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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: Parameter Conventions,  Next: Memory Management,  Prev: Variable Conventions,  Up: GMP Basics
   
    Copyright (C) 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000  Parameter Conventions
 Free Software Foundation, Inc.  =====================
   
    Permission is granted to make and distribute verbatim copies of this     When a GMP variable is used as a function parameter, it's
 manual provided the copyright notice and this permission notice are  effectively a call-by-reference, meaning if the function stores a value
 preserved on all copies.  there it will change the original in the caller.  Parameters which are
   input-only can be designated `const' to provoke a compiler error or
   warning on attempting to modify them.
   
    Permission is granted to copy and distribute modified versions of     When a function is going to return a GMP result, it should designate
 this manual under the conditions for verbatim copying, provided that  a parameter that it sets, like the library functions do.  More than one
 the entire resulting derived work is distributed under the terms of a  value can be returned by having more than one output parameter, again
 permission notice identical to this one.  like the library functions.  A `return' of an `mpz_t' etc doesn't
   return the object, only a pointer, and this is almost certainly not
   what's wanted.
   
    Permission is granted to copy and distribute translations of this     Here's an example accepting an `mpz_t' parameter, doing a
 manual into another language, under the above conditions for modified  calculation, and storing the result to the indicated parameter.
 versions, except that this permission notice may be stated in a  
 translation approved by the Foundation.  
   
        void
 File: gmp.info,  Node: Integer Exponentiation,  Next: Integer Roots,  Prev: Integer Division,  Up: Integer Functions       foo (mpz_t result, const mpz_t param, unsigned long n)
        {
          unsigned long  i;
          mpz_mul_ui (result, param, n);
          for (i = 1; i < n; i++)
            mpz_add_ui (result, result, i*7);
        }
   
        int
        main (void)
        {
          mpz_t  r, n;
          mpz_init (r);
          mpz_init_set_str (n, "123456", 0);
          foo (r, n, 20L);
          gmp_printf ("%Zd\n", r);
          return 0;
        }
   
 Exponentiation Functions     `foo' works even if the mainline passes the same variable for
 ========================  `param' and `result', just like the library functions.  But sometimes
   it's tricky to make that work, and an application might not want to
   bother supporting that sort of thing.
   
  - Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t MOD)     For interest, the GMP types `mpz_t' etc are implemented as
  - Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long int  one-element arrays of certain structures.  This is why declaring a
           EXP, mpz_t MOD)  variable creates an object with the fields GMP needs, but then using it
      Set ROP to (BASE raised to EXP) `mod' MOD.  If EXP is negative,  as a parameter passes a pointer to the object.  Note that the actual
      the result is undefined.  fields in each `mpz_t' etc are for internal use only and should not be
   accessed directly by code that expects to be compatible with future GMP
   releases.
   
   
   File: gmp.info,  Node: Memory Management,  Next: Reentrancy,  Prev: Parameter Conventions,  Up: GMP Basics
   
  - Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int  Memory Management
           EXP)  =================
  - Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,  
           unsigned long int EXP)  
      Set ROP to BASE raised to EXP.  The case of 0^0 yields 1.  
   
      The GMP types like `mpz_t' are small, containing only a couple of
 File: gmp.info,  Node: Integer Roots,  Next: Number Theoretic Functions,  Prev: Integer Exponentiation,  Up: Integer Functions  sizes, and pointers to allocated data.  Once a variable is initialized,
   GMP takes care of all space allocation.  Additional space is allocated
   whenever a variable doesn't have enough.
   
 Root Extraction Functions     `mpz_t' and `mpq_t' variables never reduce their allocated space.
 =========================  Normally this is the best policy, since it avoids frequent reallocation.
   Applications that need to return memory to the heap at some particular
   point can use `mpz_realloc2', or clear variables no longer needed.
   
  - Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)     `mpf_t' variables, in the current implementation, use a fixed amount
      Set ROP to the truncated integer part of the Nth root of OP.  of space, determined by the chosen precision and allocated at
      Return non-zero if the computation was exact, i.e., if OP is ROP  initialization, so their size doesn't change.
      to the Nth power.  
   
  - Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)     All memory is allocated using `malloc' and friends by default, but
      Set ROP to the truncated integer part of the square root of OP.  this can be changed, see *Note Custom Allocation::.  Temporary memory
   on the stack is also used (via `alloca'), but this can be changed at
   build-time if desired, see *Note Build Options::.
   
  - Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)  
      Set ROP1 to the truncated integer part of the square root of OP,  File: gmp.info,  Node: Reentrancy,  Next: Useful Macros and Constants,  Prev: Memory Management,  Up: GMP Basics
      like `mpz_sqrt'.  Set ROP2 to OP-ROP1*ROP1, (i.e., zero if OP is a  
      perfect square).  
   
      If ROP1 and ROP2 are the same variable, the results are undefined.  Reentrancy
   ==========
   
  - Function: int mpz_perfect_power_p (mpz_t OP)     GMP is reentrant and thread-safe, with some exceptions:
      Return non-zero if OP is a perfect power, i.e., if there exist  
      integers A and B, with B > 1, such that OP equals a raised to b.  
      Return zero otherwise.  
   
  - Function: int mpz_perfect_square_p (mpz_t OP)     * If configured with `--enable-alloca=malloc-notreentrant' (or with
      Return non-zero if OP is a perfect square, i.e., if the square       `--enable-alloca=notreentrant' when `alloca' is not available),
      root of OP is an integer.  Return zero otherwise.       then naturally GMP is not reentrant.
   
      * `mpf_set_default_prec' and `mpf_init' use a global variable for the
 File: gmp.info,  Node: Number Theoretic Functions,  Next: Integer Comparisons,  Prev: Integer Roots,  Up: Integer Functions       selected precision.  `mpf_init2' can be used instead.
   
 Number Theoretic Functions     * `mpz_random' and the other old random number functions use a global
 ==========================       random state and are hence not reentrant.  The newer random number
        functions that accept a `gmp_randstate_t' parameter can be used
        instead.
   
