatest common divisor of {S1P, S1N} and {S2P,
        S2N}.  Store the gcd at {R1P, RETVAL} and the first cofactor at
        {R2P, *R2N}, with *R2N negative if the cofactor is negative.  R1P
        and R2P should each have room for S1N+1 limbs, but the return
        value and value stored through R2N indicate the actual number
        produced.
   
        {S1P, S1N} >= {S2P, S2N} is required, and both must be non-zero.
        The regions {S1P, S1N+1} and {S2P, S2N+1} are destroyed (i.e. the
        operands plus an extra limb past the end of each).
   
        The cofactor R1 will satisfy R2*S1 + K*S2 = R1.  The second
        cofactor K is not calculated but can easily be obtained from (R1 -
        R2*S1) / S2.
   
    - Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
             const mp_limb_t *SP, mp_size_t N)
        Compute the square root of {SP, N} and put the result at {R1P,
        ceil(N/2)} and the remainder at {R2P, RETVAL}.  R2P needs space
        for N limbs, but the return value indicates how many are produced.
   
        The most significant limb of {SP, N} must be non-zero.  The areas
        {R1P, ceil(N/2)} and {SP, N} must be completely separate.  The
        areas {R2P, N} and {SP, N} must be either identical or completely
        separate.
   
        If the remainder is not wanted then R2P can be `NULL', and in this
        case the return value is zero or non-zero according to whether the
        remainder would have been zero or non-zero.
   
        A return value of zero indicates a perfect square.  See also
        `mpz_perfect_square_p'.
   
    - Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
             mp_limb_t *S1P, mp_size_t S1N)
        Convert {S1P, S1N} to a raw unsigned char array at STR in base
        BASE, and return the number of characters produced.  There may be
        leading zeros in the string.  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.  BASE can vary from 2 to 256.
   
        The most significant limb of the input {S1P, S1N} must be
        non-zero.  The input {S1P, S1N} is clobbered, except when BASE is
        a power of 2, in which case it's unchanged.
   
        The area at STR has to have space for the largest possible number
        represented by a S1N long limb array, plus one extra character.
   
    - Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
             *STR, size_t STRSIZE, int BASE)
        Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
   
        STR[0] is the most significant byte and STR[STRSIZE-1] is the
        least significant.  Each byte should be a value in the range 0 to
        BASE-1, not an ASCII character.  BASE can vary from 2 to 256.
   
        The return value is the number of limbs written to RP.  If the most
        significant input byte is non-zero then the high limb at RP will be
        non-zero, and only that exact number of limbs will be required
        there.
   
        If the most significant input byte is zero then there may be high
        zero limbs written to RP and included in the return value.
   
        STRSIZE must be at least 1, and no overlap is permitted between
        {STR,STRSIZE} and the result at RP.
   
    - Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P,
             unsigned long int BIT)
        Scan S1P from bit position BIT for the next clear bit.
   
        It is required that there be a clear bit within the area at S1P at
        or beyond bit position BIT, so that the function has something to
        return.
   
    - Function: unsigned long int mpn_scan1 (const mp_limb_t *S1P,
             unsigned long int BIT)
        Scan S1P from bit position BIT for the next set bit.
   
        It is required that there be a set bit within the area at S1P at or
        beyond bit position BIT, so that the function has something to
        return.
   
    - Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
    - Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
        Generate a random number of length R1N and store it at R1P.  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.
   
    - Function: unsigned long int mpn_popcount (const mp_limb_t *S1P,
             mp_size_t N)
        Count the number of set bits in {S1P, N}.
   
    - Function: unsigned long int mpn_hamdist (const mp_limb_t *S1P, const
             mp_limb_t *S2P, mp_size_t N)
        Compute the hamming distance between {S1P, N} and {S2P, N}.
   
    - Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
             N)
        Return non-zero iff {S1P, N} is a perfect square.
   
   
   Nails
   =====
   
      *Everything in this section is highly experimental and may disappear
   or be subject to incompatible changes in a future version of GMP.*
   
      Nails are an experimental feature whereby a few bits are left unused
   at the top of each `mp_limb_t'.  This can significantly improve carry
   handling on some processors.
   
      All the `mpn' functions accepting limb data will expect the nail
   bits to be zero on entry, and will return data with the nails similarly
   all zero.  This applies both to limb vectors and to single limb
   arguments.
   
      Nails can be enabled by configuring with `--enable-nails'.  By
   default the number of bits will be chosen according to what suits the
   host processor, but a particular number can be selected with
   `--enable-nails=N'.
   
      At the mpn level, a nail build is neither source nor binary
   compatible with a non-nail build, strictly speaking.  But programs
   acting on limbs only through the mpn functions are likely to work
   equally well with either build, and judicious use of the definitions
   below should make any program compatible with either build, at the
   source level.
   
      For the higher level routines, meaning `mpz' etc, a nail build
   should be fully source and binary compatible with a non-nail build.
   
    - Macro: GMP_NAIL_BITS
    - Macro: GMP_NUMB_BITS
    - Macro: GMP_LIMB_BITS
        `GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
        not in use.  `GMP_NUMB_BITS' is the number of data bits in a limb.
        `GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'.  In
        all cases
   
             GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
   
    - Macro: GMP_NAIL_MASK
    - Macro: GMP_NUMB_MASK
        Bit masks for the nail and number parts of a limb.
        `GMP_NAIL_MASK' is 0 when nails are not in use.
   
        `GMP_NAIL_MASK' is not often needed, since the nail part can be
        obtained with `x >> GMP_NUMB_BITS', and that means one less large
        constant, which can help various RISC chips.
   
    - Macro: GMP_NUMB_MAX
        The maximum value that can be stored in the number part of a limb.
        This is the same as `GMP_NUMB_MASK', but can be used for clarity
        when doing comparisons rather than bit-wise operations.
   