  - Function: int mpz_probab_prime_p (mpz_t N, int REPS)     * `mp_set_memory_functions' uses global variables to store the
      If this function returns 0, N is definitely not prime.  If it       selected memory allocation functions.
      returns 1, then N is `probably' prime.  If it returns 2, then N is  
      surely prime.  Reasonable values of reps vary from 5 to 10; a  
      higher value lowers the probability for a non-prime to pass as a  
      `probable' prime.  
   
      The function uses Miller-Rabin's probabilistic test.     * If the memory allocation functions set by a call to
        `mp_set_memory_functions' (or `malloc' and friends by default) are
        not reentrant, then GMP will not be reentrant either.
   
  - Function: int mpz_nextprime (mpz_t ROP, mpz_t OP)     * If the standard I/O functions such as `fwrite' are not reentrant
      Set ROP to the next prime greater than OP.       then the GMP I/O functions using them will not be reentrant either.
   
      This function uses a probabilistic algorithm to identify primes,     * It's safe for two threads to read from the same GMP variable
      but for for practical purposes it's adequate, since the chance of       simultaneously, but it's not safe for one to read while the
      a composite passing will be extremely small.       another might be writing, nor for two threads to write
        simultaneously.  It's not safe for two threads to generate a
        random number from the same `gmp_randstate_t' simultaneously,
        since this involves an update of that variable.
   
  - Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)     * On SCO systems the default `<ctype.h>' macros use per-file static
      Set ROP to the greatest common divisor of OP1 and OP2.  The result       variables and may not be reentrant, depending whether the compiler
      is always positive even if either of or both input operands are       optimizes away fetches from them.  The GMP text-based input
      negative.       functions are affected.
   
  - Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,  
           unsigned long int OP2)  File: gmp.info,  Node: Useful Macros and Constants,  Next: Compatibility with older versions,  Prev: Reentrancy,  Up: GMP Basics
      Compute the greatest common divisor of OP1 and OP2.  If ROP is not  
      `NULL', store the result there.  
   
      If the result is small enough to fit in an `unsigned long int', it  Useful Macros and Constants
      is returned.  If the result does not fit, 0 is returned, and the  ===========================
      result is equal to the argument OP1.  Note that the result will  
      always fit if OP2 is non-zero.  
   
  - Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A, mpz_t   - Global Constant: const int mp_bits_per_limb
           B)       The number of bits per limb.
      Compute G, S, and T, such that AS + BT = G = `gcd'(A, B).  If T is  
      `NULL', that argument is not computed.  
   
  - Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)   - Macro: __GNU_MP_VERSION
      Set ROP to the least common multiple of OP1 and OP2.   - Macro: __GNU_MP_VERSION_MINOR
    - Macro: __GNU_MP_VERSION_PATCHLEVEL
        The major and minor GMP version, and patch level, respectively, as
        integers.  For GMP i.j, these numbers will be i, j, and 0,
        respectively.  For GMP i.j.k, these numbers will be i, j, and k,
        respectively.
   
  - Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)   - Global Constant: const char * const gmp_version
      Compute the inverse of OP1 modulo OP2 and put the result in ROP.       The GMP version number, as a null-terminated string, in the form
      Return non-zero if an inverse exists, zero otherwise.  When the       "i.j" or "i.j.k".  This release is "4.1.2".
      function returns zero, ROP is undefined.  
   
  - Function: int mpz_jacobi (mpz_t OP1, mpz_t OP2)  
  - Function: int mpz_legendre (mpz_t OP1, mpz_t OP2)  File: gmp.info,  Node: Compatibility with older versions,  Next: Demonstration Programs,  Prev: Useful Macros and Constants,  Up: GMP Basics
      Compute the Jacobi and Legendre symbols, respectively.  OP2 should  
      be odd and must be positive.  
   
  - Function: int mpz_si_kronecker (long A, mpz_t B);  Compatibility with older versions
  - Function: int mpz_ui_kronecker (unsigned long A, mpz_t B);  =================================
  - Function: int mpz_kronecker_si (mpz_t A, long B);  
  - Function: int mpz_kronecker_ui (mpz_t A, unsigned long B);  
      Calculate the value of the Kronecker/Jacobi symbol (A/B), with the  
      Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.  
      All values of A and B give a well-defined result.  See Henri  
      Cohen, section 1.4.2, for more information (*note References::).  
      See also the example program `demos/qcn.c' which uses  
      `mpz_kronecker_ui'.  
   
  - Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)     This version of GMP is upwardly binary compatible with all 4.x and
      Remove all occurrences of the factor F from OP and store the  3.x versions, and upwardly compatible at the source level with all 2.x
      result in ROP.  Return the multiplicity of F in OP.  versions, with the following exceptions.
   
  - Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)     * `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
      Set ROP to OP!, the factorial of OP.       consistency with other `mpn' functions.
   
  - Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)     * `mpf_get_prec' counted precision slightly differently in GMP 3.0
  - Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,       and 3.0.1, but in 3.1 reverted to the 2.x style.
           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: void mpz_fib_ui (mpz_t ROP, unsigned long int N)     There are a number of compatibility issues between GMP 1 and GMP 2
      Compute the Nth Fibonacci number and store the result in ROP.  that of course also apply when porting applications from GMP 1 to GMP
   4.  Please see the GMP 2 manual for details.
   
      The Berkeley MP compatibility library (*note BSD Compatible
   Functions::) is source and binary compatible with the standard `libmp'.
   
   
 File: gmp.info,  Node: Integer Comparisons,  Next: Integer Logic and Bit Fiddling,  Prev: Number Theoretic Functions,  Up: Integer Functions  File: gmp.info,  Node: Demonstration Programs,  Next: Efficiency,  Prev: Compatibility with older versions,  Up: GMP Basics
   
 Comparison Functions  Demonstration programs
 ====================  ======================
   
  - Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)     The `demos' subdirectory has some sample programs using GMP.  These
      Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero  aren't built or installed, but there's a `Makefile' with rules for them.
      if OP1 = OP2, and a negative value if OP1 < OP2.  For instance,
   
  - Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)       make pexpr
  - Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)       ./pexpr 68^975+10
      Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero  
      if OP1 = OP2, and a negative value if OP1 < OP2.  
   
      These functions are actually implemented as macros.  They evaluate  The following programs are provided
      their arguments multiple times.  
   
  - Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)     * `pexpr' is an expression evaluator, the program used on the GMP
  - Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)       web page.
      Compare the absolute values of OP1 and OP2.  Return a positive  
      value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1  
      < OP2.  
   
  - Macro: int mpz_sgn (mpz_t OP)     * The `calc' subdirectory has a similar but simpler evaluator using
      Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.       `lex' and `yacc'.
   
      This function is actually implemented as a macro.  It evaluates its     * The `expr' subdirectory is yet another expression evaluator, a
      arguments multiple times.       library designed for ease of use within a C program.  See
        `demos/expr/README' for more information.
   
      * `factorize' is a Pollard-Rho factorization program.
 File: gmp.info,  Node: Integer Logic and Bit Fiddling,  Next: I/O of Integers,  Prev: Integer Comparisons,  Up: Integer Functions  
   
 Logical and Bit Manipulation Functions     * `isprime' is a command-line interface to the `mpz_probab_prime_p'
 ======================================       function.
   
    These functions behave as if two's complement arithmetic were used     * `primes' counts or lists primes in an interval, using a sieve.
 (although sign-magnitude is used by the actual implementation).  
   
  - Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)     * `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic
      Set ROP to OP1 logical-and OP2.       class numbers.
   
  - Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)     * The `perl' subdirectory is a comprehensive perl interface to GMP.
      Set ROP to OP1 inclusive-or OP2.       See `demos/perl/INSTALL' for more information.  Documentation is
        in POD format in `demos/perl/GMP.pm'.
   
  - Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)  
      Set ROP to OP1 exclusive-or OP2.  File: gmp.info,  Node: Efficiency,  Next: Debugging,  Prev: Demonstration Programs,  Up: GMP Basics
   
  - Function: void mpz_com (mpz_t ROP, mpz_t OP)  Efficiency
      Set ROP to the one's complement of OP.  ==========
   
  - Function: unsigned long int mpz_popcount (mpz_t OP)  Small operands
      For non-negative numbers, return the population count of OP.  For       On small operands, the time for function call overheads and memory
      negative numbers, return the largest possible value (MAX_ULONG).       allocation can be significant in comparison to actual calculation.
        This is unavoidable in a general purpose variable precision
        library, although GMP attempts to be as efficient as it can on
        both large and small operands.
   
  - Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)  Static Linking
      If OP1 and OP2 are both non-negative, return the hamming distance       On some CPUs, in particular the x86s, the static `libgmp.a' should
      between the two operands.  Otherwise, return the largest possible       be used for maximum speed, since the PIC code in the shared
      value (MAX_ULONG).       `libgmp.so' will have a small overhead on each function call and
        global data address.  For many programs this will be
        insignificant, but for long calculations there's a gain to be had.
   
      It is possible to extend this function to return a useful value  Initializing and clearing
      when the operands are both negative, but the current       Avoid excessive initializing and clearing of variables, since this
      implementation returns MAX_ULONG in this case.  *Do not depend on       can be quite time consuming, especially in comparison to otherwise
      this behavior, since it will change in a future release.*       fast operations like addition.
   
  - Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int       A language interpreter might want to keep a free list or stack of
           STARTING_BIT)       initialized variables ready for use.  It should be possible to
      Scan OP, starting with bit STARTING_BIT, towards more significant       integrate something like that with a garbage collector too.
      bits, until the first clear bit is found.  Return the index of the  
      found bit.  
   
  - Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int  Reallocations
           STARTING_BIT)       An `mpz_t' or `mpq_t' variable used to hold successively increasing
      Scan OP, starting with bit STARTING_BIT, towards more significant       values will have its memory repeatedly `realloc'ed, which could be
      bits, until the first set bit is found.  Return the index of the       quite slow or could fragment memory, depending on the C library.
      found bit.       If an application can estimate the final size then `mpz_init2' or
        `mpz_realloc2' can be called to allocate the necessary space from
        the beginning (*note Initializing Integers::).
   
  - Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX)       It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
      Set bit BIT_INDEX in ROP.       is too small, since all functions will do a further reallocation
        if necessary.  Badly overestimating memory required will waste
        space though.
   
  - Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)  `2exp' functions
      Clear bit BIT_INDEX in ROP.       It's up to an application to call functions like `mpz_mul_2exp'
        when appropriate.  General purpose functions like `mpz_mul' make
        no attempt to identify powers of two or other special forms,
        because such inputs will usually be very rare and testing every
        time would be wasteful.
   
  - Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX)  `ui' and `si' functions
      Check bit BIT_INDEX in OP and return 0 or 1 accordingly.       The `ui' functions and the small number of `si' functions exist for
        convenience and should be used where applicable.  But if for
        example an `mpz_t' contains a value that fits in an `unsigned
        long' there's no need extract it and call a `ui' function, just
        use the regular `mpz' function.
   
   In-Place Operations
 File: gmp.info,  Node: I/O of Integers,  Next: Integer Random Numbers,  Prev: Integer Logic and Bit Fiddling,  Up: Integer Functions       `mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
        `mpf_neg' are fast when used for in-place operations like
        `mpz_abs(x,x)', since in the current implementation only a single
        field of `x' needs changing.  On suitable compilers (GCC for
        instance) this is inlined too.
   
 Input and Output Functions       `mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
 ==========================       from an in-place operation like `mpz_add_ui(x,x,y)', since usually
        only one or two limbs of `x' will need to be changed.  The same
        applies to the full precision `mpz_add' etc if `y' is small.  If
        `y' is big then cache locality may be helped, but that's all.
   