      The term "nails" comes from finger or toe nails, which are at the
   ends of a limb (arm or leg).  "numb" is short for number, but is also
   how the developers felt after trying for a long time to come up with
   sensible names for these things.
   
      In the future (the distant future most likely) a non-zero nail might
   be permitted, giving non-unique representations for numbers in a limb
   vector.  This would help vector processors since carries would only
   ever need to propagate one or two limbs.
   
   
   File: gmp.info,  Node: Random Number Functions,  Next: Formatted Output,  Prev: Low-level Functions,  Up: Top
   
   Random Number Functions
 ***********************  ***********************
   
      Sequences of pseudo-random numbers in GMP are generated using a
   variable of type `gmp_randstate_t', which holds an algorithm selection
   and a current state.  Such a variable must be initialized by a call to
   one of the `gmp_randinit' functions, and can be seeded with one of the
   `gmp_randseed' functions.
   
      The functions actually generating random numbers are described in
   *Note Integer Random Numbers::, and *Note Miscellaneous Float
   Functions::.
   
      The older style random number functions don't accept a
   `gmp_randstate_t' parameter but instead share a global variable of that
   type.  They use a default algorithm and are currently not seeded
   (though perhaps that will change in the future).  The new functions
   accepting a `gmp_randstate_t' are recommended for applications that
   care about randomness.
   
 * Menu:  * Menu:
   
 * __GNU_MP_VERSION:                      Useful Macros and Constants.  * Random State Initialization::
 * __GNU_MP_VERSION_MINOR:                Useful Macros and Constants.  * Random State Seeding::
 * __GNU_MP_VERSION_PATCHLEVEL:           Useful Macros and Constants.  
 * _mpz_realloc:                          Initializing Integers.  
 * allocate_function:                     Custom Allocation.  
 * deallocate_function:                   Custom Allocation.  
 * gcd:                                   BSD Compatible Functions.  
 * gmp_randclear:                         Random State Initialization.  
 * gmp_randinit:                          Random State Initialization.  
 * gmp_randinit_lc_2exp:                  Random State Initialization.  
 * gmp_randseed:                          Random State Initialization.  
 * gmp_randseed_ui:                       Random State Initialization.  
 * itom:                                  BSD Compatible Functions.  
 * madd:                                  BSD Compatible Functions.  
 * mcmp:                                  BSD Compatible Functions.  
 * mdiv:                                  BSD Compatible Functions.  
 * mfree:                                 BSD Compatible Functions.  
 * min:                                   BSD Compatible Functions.  
 * mout:                                  BSD Compatible Functions.  
 * move:                                  BSD Compatible Functions.  
 * mp_limb_t:                             Nomenclature and Types.  
 * mp_set_memory_functions:               Custom Allocation.  
 * mpf_abs:                               Float Arithmetic.  
 * mpf_add:                               Float Arithmetic.  
 * mpf_add_ui:                            Float Arithmetic.  
 * mpf_ceil:                              Miscellaneous Float Functions.  
 * mpf_clear:                             Initializing Floats.  
 * mpf_cmp:                               Float Comparison.  
 * mpf_cmp_si:                            Float Comparison.  
 * mpf_cmp_ui:                            Float Comparison.  
 * mpf_div:                               Float Arithmetic.  
 * mpf_div_2exp:                          Float Arithmetic.  
 * mpf_div_ui:                            Float Arithmetic.  
 * mpf_eq:                                Float Comparison.  
 * mpf_floor:                             Miscellaneous Float Functions.  
 * mpf_get_d:                             Converting Floats.  
 * mpf_get_prec:                          Initializing Floats.  
 * mpf_get_str:                           Converting Floats.  
 * mpf_init:                              Initializing Floats.  
 * mpf_init2:                             Initializing Floats.  
 * mpf_init_set:                          Simultaneous Float Init & Assign.  
 * mpf_init_set_d:                        Simultaneous Float Init & Assign.  
 * mpf_init_set_si:                       Simultaneous Float Init & Assign.  
 * mpf_init_set_str:                      Simultaneous Float Init & Assign.  
 * mpf_init_set_ui:                       Simultaneous Float Init & Assign.  
 * mpf_inp_str:                           I/O of Floats.  
 * mpf_mul:                               Float Arithmetic.  
 * mpf_mul_2exp:                          Float Arithmetic.  
 * mpf_mul_ui:                            Float Arithmetic.  
 * mpf_neg:                               Float Arithmetic.  
 * mpf_out_str:                           I/O of Floats.  
 * mpf_pow_ui:                            Float Arithmetic.  
 * mpf_random2:                           Miscellaneous Float Functions.  
 * mpf_reldiff:                           Float Comparison.  
 * mpf_set:                               Assigning Floats.  
 * mpf_set_d:                             Assigning Floats.  
 * mpf_set_default_prec:                  Initializing Floats.  
 * mpf_set_prec:                          Initializing Floats.  
 * mpf_set_prec_raw:                      Initializing Floats.  
 * mpf_set_q:                             Assigning Floats.  
 * mpf_set_si:                            Assigning Floats.  
 * mpf_set_str:                           Assigning Floats.  
 * mpf_set_ui:                            Assigning Floats.  
 * mpf_set_z:                             Assigning Floats.  
 * mpf_sgn:                               Float Comparison.  
 * mpf_sqrt:                              Float Arithmetic.  
 * mpf_sqrt_ui:                           Float Arithmetic.  
 * mpf_sub:                               Float Arithmetic.  
 * mpf_sub_ui:                            Float Arithmetic.  
 * mpf_swap:                              Assigning Floats.  
 * mpf_t:                                 Nomenclature and Types.  
 * mpf_trunc:                             Miscellaneous Float Functions.  
 * mpf_ui_div:                            Float Arithmetic.  
 * mpf_ui_sub:                            Float Arithmetic.  
 * mpf_urandomb:                          Miscellaneous Float Functions.  
 * mpn_add:                               Low-level Functions.  
 * mpn_add_1:                             Low-level Functions.  
 * mpn_add_n:                             Low-level Functions.  
 * mpn_addmul_1:                          Low-level Functions.  
 * mpn_bdivmod:                           Low-level Functions.  
 * mpn_cmp:                               Low-level Functions.  
 * mpn_divexact_by3:                      Low-level Functions.  
 * mpn_divexact_by3c:                     Low-level Functions.  
 * mpn_divmod:                            Low-level Functions.  
 * mpn_divmod_1:                          Low-level Functions.  
 * mpn_divrem:                            Low-level Functions.  
 * mpn_divrem_1:                          Low-level Functions.  
 * mpn_gcd:                               Low-level Functions.  
 * mpn_gcd_1:                             Low-level Functions.  
 * mpn_gcdext:                            Low-level Functions.  
 * mpn_get_str:                           Low-level Functions.  
 * mpn_hamdist:                           Low-level Functions.  
 * mpn_lshift:                            Low-level Functions.  
 * mpn_mod_1:                             Low-level Functions.  
 * mpn_mul:                               Low-level Functions.  
 * mpn_mul_1:                             Low-level Functions.  
 * mpn_mul_n:                             Low-level Functions.  
 * mpn_perfect_square_p:                  Low-level Functions.  
 * mpn_popcount:                          Low-level Functions.  
 * mpn_preinv_mod_1:                      Low-level Functions.  
 * mpn_random:                            Low-level Functions.  
 * mpn_random2:                           Low-level Functions.  
 * mpn_rshift:                            Low-level Functions.  
 * mpn_scan0:                             Low-level Functions.  
 * mpn_scan1:                             Low-level Functions.  
 * mpn_set_str:                           Low-level Functions.  
 * mpn_sqrtrem:                           Low-level Functions.  
 * mpn_sub:                               Low-level Functions.  
 * mpn_sub_1:                             Low-level Functions.  
 * mpn_sub_n:                             Low-level Functions.  
 * mpn_submul_1:                          Low-level Functions.  
 * mpn_tdiv_qr:                           Low-level Functions.  
 * mpq_add:                               Rational Arithmetic.  
 * mpq_canonicalize:                      Rational Number Functions.  
 * mpq_clear:                             Initializing Rationals.  
 * mpq_cmp:                               Comparing Rationals.  
 * mpq_cmp_ui:                            Comparing Rationals.  
 * mpq_denref:                            Applying Integer Functions.  
 * mpq_div:                               Rational Arithmetic.  
 * mpq_equal:                             Comparing Rationals.  
 * mpq_get_d:                             Miscellaneous Rational Functions.  
 * mpq_get_den:                           Miscellaneous Rational Functions.  
 * mpq_get_num:                           Miscellaneous Rational Functions.  
 * mpq_init:                              Initializing Rationals.  
 * mpq_inv:                               Rational Arithmetic.  
 * mpq_mul:                               Rational Arithmetic.  
 * mpq_neg:                               Rational Arithmetic.  
 * mpq_numref:                            Applying Integer Functions.  
 * mpq_out_str:                           I/O of Rationals.  
 * mpq_set:                               Initializing Rationals.  
 * mpq_set_d:                             Miscellaneous Rational Functions.  
 * mpq_set_den:                           Miscellaneous Rational Functions.  
 * mpq_set_num:                           Miscellaneous Rational Functions.  
 * mpq_set_si:                            Initializing Rationals.  
 * mpq_set_ui:                            Initializing Rationals.  
 * mpq_set_z:                             Initializing Rationals.  
 * mpq_sgn:                               Comparing Rationals.  