    Functions that perform input from a stdio stream, and functions that       `mpz_mul' is currently the opposite, a separate destination is
 output to a stdio stream.  Passing a `NULL' pointer for a STREAM       slightly better.  A call like `mpz_mul(x,x,y)' will, unless `y' is
 argument to any of these functions will make them read from `stdin' and       only one limb, make a temporary copy of `x' before forming the
 write to `stdout', respectively.       result.  Normally that copying will only be a tiny fraction of the
        time for the multiply, so this is not a particularly important
        consideration.
   
    When using any of these functions, it is a good idea to include       `mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
 `stdio.h' before `gmp.h', since that will allow `gmp.h' to define       attempt to recognise a copy of something to itself, so a call like
 prototypes for these functions.       `mpz_set(x,x)' will be wasteful.  Naturally that would never be
        written deliberately, but if it might arise from two pointers to
        the same object then a test to avoid it might be desirable.
   
  - Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)            if (x != y)
      Output OP on stdio stream STREAM, as a string of digits in base              mpz_set (x, y);
      BASE.  The base may vary from 2 to 36.  
   
      Return the number of bytes written, or if an error occurred,       Note that it's never worth introducing extra `mpz_set' calls just
      return 0.       to get in-place operations.  If a result should go to a particular
        variable then just direct it there and let GMP take care of data
        movement.
   
  - Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)  Divisibility Testing (Small Integers)
      Input a possibly white-space preceded string in base BASE from       `mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best
      stdio stream STREAM, and put the read integer in ROP.  The base       functions for testing whether an `mpz_t' is divisible by an
      may vary from 2 to 36.  If BASE is 0, the actual base is       individual small integer.  They use an algorithm which is faster
      determined from the leading characters: if the first two       than `mpz_tdiv_ui', but which gives no useful information about
      characters are `0x' or `0X', hexadecimal is assumed, otherwise if       the actual remainder, only whether it's zero (or a particular
      the first character is `0', octal is assumed, otherwise decimal is       value).
      assumed.  
   
      Return the number of bytes read, or if an error occurred, return 0.       However when testing divisibility by several small integers, it's
        best to take a remainder modulo their product, to save
        multi-precision operations.  For instance to test whether a number
        is divisible by any of 23, 29 or 31 take a remainder modulo
        23*29*31 = 20677 and then test that.
   
  - Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)       The division functions like `mpz_tdiv_q_ui' which give a quotient
      Output OP on stdio stream STREAM, in raw binary format.  The       as well as a remainder are generally a little slower than the
      integer is written in a portable format, with 4 bytes of size       remainder-only functions like `mpz_tdiv_ui'.  If the quotient is
      information, and that many bytes of limbs.  Both the size and the       only rarely wanted then it's probably best to just take a
      limbs are written in decreasing significance order (i.e., in       remainder and then go back and calculate the quotient if and when
      big-endian).       it's wanted (`mpz_divexact_ui' can be used if the remainder is
        zero).
   
      The output can be read with `mpz_inp_raw'.  Rational Arithmetic
        The `mpq' functions operate on `mpq_t' values with no common
        factors in the numerator and denominator.  Common factors are
        checked-for and cast out as necessary.  In general, cancelling
        factors every time is the best approach since it minimizes the
        sizes for subsequent operations.
   
      Return the number of bytes written, or if an error occurred,       However, applications that know something about the factorization
      return 0.       of the values they're working with might be able to avoid some of
        the GCDs used for canonicalization, or swap them for divisions.
        For example when multiplying by a prime it's enough to check for
        factors of it in the denominator instead of doing a full GCD.  Or
        when forming a big product it might be known that very little
        cancellation will be possible, and so canonicalization can be left
        to the end.
   
      The output of this can not be read by `mpz_inp_raw' from GMP 1,       The `mpq_numref' and `mpq_denref' macros give access to the
      because of changes necessary for compatibility between 32-bit and       numerator and denominator to do things outside the scope of the
      64-bit machines.       supplied `mpq' functions.  *Note Applying Integer Functions::.
   
  - Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)       The canonical form for rationals allows mixed-type `mpq_t' and
      Input from stdio stream STREAM in the format written by       integer additions or subtractions to be done directly with
      `mpz_out_raw', and put the result in ROP.  Return the number of       multiples of the denominator.  This will be somewhat faster than
      bytes read, or if an error occurred, return 0.       `mpq_add'.  For example,
   
      This routine can read the output from `mpz_out_raw' also from GMP            /* mpq increment */
      1, in spite of changes necessary for compatibility between 32-bit            mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
      and 64-bit machines.  
             /* mpq += unsigned long */
             mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
   
             /* mpq -= mpz */
             mpz_submul (mpq_numref(q), mpq_denref(q), z);
   
   Number Sequences
 File: gmp.info,  Node: Integer Random Numbers,  Next: Miscellaneous Integer Functions,  Prev: I/O of Integers,  Up: Integer Functions       Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
        designed for calculating isolated values.  If a range of values is
        wanted it's probably best to call to get a starting point and
        iterate from there.
   
 Random Number Functions  Text Input/Output
 =======================       Hexadecimal or octal are suggested for input or output in text
        form.  Power-of-2 bases like these can be converted much more
        efficiently than other bases, like decimal.  For big numbers
        there's usually nothing of particular interest to be seen in the
        digits, so the base doesn't matter much.
   
    The random number functions of GMP come in two groups; older function       Maybe we can hope octal will one day become the normal base for
 that rely on a global state, and newer functions that accept a state       everyday use, as proposed by King Charles XII of Sweden and later
 parameter that is read and modified.  Please see the *Note Random       reformers.
 Number Functions:: for more information on how to use and not to use  
 random number functions.  
   
  - Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,  
      unsigned long int N) Generate a uniformly distributed random  File: gmp.info,  Node: Debugging,  Next: Profiling,  Prev: Efficiency,  Up: GMP Basics
      integer in the range 0 to 2^N - 1, inclusive.  
   
      The variable STATE must be initialized by calling one of the  Debugging
      `gmp_randinit' functions (*Note Random State Initialization::)  =========
      before invoking this function.  
   
  - Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,  Stack Overflow
      mpz_t N) Generate a uniform random integer in the range 0 to N -       Depending on the system, a segmentation violation or bus error
      1, inclusive.       might be the only indication of stack overflow.  See
        `--enable-alloca' choices in *Note Build Options::, for how to
        address this.
   
      The variable STATE must be initialized by calling one of the       In new enough versions of GCC, `-fstack-check' may be able to
      `gmp_randinit' functions (*Note Random State Initialization::)       ensure an overflow is recognised by the system before too much
      before invoking this function.       damage is done, or `-fstack-limit-symbol' or
        `-fstack-limit-register' may be able to add checking if the system
        itself doesn't do any (*note Options for Code Generation:
        (gcc)Code Gen Options.).  These options must be added to the
        `CFLAGS' used in the GMP build (*note Build Options::), adding
        them just to an application will have no effect.  Note also
        they're a slowdown, adding overhead to each function call and each
        stack allocation.
   
  - Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,  Heap Problems
           unsigned long int N)       The most likely cause of application problems with GMP is heap
      Generate a random integer with long strings of zeros and ones in       corruption.  Failing to `init' GMP variables will have
      the binary representation.  Useful for testing functions and       unpredictable effects, and corruption arising elsewhere in a
      algorithms, since this kind of random numbers have proven to be       program may well affect GMP.  Initializing GMP variables more than
      more likely to trigger corner-case bugs.  The random number will       once or failing to clear them will cause memory leaks.
      be in the range 0 to 2^N - 1, inclusive.  
   
      The variable STATE must be initialized by calling one of the       In all such cases a malloc debugger is recommended.  On a GNU or
      `gmp_randinit' functions (*Note Random State Initialization::)       BSD system the standard C library `malloc' has some diagnostic
      before invoking this function.       facilities, see *Note Allocation Debugging: (libc)Allocation
        Debugging, or `man 3 malloc'.  Other possibilities, in no
        particular order, include
   
  - Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)            `http://www.inf.ethz.ch/personal/biere/projects/ccmalloc'
      Generate a random integer of at most MAX_SIZE limbs.  The generated            `http://quorum.tamu.edu/jon/gnu'  (debauch)
      random number doesn't satisfy any particular requirements of            `http://dmalloc.com'
      randomness.  Negative random numbers are generated when MAX_SIZE            `http://www.perens.com/FreeSoftware'  (electric fence)
      is negative.            `http://packages.debian.org/fda'
             `http://www.gnupdate.org/components/leakbug'
             `http://people.redhat.com/~otaylor/memprof'
             `http://www.cbmamiga.demon.co.uk/mpatrol'
   
      This function is obsolete.  Use `mpz_urandomb' or `mpz_urandomm'       The GMP default allocation routines in `memory.c' also have a
      instead.       simple sentinel scheme which can be enabled with `#define DEBUG'
        in that file.  This is mainly designed for detecting buffer
        overruns during GMP development, but might find other uses.
   
  - Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)  Stack Backtraces
      Generate a random integer of at most MAX_SIZE limbs, with long       On some systems the compiler options GMP uses by default can
      strings of zeros and ones in the binary representation.  Useful       interfere with debugging.  In particular on x86 and 68k systems
      for testing functions and algorithms, since this kind of random       `-fomit-frame-pointer' is used and this generally inhibits stack
      numbers have proven to be more likely to trigger corner-case bugs.       backtracing.  Recompiling without such options may help while
      Negative random numbers are generated when MAX_SIZE is negative.       debugging, though the usual caveats about it potentially moving a
        memory problem or hiding a compiler bug will apply.
   
      This function is obsolete.  Use `mpz_rrandomb' instead.  GNU Debugger
        A sample `.gdbinit' is included in the distribution, showing how
        to call some undocumented dump functions to print GMP variables
        from within GDB.  Note that these functions shouldn't be used in
        final application code since they're undocumented and may be
        subject to incompatible changes in future versions of GMP.
   
   Source File Paths
 File: gmp.info,  Node: Miscellaneous Integer Functions,  Prev: Integer Random Numbers,  Up: Integer Functions       GMP has multiple source files with the same name, in different
        directories.  For example `mpz', `mpq', `mpf' and `mpfr' each have
        an `init.c'.  If the debugger can't already determine the right
        one it may help to build with absolute paths on each C file.  One
        way to do that is to use a separate object directory with an
        absolute path to the source directory.
   
 Miscellaneous Functions            cd /my/build/dir
 =======================            /my/source/dir/gmp-4.1.2/configure
   
  - Function: int mpz_fits_ulong_p (mpz_t OP)       This works via `VPATH', and might require GNU `make'.  Alternately
  - Function: int mpz_fits_slong_p (mpz_t OP)       it might be possible to change the `.c.lo' rules appropriately.
  - 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.  
   
  - Macro: int mpz_odd_p (mpz_t OP)  Assertion Checking
  - Macro: int mpz_even_p (mpz_t OP)       The build option `--enable-assert' is available to add some
      Determine whether OP is odd or even, respectively.  Return       consistency checks to the library (see *Note Build Options::).
      non-zero if yes, zero if no.  These macros evaluate their       These are likely to be of limited value to most applications.
      arguments more than once.       Assertion failures are just as likely to indicate memory
        corruption as a library or compiler bug.
   
  - Function: size_t mpz_size (mpz_t OP)       Applications using the low-level `mpn' functions, however, will
      Return the size of OP measured in number of limbs.  If OP is zero,       benefit from `--enable-assert' since it adds checks on the
      the returned value will be zero.       parameters of most such functions, many of which have subtle
        restrictions on their usage.  Note however that only the generic C
        code has checks, not the assembler code, so CPU `none' should be
        used for maximum checking.
   
  - Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)  Temporary Memory Checking
      Return the size of OP measured in number of digits in base BASE.       The build option `--enable-alloca=debug' arranges that each block
      The base may vary from 2 to 36.  The returned value will be exact       of temporary memory in GMP is allocated with a separate call to
      or 1 too big.  If BASE is a power of 2, the returned value will       `malloc' (or the allocation function set with
      always be exact.       `mp_set_memory_functions').
   
      This function is useful in order to allocate the right amount of       This can help a malloc debugger detect accesses outside the
      space before converting OP to a string.  The right amount of       intended bounds, or detect memory not released.  In a normal
      allocation is normally two more than the value returned by       build, on the other hand, temporary memory is allocated in blocks
      `mpz_sizeinbase' (one extra for a minus sign and one for the       which GMP divides up for its own use, or may be allocated with a
      terminating '\0').       compiler builtin `alloca' which will go nowhere near any malloc
        debugger hooks.
   
   Maximum Debuggability
 File: gmp.info,  Node: Rational Number Functions,  Next: Floating-point Functions,  Prev: Integer Functions,  Up: Top       To summarize the above, a GMP build for maximum debuggability
        would be
   
 Rational Number Functions            ./configure --disable-shared --enable-assert \
 *************************              --enable-alloca=debug --host=none CFLAGS=-g
   
    This chapter describes the GMP functions for performing arithmetic       For C++, add `--enable-cxx CXXFLAGS=-g'.
 on rational numbers.  These functions start with the prefix `mpq_'.  
   
    Rational numbers are stored in objects of type `mpq_t'.  Checker
        The checker program (`http://savannah.gnu.org/projects/checker')
        can be used with GMP.  It contains a stub library which means GMP
        applications compiled with checker can use a normal GMP build.
   
    All rational arithmetic functions assume operands have a canonical       A build of GMP with checking within GMP itself can be made.  This
 form, and canonicalize their result.  The canonical from means that the       will run very very slowly.  Configure with
 denominator and the numerator have no common factors, and that the  
 denominator is positive.  Zero has the unique representation 0/1.  
   
    Pure assignment functions do not canonicalize the assigned variable.            ./configure --host=none-pc-linux-gnu CC=checkergcc
 It is the responsibility of the user to canonicalize the assigned  
 variable before any arithmetic operations are performed on that  
 variable.  *Note that this is an incompatible change from version 1 of  
 the library.*  
   
  - Function: void mpq_canonicalize (mpq_t OP)       `--host=none' must be used, since the GMP assembler code doesn't
      Remove any factors that are common to the numerator and       support the checking scheme.  The GMP C++ features cannot be used,
      denominator of OP, and make the denominator positive.       since current versions of checker (0.9.9.1) don't yet support the
        standard C++ library.
   
 * Menu:  Valgrind
        The valgrind program (`http://devel-home.kde.org/~sewardj') is a
        memory checker for x86s.  It translates and emulates machine
        instructions to do strong checks for uninitialized data (at the
        level of individual bits), memory accesses through bad pointers,
        and memory leaks.
   
 * Initializing Rationals::       Current versions (20020226 snapshot) don't support MMX or SSE, so
 * Rational Arithmetic::       GMP must be configured for an x86 without those (eg. plain
 * Comparing Rationals::       `i386'), or with a special `MPN_PATH' that excludes those
 * Applying Integer Functions::       subdirectories (*note Build Options::).
 * I/O of Rationals::  
 * Miscellaneous Rational Functions::  
   
   Other Problems
        Any suspected bug in GMP itself should be isolated to make sure
        it's not an application problem, see *Note Reporting Bugs::.
   
   
 File: gmp.info,  Node: Initializing Rationals,  Next: Rational Arithmetic,  Prev: Rational Number Functions,  Up: Rational Number Functions  File: gmp.info,  Node: Profiling,  Next: Autoconf,  Prev: Debugging,  Up: GMP Basics
   
 Initialization and Assignment Functions  Profiling
 =======================================  =========
   
  - Function: void mpq_init (mpq_t DEST_RATIONAL)     Running a program under a profiler is a good way to find where it's
      Initialize DEST_RATIONAL and set it to 0/1.  Each variable should  spending most time and where improvements can be best sought.
      normally only be initialized once, or at least cleared out (using  
      the function `mpq_clear') between each initialization.  
   
  - Function: void mpq_clear (mpq_t RATIONAL_NUMBER)     Depending on the system, it may be possible to get a flat profile,
      Free the space occupied by RATIONAL_NUMBER.  Make sure to call this  meaning simple timer sampling of the program counter, with no special
      function for all `mpq_t' variables when you are done with them.  GMP build options, just a `-p' when compiling the mainline.  This is a
   good way to ensure minimum interference with normal operation.  The
   necessary symbol type and size information exists in most of the GMP
   assembler code.
   
  - Function: void mpq_set (mpq_t ROP, mpq_t OP)     The `--enable-profiling' build option can be used to add suitable
  - Function: void mpq_set_z (mpq_t ROP, mpz_t OP)  compiler flags, either for `prof' (`-p') or `gprof' (`-pg'), see *Note
      Assign ROP from OP.  Build Options::.  Which of the two is available and what they do will
   depend on the system, and possibly on support available in `libc'.  For
   some systems appropriate corresponding `mcount' calls are added to the
   assembler code too.
   
  - Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,     On x86 systems `prof' gives call counting, so that average time spent
           unsigned long int OP2)  in a function can be determined.  `gprof', where supported, adds call
  - Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned  graph construction, so for instance calls to `mpn_add_n' from `mpz_add'
           long int OP2)  and from `mpz_mul' can be differentiated.
      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.  
   
  - Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)     On x86 and 68k systems `-pg' and `-fomit-frame-pointer' are
      Swap the values ROP1 and ROP2 efficiently.  incompatible, so the latter is not used when `gprof' profiling is
   selected, which may result in poorer code generation.  If `prof'
   profiling is selected instead it should still be possible to use
   `gprof', but only the `gprof -p' flat profile and call counts can be
   expected to be valid, not the `gprof -q' call graph.
   