 * mpq_sub:                               Rational Arithmetic.  
 * mpq_swap:                              Initializing Rationals.  
 * mpq_t:                                 Nomenclature and Types.  
 * mpz_abs:                               Integer Arithmetic.  
 * mpz_add:                               Integer Arithmetic.  
 * mpz_add_ui:                            Integer Arithmetic.  
 * mpz_addmul_ui:                         Integer Arithmetic.  
 * mpz_and:                               Integer Logic and Bit Fiddling.  
 * mpz_array_init:                        Initializing Integers.  
 * mpz_bin_ui:                            Number Theoretic Functions.  
 * mpz_bin_uiui:                          Number Theoretic Functions.  
 * mpz_cdiv_q:                            Integer Division.  
 * mpz_cdiv_q_ui:                         Integer Division.  
 * mpz_cdiv_qr:                           Integer Division.  
 * mpz_cdiv_qr_ui:                        Integer Division.  
 * mpz_cdiv_r:                            Integer Division.  
 * mpz_cdiv_r_ui:                         Integer Division.  
 * mpz_cdiv_ui:                           Integer Division.  
 * mpz_clear:                             Initializing Integers.  
 * mpz_clrbit:                            Integer Logic and Bit Fiddling.  
 * mpz_cmp:                               Integer Comparisons.  
 * mpz_cmp_si:                            Integer Comparisons.  
 * mpz_cmp_ui:                            Integer Comparisons.  
 * mpz_cmpabs:                            Integer Comparisons.  
 * mpz_cmpabs_ui:                         Integer Comparisons.  
 * mpz_com:                               Integer Logic and Bit Fiddling.  
 * mpz_divexact:                          Integer Division.  
 * mpz_even_p:                            Miscellaneous Integer Functions.  
 * mpz_fac_ui:                            Number Theoretic Functions.  
 * mpz_fdiv_q:                            Integer Division.  
 * mpz_fdiv_q_2exp:                       Integer Division.  
 * mpz_fdiv_q_ui:                         Integer Division.  
 * mpz_fdiv_qr:                           Integer Division.  
 * mpz_fdiv_qr_ui:                        Integer Division.  
 * mpz_fdiv_r:                            Integer Division.  
 * mpz_fdiv_r_2exp:                       Integer Division.  
 * mpz_fdiv_r_ui:                         Integer Division.  
 * mpz_fdiv_ui:                           Integer Division.  
 * mpz_fib_ui:                            Number Theoretic Functions.  
 * mpz_fits_sint_p:                       Miscellaneous Integer Functions.  
 * mpz_fits_slong_p:                      Miscellaneous Integer Functions.  
 * mpz_fits_sshort_p:                     Miscellaneous Integer Functions.  
 * mpz_fits_uint_p:                       Miscellaneous Integer Functions.  
 * mpz_fits_ulong_p:                      Miscellaneous Integer Functions.  
 * mpz_fits_ushort_p:                     Miscellaneous Integer Functions.  
 * mpz_gcd:                               Number Theoretic Functions.  
 * mpz_gcd_ui:                            Number Theoretic Functions.  
 * mpz_gcdext:                            Number Theoretic Functions.  
 * mpz_get_d:                             Converting Integers.  
 * mpz_get_si:                            Converting Integers.  
 * mpz_get_str:                           Converting Integers.  
 * mpz_get_ui:                            Converting Integers.  
 * mpz_getlimbn:                          Converting Integers.  
 * mpz_hamdist:                           Integer Logic and Bit Fiddling.  
 * mpz_init:                              Initializing Integers.  
 * mpz_init_set:                          Simultaneous Integer Init & Assign.  
 * mpz_init_set_d:                        Simultaneous Integer Init & Assign.  
 * mpz_init_set_si:                       Simultaneous Integer Init & Assign.  
 * mpz_init_set_str:                      Simultaneous Integer Init & Assign.  
 * mpz_init_set_ui:                       Simultaneous Integer Init & Assign.  
 * mpz_inp_raw:                           I/O of Integers.  
 * mpz_inp_str:                           I/O of Integers.  
 * mpz_invert:                            Number Theoretic Functions.  
 * mpz_ior:                               Integer Logic and Bit Fiddling.  
 * mpz_jacobi:                            Number Theoretic Functions.  
 * mpz_kronecker_si:                      Number Theoretic Functions.  
 * mpz_kronecker_ui:                      Number Theoretic Functions.  
 * mpz_lcm:                               Number Theoretic Functions.  
 * mpz_legendre:                          Number Theoretic Functions.  
 * mpz_mod:                               Integer Division.  
 * mpz_mod_ui:                            Integer Division.  
 * mpz_mul:                               Integer Arithmetic.  
 * mpz_mul_2exp:                          Integer Arithmetic.  
 * mpz_mul_si:                            Integer Arithmetic.  
 * mpz_mul_ui:                            Integer Arithmetic.  
 * mpz_neg:                               Integer Arithmetic.  
 * mpz_nextprime:                         Number Theoretic Functions.  
 * mpz_odd_p:                             Miscellaneous Integer Functions.  
 * mpz_out_raw:                           I/O of Integers.  
 * mpz_out_str:                           I/O of Integers.  
 * mpz_perfect_power_p:                   Integer Roots.  
 * mpz_perfect_square_p:                  Integer Roots.  
 * mpz_popcount:                          Integer Logic and Bit Fiddling.  
 * mpz_pow_ui:                            Integer Exponentiation.  
 * mpz_powm:                              Integer Exponentiation.  
 * mpz_powm_ui:                           Integer Exponentiation.  
 * mpz_probab_prime_p:                    Number Theoretic Functions.  
 * mpz_random:                            Integer Random Numbers.  
 * mpz_random2:                           Integer Random Numbers.  
 * mpz_remove:                            Number Theoretic Functions.  
 * mpz_root:                              Integer Roots.  
 * mpz_rrandomb:                          Integer Random Numbers.  
 * mpz_scan0:                             Integer Logic and Bit Fiddling.  
 * mpz_scan1:                             Integer Logic and Bit Fiddling.  
 * mpz_set:                               Assigning Integers.  
 * mpz_set_d:                             Assigning Integers.  
 * mpz_set_f:                             Assigning Integers.  
 * mpz_set_q:                             Assigning Integers.  
 * mpz_set_si:                            Assigning Integers.  
 * mpz_set_str:                           Assigning Integers.  
 * mpz_set_ui:                            Assigning Integers.  
 * mpz_setbit:                            Integer Logic and Bit Fiddling.  
 * mpz_sgn:                               Integer Comparisons.  
 * mpz_si_kronecker:                      Number Theoretic Functions.  
 * mpz_size:                              Miscellaneous Integer Functions.  
 * mpz_sizeinbase:                        Miscellaneous Integer Functions.  
 * mpz_sqrt:                              Integer Roots.  
 * mpz_sqrtrem:                           Integer Roots.  
 * mpz_sub:                               Integer Arithmetic.  
 * mpz_sub_ui:                            Integer Arithmetic.  
 * mpz_swap:                              Assigning Integers.  
 * mpz_t:                                 Nomenclature and Types.  
 * mpz_tdiv_q:                            Integer Division.  
 * mpz_tdiv_q_2exp:                       Integer Division.  
 * mpz_tdiv_q_ui:                         Integer Division.  
 * mpz_tdiv_qr:                           Integer Division.  
 * mpz_tdiv_qr_ui:                        Integer Division.  
 * mpz_tdiv_r:                            Integer Division.  
 * mpz_tdiv_r_2exp:                       Integer Division.  
 * mpz_tdiv_r_ui:                         Integer Division.  
 * mpz_tdiv_ui:                           Integer Division.  
 * mpz_tstbit:                            Integer Logic and Bit Fiddling.  
 * mpz_ui_kronecker:                      Number Theoretic Functions.  
 * mpz_ui_pow_ui:                         Integer Exponentiation.  
 * mpz_urandomb:                          Integer Random Numbers.  
 * mpz_urandomm:                          Integer Random Numbers.  
 * mpz_xor:                               Integer Logic and Bit Fiddling.  
 * msqrt:                                 BSD Compatible Functions.  
 * msub:                                  BSD Compatible Functions.  
 * mtox:                                  BSD Compatible Functions.  
 * mult:                                  BSD Compatible Functions.  
 * pow:                                   BSD Compatible Functions.  
 * reallocate_function:                   Custom Allocation.  
 * rpow:                                  BSD Compatible Functions.  
 * sdiv:                                  BSD Compatible Functions.  
 * xtom:                                  BSD Compatible Functions.  
   
   
   File: gmp.info,  Node: Random State Initialization,  Next: Random State Seeding,  Prev: Random Number Functions,  Up: Random Number Functions
   
   Random State Initialization
   ===========================
   
    - Function: void gmp_randinit_default (gmp_randstate_t STATE)
        Initialize STATE with a default algorithm.  This will be a
        compromise between speed and randomness, and is recommended for
        applications with no special requirements.
   
    - Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t A,
             unsigned long C, unsigned long M2EXP)
        Initialize STATE with a linear congruential algorithm X = (A*X +
        C) mod 2^M2EXP.
   
        The low bits of X in this algorithm are not very random.  The least
        significant bit will have a period no more than 2, and the second
        bit no more than 4, etc.  For this reason only the high half of
        each X is actually used.
   
        When a random number of more than M2EXP/2 bits is to be generated,
        multiple iterations of the recurrence are used and the results
        concatenated.
   
    - Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
             unsigned long SIZE)
        Initialize STATE for a linear congruential algorithm as per
        `gmp_randinit_lc_2exp'.  A, C and M2EXP are selected from a table,
        chosen so that SIZE bits (or more) of each X will be used, ie.
        M2EXP/2 >= SIZE.
   
        If successful the return value is non-zero.  If SIZE is bigger
        than the table data provides then the return value is zero.  The
        maximum SIZE currently supported is 128.
   
    - Function: void gmp_randinit (gmp_randstate_t STATE,
             gmp_randalg_t ALG, ...)
        *This function is obsolete.*
   
        Initialize STATE with an algorithm selected by ALG.  The only
        choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size'.
        A third parameter of type `unsigned long' is required, this is the
        SIZE for that function.  `GMP_RAND_ALG_DEFAULT' or 0 are the same
        as `GMP_RAND_ALG_LC'.
   
        `gmp_randinit' sets bits in `gmp_errno' to indicate an error.
        `GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is unsupported, or
        `GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter is too big.
   
    - Function: void gmp_randclear (gmp_randstate_t STATE)
        Free all memory occupied by STATE.
   
   
   File: gmp.info,  Node: Random State Seeding,  Prev: Random State Initialization,  Up: Random Number Functions
   
   Random State Seeding
   ====================
   
    - Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
    - Function: void gmp_randseed_ui (gmp_randstate_t STATE,
             unsigned long int SEED)
        Set an initial seed value into STATE.
   
        The size of a seed determines how many different sequences of
        random numbers that it's possible to generate.  The "quality" of
        the seed is the randomness of a given seed compared to the
        previous seed used, and this affects the randomness of separate
        number sequences.  The method for choosing a seed is critical if
        the generated numbers are to be used for important applications,
        such as generating cryptographic keys.
   
        Traditionally the system time has been used to seed, but care
        needs to be taken with this.  If an application seeds often and
        the resolution of the system clock is low, then the same sequence
        of numbers might be repeated.  Also, the system time is quite easy
        to guess, so if unpredictability is required then it should
        definitely not be the only source for the seed value.  On some
        systems there's a special device `/dev/random' which provides
        random data better suited for use as a seed.
   
   
   File: gmp.info,  Node: Formatted Output,  Next: Formatted Input,  Prev: Random Number Functions,  Up: Top
   
   Formatted Output
   ****************
   
   * Menu:
   
   * Formatted Output Strings::
   * Formatted Output Functions::
   * C++ Formatted Output::
   
   
   File: gmp.info,  Node: Formatted Output Strings,  Next: Formatted Output Functions,  Prev: Formatted Output,  Up: Formatted Output
   
   Format Strings
   ==============
   
      `gmp_printf' and friends accept format strings similar to the
   standard C `printf' (*note Formatted Output: (libc)Formatted Output.).
   A format specification is of the form
   
        % [flags] [width] [.[precision]] [type] conv
   
      GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
   respectively, and `N' for an `mp_limb_t' array.  `Z', `Q' and `N'
   behave like integers.  `Q' will print a `/' and a denominator, if
   needed.  `F' behaves like a float.  For example,
   
        mpz_t z;
        gmp_printf ("%s is an mpz %Zd\n", "here", z);
   
        mpq_t q;
        gmp_printf ("a hex rational: %#40Qx\n", q);
   
        mpf_t f;
        int   n;
        gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
   
        const mp_limb_t *ptr;
        mp_size_t       size;
        gmp_printf ("limb array %Nx\n", ptr, size);
   
      For `N' the limbs are expected least significant first, as per the
   `mpn' functions (*note Low-level Functions::).  A negative size can be
   given to print the value as a negative.
   
      All the standard C `printf' types behave the same as the C library
   `printf', and can be freely intermixed with the GMP extensions.  In the
   current implementation the standard parts of the format string are
   simply handed to `printf' and only the GMP extensions handled directly.
   