   
 File: gmp.info,  Node: Rational Arithmetic,  Next: Comparing Rationals,  Prev: Initializing Rationals,  Up: Rational Number Functions  File: gmp.info,  Node: Autoconf,  Next: Emacs,  Prev: Profiling,  Up: GMP Basics
   
 Arithmetic Functions  Autoconf
 ====================  ========
   
  - Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)     Autoconf based applications can easily check whether GMP is
      Set SUM to ADDEND1 + ADDEND2.  installed.  The only thing to be noted is that GMP library symbols from
   version 3 onwards have prefixes like `__gmpz'.  The following therefore
   would be a simple test,
   
  - Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t       AC_CHECK_LIB(gmp, __gmpz_init)
           SUBTRAHEND)  
      Set DIFFERENCE to MINUEND - SUBTRAHEND.  
   
  - Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t     This just uses the default `AC_CHECK_LIB' actions for found or not
           MULTIPLICAND)  found, but an application that must have GMP would want to generate an
      Set PRODUCT to MULTIPLIER times MULTIPLICAND.  error if not found.  For example,
   
  - Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t       AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR(
           DIVISOR)       [GNU MP not found, see http://swox.com/gmp])])
      Set QUOTIENT to DIVIDEND/DIVISOR.  
   
  - Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)     If functions added in some particular version of GMP are required,
      Set NEGATED_OPERAND to -OPERAND.  then one of those can be used when checking.  For example `mpz_mul_si'
   was added in GMP 3.1,
   
  - Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)       AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR(
      Set INVERTED_NUMBER to 1/NUMBER.  If the new denominator is zero,       [GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])])
      this routine will divide by zero.  
   
      An alternative would be to test the version number in `gmp.h' using
 File: gmp.info,  Node: Comparing Rationals,  Next: Applying Integer Functions,  Prev: Rational Arithmetic,  Up: Rational Number Functions  say `AC_EGREP_CPP'.  That would make it possible to test the exact
   version, if some particular sub-minor release is known to be necessary.
   
 Comparison Functions     An application that can use either GMP 2 or 3 will need to test for
 ====================  `__gmpz_init' (GMP 3 and up) or `mpz_init' (GMP 2), and it's also worth
   checking for `libgmp2' since Debian GNU/Linux systems used that name in
   the past.  For example,
   
  - Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)       AC_CHECK_LIB(gmp, __gmpz_init, ,
      Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero         [AC_CHECK_LIB(gmp, mpz_init, ,
      if OP1 = OP2, and a negative value if OP1 < OP2.           [AC_CHECK_LIB(gmp2, mpz_init)])])
   
      To determine if two rationals are equal, `mpq_equal' is faster than     In general it's suggested that applications should simply demand a
      `mpq_cmp'.  new enough GMP rather than trying to provide supplements for features
   not available in past versions.
   
  - Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned     Occasionally an application will need or want to know the size of a
           long int DEN2)  type at configuration or preprocessing time, not just with `sizeof' in
      Compare OP1 and NUM2/DEN2.  Return a positive value if OP1 >  the code.  This can be done in the normal way with `mp_limb_t' etc, but
      NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <  GMP 4.0 or up is best for this, since prior versions needed certain
      NUM2/DEN2.  `-D' defines on systems using a `long long' limb.  The following would
   suit Autoconf 2.50 or up,
   
      This routine allows that NUM2 and DEN2 have common factors.       AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
   
      This function is actually implemented as a macro.  It evaluates its     The optional `mpfr' functions are provided in a separate
      arguments multiple times.  `libmpfr.a', and this might be from GMP with `--enable-mpfr' or from
   MPFR installed separately.  Either way `libmpfr' depends on `libgmp',
   it doesn't stand alone.  Currently only a static `libmpfr.a' will be
   available, not a shared library, since upward binary compatibility is
   not guaranteed.
   
  - Macro: int mpq_sgn (mpq_t OP)       AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR(
      Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.       [Need MPFR either from GNU MP 4 or separate MPFR package.
        See http://www.mpfr.org or http://swox.com/gmp])
   
      This function is actually implemented as a macro.  It evaluates its  
      arguments multiple times.  File: gmp.info,  Node: Emacs,  Prev: Autoconf,  Up: GMP Basics
   
  - Function: int mpq_equal (mpq_t OP1, mpq_t OP2)  Emacs
      Return non-zero if OP1 and OP2 are equal, zero if they are  =====
      non-equal.  Although `mpq_cmp' can be used for the same purpose,  
      this function is much faster.  
   
      <C-h C-i> (`info-lookup-symbol') is a good way to find documentation
 File: gmp.info,  Node: Applying Integer Functions,  Next: I/O of Rationals,  Prev: Comparing Rationals,  Up: Rational Number Functions  on C functions while editing (*note Info Documentation Lookup:
   (emacs)Info Lookup.).
   
 Applying Integer Functions to Rationals     The GMP manual can be included in such lookups by putting the
 =======================================  following in your `.emacs',
   
    The set of `mpq' functions is quite small.  In particular, there are       (eval-after-load "info-look"
 few functions for either input or output.  But there are two macros         '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
 that allow us to apply any `mpz' function on the numerator or            (setcar (nthcdr 3 mode-value)
 denominator of a rational number.  If these macros are used to assign                    (cons '("(gmp)Function Index" nil "^ -.* " "\\>")
 to the rational number, `mpq_canonicalize' normally need to be called                          (nth 3 mode-value)))))
 afterwards.  
   
  - Macro: mpz_t mpq_numref (mpq_t OP)     The same can be done for MPFR, with `(mpfr)' in place of `(gmp)'.
  - 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.  
   
   
 File: gmp.info,  Node: I/O of Rationals,  Next: Miscellaneous Rational Functions,  Prev: Applying Integer Functions,  Up: Rational Number Functions  File: gmp.info,  Node: Reporting Bugs,  Next: Integer Functions,  Prev: GMP Basics,  Up: Top
   
 Input and Output Functions  Reporting Bugs
 ==========================  **************
   
    Functions that perform input from a stdio stream, and functions that     If you think you have found a bug in the GMP library, please
 output to a stdio stream.  Passing a `NULL' pointer for a STREAM  investigate it and report it.  We have made this library available to
 argument to any of these functions will make them read from `stdin' and  you, and it is not too much to ask you to report the bugs you find.
 write to `stdout', respectively.  
   