      The flags accepted are as follows.  GLIBC style ' is only for the
   standard C types (not the GMP types), and only if the C library
   supports it.
   
        0         pad with zeros (rather than spaces)
        #         show the base with `0x', `0X' or `0'
        +         always show a sign
        (space)   show a space or a `-' sign
        '         group digits, GLIBC style (not GMP types)
   
      The optional width and precision can be given as a number within the
   format string, or as a `*' to take an extra parameter of type `int', the
   same as the standard `printf'.
   
      The standard types accepted are as follows.  `h' and `l' are
   portable, the rest will depend on the compiler (or include files) for
   the type and the C library for the output.
   
        h         short
        hh        char
        j         intmax_t or uintmax_t
        l         long or wchar_t
        ll        long long
        L         long double
        q         quad_t or u_quad_t
        t         ptrdiff_t
        z         size_t
   
   The GMP types are
   
        F         mpf_t, float conversions
        Q         mpq_t, integer conversions
        N         mp_limb_t array, integer conversions
        Z         mpz_t, integer conversions
   
      The conversions accepted are as follows.  `a' and `A' are always
   supported for `mpf_t' but depend on the C library for standard C float
   types.  `m' and `p' depend on the C library.
   
        a A       hex floats, C99 style
        c         character
        d         decimal integer
        e E       scientific format float
        f         fixed point float
        i         same as d
        g G       fixed or scientific float
        m         `strerror' string, GLIBC style
        n         store characters written so far
        o         octal integer
        p         pointer
        s         string
        u         unsigned integer
        x X       hex integer
   
      `o', `x' and `X' are unsigned for the standard C types, but for
   types `Z', `Q' and `N' they are signed.  `u' is not meaningful for `Z',
   `Q' and `N'.
   
      `n' can be used with any type, even the GMP types.
   
      Other types or conversions that might be accepted by the C library
   `printf' cannot be used through `gmp_printf', this includes for
   instance extensions registered with GLIBC `register_printf_function'.
   Also currently there's no support for POSIX `$' style numbered arguments
   (perhaps this will be added in the future).
   
      The precision field has it's usual meaning for integer `Z' and float
   `F' types, but is currently undefined for `Q' and should not be used
   with that.
   
      `mpf_t' conversions only ever generate as many digits as can be
   accurately represented by the operand, the same as `mpf_get_str' does.
   Zeros will be used if necessary to pad to the requested precision.  This
   happens even for an `f' conversion of an `mpf_t' which is an integer,
   for instance 2^1024 in an `mpf_t' of 128 bits precision will only
   produce about 40 digits, then pad with zeros to the decimal point.  An
   empty precision field like `%.Fe' or `%.Ff' can be used to specifically
   request just the significant digits.
   
      The decimal point character (or string) is taken from the current
   locale settings on systems which provide `localeconv' (*note Locales
   and Internationalization: (libc)Locales.).  The C library will normally
   do the same for standard float output.
   
      The format string is only interpreted as plain `char's, multibyte
   characters are not recognised.  Perhaps this will change in the future.
   
   
   File: gmp.info,  Node: Formatted Output Functions,  Next: C++ Formatted Output,  Prev: Formatted Output Strings,  Up: Formatted Output
   
   Functions
   =========
   
      Each of the following functions is similar to the corresponding C
   library function.  The basic `printf' forms take a variable argument
   list.  The `vprintf' forms take an argument pointer, see *Note Variadic
   Functions: (libc)Variadic Functions, or `man 3 va_start'.
   
      It should be emphasised that if a format string is invalid, or the
   arguments don't match what the format specifies, then the behaviour of
   any of these functions will be unpredictable.  GCC format string
   checking is not available, since it doesn't recognise the GMP
   extensions.
   
      The file based functions `gmp_printf' and `gmp_fprintf' will return
   -1 to indicate a write error.  All the functions can return -1 if the C
   library `printf' variant in use returns -1, but this shouldn't normally
   occur.
   
    - Function: int gmp_printf (const char *FMT, ...)
    - Function: int gmp_vprintf (const char *FMT, va_list AP)
        Print to the standard output `stdout'.  Return the number of
        characters written, or -1 if an error occurred.
   
    - Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
    - Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
        Print to the stream FP.  Return the number of characters written,
        or -1 if an error occurred.
   
    - Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
    - Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
        Form a null-terminated string in BUF.  Return the number of
        characters written, excluding the terminating null.
   
        No overlap is permitted between the space at BUF and the string
        FMT.
   
        These functions are not recommended, since there's no protection
        against exceeding the space available at BUF.
   
    - Function: int gmp_snprintf (char *BUF, size_t SIZE, const char *FMT,
             ...)
    - Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
             *FMT, va_list AP)
        Form a null-terminated string in BUF.  No more than SIZE bytes
        will be written.  To get the full output, SIZE must be enough for
        the string and null-terminator.
   
        The return value is the total number of characters which ought to
        have been produced, excluding the terminating null.  If RETVAL >=
        SIZE then the actual output has been truncated to the first SIZE-1
        characters, and a null appended.
   
        No overlap is permitted between the region {BUF,SIZE} and the FMT
        string.
   
        Notice the return value is in ISO C99 `snprintf' style.  This is
        so even if the C library `vsnprintf' is the older GLIBC 2.0.x
        style.
   
    - Function: int gmp_asprintf (char **PP, const char *FMT, ...)
    - Function: int gmp_vasprintf (char *PP, const char *FMT, va_list AP)
        Form a null-terminated string in a block of memory obtained from
        the current memory allocation function (*note Custom
        Allocation::).  The block will be the size of the string and
        null-terminator.  Put the address of the block in *PP.  Return the
        number of characters produced, excluding the null-terminator.
   
        Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1
        if there's no more memory available, it lets the current allocation
        function handle that.
   
    - Function: int gmp_obstack_printf (struct obstack *OB, const char
             *FMT, ...)
    - Function: int gmp_obstack_vprintf (struct obstack *OB, const char
             *FMT, va_list AP)
        Append to the current obstack object, in the same style as
        `obstack_printf'.  Return the number of characters written.  A
        null-terminator is not written.
   
        FMT cannot be within the current obstack object, since the object
        might move as it grows.
   
        These functions are available only when the C library provides the
        obstack feature, which probably means only on GNU systems, see
        *Note Obstacks: (libc)Obstacks.
   
   
   File: gmp.info,  Node: C++ Formatted Output,  Prev: Formatted Output Functions,  Up: Formatted Output
   
   C++ Formatted Output
   ====================
   
      The following functions are provided in `libgmpxx', which is built
   if C++ support is enabled (*note Build Options::).  Prototypes are
   available from `<gmp.h>'.
   
    - Function: ostream& operator<< (ostream& STREAM, mpz_t OP)
        Print OP to STREAM, using its `ios' formatting settings.
        `ios::width' is reset to 0 after output, the same as the standard
        `ostream operator<<' routines do.
   
        In hex or octal, OP is printed as a signed number, the same as for
        decimal.  This is unlike the standard `operator<<' routines on
        `int' etc, which instead give twos complement.
   
    - Function: ostream& operator<< (ostream& STREAM, mpq_t OP)
        Print OP to STREAM, using its `ios' formatting settings.
        `ios::width' is reset to 0 after output, the same as the standard
        `ostream operator<<' routines do.
   
        Output will be a fraction like `5/9', or if the denominator is 1
        then just a plain integer like `123'.
   
        In hex or octal, OP is printed as a signed value, the same as for
        decimal.  If `ios::showbase' is set then a base indicator is shown
        on both the numerator and denominator (if the denominator is
        required).
   
    - Function: ostream& operator<< (ostream& STREAM, mpf_t OP)
        Print OP to STREAM, using its `ios' formatting settings.
        `ios::width' is reset to 0 after output, the same as the standard
        `ostream operator<<' routines do.  The decimal point follows the
        current locale, on systems providing `localeconv'.
   
        Hex and octal are supported, unlike the standard `operator<<' on
        `double'.  The mantissa will be in hex or octal, the exponent will
        be in decimal.  For hex the exponent delimiter is an `@'.  This is
        as per `mpf_out_str'.
   
        `ios::showbase' is supported, and will put a base on the mantissa,
        for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
        This last form is slightly strange, but at least differentiates
        itself from decimal.
   
      These operators mean that GMP types can be printed in the usual C++
   way, for example,
   
        mpz_t  z;
        int    n;
        ...
        cout << "iteration " << n << " value " << z << "\n";
   
      But note that `ostream' output (and `istream' input, *note C++
   Formatted Input::) is the only overloading available and using for
   instance `+' with an `mpz_t' will have unpredictable results.
   
   
   File: gmp.info,  Node: Formatted Input,  Next: C++ Class Interface,  Prev: Formatted Output,  Up: Top
   
   Formatted Input
   ***************
   
   * Menu:
   
   * Formatted Input Strings::
   * Formatted Input Functions::
   * C++ Formatted Input::
   
   
   File: gmp.info,  Node: Formatted Input Strings,  Next: Formatted Input Functions,  Prev: Formatted Input,  Up: Formatted Input
   
   Formatted Input Strings
   =======================
   
      `gmp_scanf' and friends accept format strings similar to the
   standard C `scanf' (*note Formatted Input: (libc)Formatted Input.).  A
   format specification is of the form
   
        % [flags] [width] [type] conv
   
      GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
   respectively.  `Z' and `Q' behave like integers.  `Q' will read a `/'
   and a denominator, if present.  `F' behaves like a float.
   
      GMP variables don't require an `&' when passed to `gmp_scanf', since
   they're already "call-by-reference".  For example,
   
        /* to read say "a(5) = 1234" */
        int   n;
        mpz_t z;
        gmp_scanf ("a(%d) = %Zd\n", &n, z);
   
        mpq_t q1, q2;
        gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
   
        /* to read say "topleft (1.55,-2.66)" */
        mpf_t x, y;
        char  buf[32];
        gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
   
      All the standard C `scanf' types behave the same as in the C library
   `scanf', and can be freely intermixed with the GMP extensions.  In the
   current implementation the standard parts of the format string are
   simply handed to `scanf' and only the GMP extensions handled directly.
   
      The flags accepted are as follows.  `a' and `'' will depend on
   support from the C library, and `'' cannot be used with GMP types.
   
        *         read but don't store
        a         allocate a buffer (string conversions)
        '         group digits, GLIBC style (not GMP types)
   
      The standard types accepted are as follows.  `h' and `l' are
   portable, the rest will depend on the compiler (or include files) for
   the type and the C library for the input.
   
        h         short
        hh        char
        j         intmax_t or uintmax_t
        l         long int, double or wchar_t
        ll        long long
        L         long double
        q         quad_t or u_quad_t
        t         ptrdiff_t
        z         size_t
   
   The GMP types are
   
        F         mpf_t, float conversions
        Q         mpq_t, integer conversions
        Z         mpz_t, integer conversions
   
      The conversions accepted are as follows.  `p' and `[' will depend on
   support from the C library, the rest are standard.
   
        c         character or characters
        d         decimal integer
        e E f g G float
        i         integer with base indicator
        n         characters read so far
        o         octal integer
        p         pointer
        s         string of non-whitespace characters
        u         decimal integer
        x X       hex integer
        [         string of characters in a set
   
      `e', `E', `f', `g' and `G' are identical, they all read either fixed
   point or scientific format, and either `e' or `E' for the exponent in
   scientific format.
   
      `x' and `X' are identical, both accept both upper and lower case
   hexadecimal.
   
      `o', `u', `x' and `X' all read positive or negative values.  For the
   standard C types these are described as "unsigned" conversions, but
   that merely affects certain overflow handling, negatives are still
   allowed (see `strtoul', *Note Parsing of Integers: (libc)Parsing of
   Integers).  For GMP types there are no overflows, and `d' and `u' are
   identical.
   
      `Q' type reads the numerator and (optional) denominator as given.
   If the value might not be in canonical form then `mpq_canonicalize'
   must be called before using it in any calculations (*note Rational
   Number Functions::).
   
      `Qi' will read a base specification separately for the numerator and
   denominator.  For example `0x10/11' would be 16/11, whereas `0x10/0x11'
   would be 16/17.
   
      `n' can be used with any of the types above, even the GMP types.
   `*' to suppress assignment is allowed, though the field would then do
   nothing at all.
   
      Other conversions or types that might be accepted by the C library
   `scanf' cannot be used through `gmp_scanf'.
   
      Whitespace is read and discarded before a field, except for `c' and
   `[' conversions.
   
      For float conversions, the decimal point character (or string)
   expected is taken from the current locale settings on systems which
   provide `localeconv' (*note Locales and Internationalization:
   (libc)Locales.).  The C library will normally do the same for standard
   float input.
   
      The format string is only interpreted as plain `char's, multibyte
   characters are not recognised.  Perhaps this will change in the future.
   
   
   File: gmp.info,  Node: Formatted Input Functions,  Next: C++ Formatted Input,  Prev: Formatted Input Strings,  Up: Formatted Input
   
   Formatted Input Functions
   =========================
   
      Each of the following functions is similar to the corresponding C
   library function.  The plain `scanf' forms take a variable argument
   list.  The `vscanf' forms take an argument pointer, see *Note Variadic
   Functions: (libc)Variadic Functions, or `man 3 va_start'.
   
      It should be emphasised that if a format string is invalid, or the
   arguments don't match what the format specifies, then the behaviour of
   any of these functions will be unpredictable.  GCC format string
   checking is not available, since it doesn't recognise the GMP
   extensions.
   
      No overlap is permitted between the FMT string and any of the results
   produced.
   
    - Function: int gmp_scanf (const char *FMT, ...)
    - Function: int gmp_vscanf (const char *FMT, va_list AP)
        Read from the standard input `stdin'.
   
    - Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
    - Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
        Read from the stream FP.
   
    - Function: int gmp_sscanf (const char *S, const char *FMT, ...)
    - Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
             AP)
        Read from a null-terminated string S.
   
      The return value from each of these functions is the same as the
   standard C99 `scanf', namely the number of fields successfully parsed
   and stored.  `%n' fields and fields read but suppressed by `*' don't
   count towards the return value.
   
      If end of file or file error, or end of string, is reached when a
   match is required, and when no previous non-suppressed fields have
   matched, then the return value is EOF instead of 0.  A match is
   required for a literal character in the format string or a field other
   than `%n'.  Whitespace in the format string is only an optional match
   and won't induce an EOF in this fashion.  Leading whitespace read and
   discarded for a field doesn't count as a match.
   
   
   File: gmp.info,  Node: C++ Formatted Input,  Prev: Formatted Input Functions,  Up: Formatted Input
   
   C++ Formatted Input
   ===================
   
      The following functions are provided in `libgmpxx', which is built
   only if C++ support is enabled (*note Build Options::).  Prototypes are
   available from `<gmp.h>'.
   
    - Function: istream& operator>> (istream& STREAM, mpz_t ROP)
        Read ROP from STREAM, using its `ios' formatting settings.
   
    - Function: istream& operator>> (istream& STREAM, mpq_t ROP)
        Read ROP from STREAM, using its `ios' formatting settings.
   
        An integer like `123' will be read, or a fraction like `5/9'.  If
        the fraction is not in canonical form then `mpq_canonicalize' must
        be called (*note Rational Number Functions::).
   
    - Function: istream& operator>> (istream& STREAM, mpf_t ROP)
        Read ROP from STREAM, using its `ios' formatting settings.
   
        Hex or octal floats are not supported, but might be in the future.
   
      These operators mean that GMP types can be read in the usual C++
   way, for example,
   
        mpz_t  z;
        ...
        cin >> z;
   
      But note that `istream' input (and `ostream' output, *note C++
   Formatted Output::) is the only overloading available and using for
   instance `+' with an `mpz_t' will have unpredictable results.
   
   
   File: gmp.info,  Node: C++ Class Interface,  Next: BSD Compatible Functions,  Prev: Formatted Input,  Up: Top
   
   C++ Class Interface
   *******************
   
      This chapter describes the C++ class based interface to GMP.
   
      All GMP C language types and functions can be used in C++ programs,
   since `gmp.h' has `extern "C"' qualifiers, but the class interface
   offers overloaded functions and operators which may be more convenient.
   
      Due to the implementation of this interface, a reasonably recent C++
   compiler is required, one supporting namespaces, partial specialization
   of templates and member templates.  For GCC this means version 2.91 or
   later.
   
      *Everything described in this chapter is to be considered preliminary
   and might be subject to incompatible changes if some unforeseen
   difficulty reveals itself.*
   
   * Menu:
   
   * C++ Interface General::
   * C++ Interface Integers::
   * C++ Interface Rationals::
   * C++ Interface Floats::
   * C++ Interface MPFR::
   * C++ Interface Random Numbers::
   * C++ Interface Limitations::
   
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