    When using any of these functions, it is a good idea to include     Before you report a bug, check it's not already addressed in *Note
 `stdio.h' before `gmp.h', since that will allow `gmp.h' to define  Known Build Problems::, or perhaps *Note Notes for Particular
 prototypes for these functions.  Systems::.  You may also want to check `http://swox.com/gmp/' for
   patches for this release.
   
  - Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)     Please include the following in any report,
      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'.  
   
      Return the number of bytes written, or if an error occurred,     * The GMP version number, and if pre-packaged or patched then say so.
      return 0.  
   
      * A test program that makes it possible for us to reproduce the bug.
 File: gmp.info,  Node: Miscellaneous Rational Functions,  Prev: I/O of Rationals,  Up: Rational Number Functions       Include instructions on how to run the program.
   
 Miscellaneous Functions     * A description of what is wrong.  If the results are incorrect, in
 =======================       what way.  If you get a crash, say so.
   
  - Function: double mpq_get_d (mpq_t OP)     * If you get a crash, include a stack backtrace from the debugger if
      Convert OP to a double.       it's informative (`where' in `gdb', or `$C' in `adb').
   
  - Function: double mpq_set_d (mpq_t ROP, double D)     * Please do not send core dumps, executables or `strace's.
      Set ROP to the value of d, without rounding.  
   
    These functions assign between either the numerator or denominator     * The configuration options you used when building GMP, if any.
 of a rational, and an integer.  Instead of using these functions, it is  
 preferable to use the more general mechanisms `mpq_numref' and  
 `mpq_denref', together with `mpz_set'.  
   
  - Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)     * The name of the compiler and its version.  For `gcc', get the
      Copy NUMERATOR to the numerator of RATIONAL.  When this risks to       version with `gcc -v', otherwise perhaps `what `which cc`', or
      make the numerator and denominator of RATIONAL have common       similar.
      factors, you have to pass RATIONAL to `mpq_canonicalize' before  
      any operations are performed on RATIONAL.  
   
      This function is equivalent to `mpz_set (mpq_numref (RATIONAL),     * The output from running `uname -a'.
      NUMERATOR)'.  
   
  - Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)     * The output from running `./config.guess', and from running
      Copy DENOMINATOR to the denominator of RATIONAL.  When this risks       `./configfsf.guess' (might be the same).
      to make the numerator and denominator of RATIONAL have common  
      factors, or if the denominator might be negative, you have to pass  
      RATIONAL to `mpq_canonicalize' before any operations are performed  
      on RATIONAL.  
   
      *In version 1 of the library, negative denominators were handled by     * If the bug is related to `configure', then the contents of
      copying the sign to the numerator.  That is no longer done.*       `config.log'.
   
      This function is equivalent to `mpz_set (mpq_denref (RATIONAL),     * If the bug is related to an `asm' file not assembling, then the
      DENOMINATORS)'.       contents of `config.m4' and the offending line or lines from the
        temporary `mpn/tmp-<file>.s'.
   
  - Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)     Please make an effort to produce a self-contained report, with
      Copy the numerator of RATIONAL to the integer NUMERATOR, to  something definite that can be tested or debugged.  Vague queries or
      prepare for integer operations on the numerator.  piecemeal messages are difficult to act on and don't help the
   development effort.
   
      This function is equivalent to `mpz_set (NUMERATOR, mpq_numref     It is not uncommon that an observed problem is actually due to a bug
      (RATIONAL))'.  in the compiler; the GMP code tends to explore interesting corners in
   compilers.
   
  - Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)     If your bug report is good, we will do our best to help you get a
      Copy the denominator of RATIONAL to the integer DENOMINATOR, to  corrected version of the library; if the bug report is poor, we won't
      prepare for integer operations on the denominator.  do anything about it (except maybe ask you to send a better report).
   
      This function is equivalent to `mpz_set (DENOMINATOR, mpq_denref     Send your report to: <bug-gmp@gnu.org>.
      (RATIONAL))'.  
   
      If you think something in this manual is unclear, or downright
   incorrect, or if the language needs to be improved, please send a note
   to the same address.
   
   
 File: gmp.info,  Node: Floating-point Functions,  Next: Low-level Functions,  Prev: Rational Number Functions,  Up: Top  File: gmp.info,  Node: Integer Functions,  Next: Rational Number Functions,  Prev: Reporting Bugs,  Up: Top
   
 Floating-point Functions  Integer Functions
 ************************  *****************
   
    This chapter describes the GMP functions for performing floating     This chapter describes the GMP functions for performing integer
 point arithmetic.  These functions start with the prefix `mpf_'.  arithmetic.  These functions start with the prefix `mpz_'.
   
    GMP floating point numbers are stored in objects of type `mpf_t'.     GMP integers are stored in objects of type `mpz_t'.
   
    The GMP floating-point functions have an interface that is similar  
 to the GMP integer functions.  The function prefix for floating-point  
 operations is `mpf_'.  
   
    There is one significant characteristic of floating-point numbers  
 that has motivated a difference between this function class and other  
 GMP function classes: the inherent inexactness of floating point  
 arithmetic.  The user has to specify the precision of each variable.  A  
 computation that assigns a variable will take place with the precision  
 of the assigned variable; the precision of variables used as input is  
 ignored.  
   
    The precision of a calculation is defined as follows: Compute the  
 requested operation exactly (with "infinite precision"), and truncate  
 the result to the destination variable precision.  Even if the user has  
 asked for a very high precision, GMP will not calculate with  
 superfluous digits.  For example, if two low-precision numbers of  
 nearly equal magnitude are added, the precision of the result will be  
 limited to what is required to represent the result accurately.  
   
    The GMP floating-point functions are _not_ intended as a smooth  
 extension to the IEEE P754 arithmetic.  Specifically, the results  
 obtained on one computer often differs from the results obtained on a  
 computer with a different word size.  
   
 * Menu:  * Menu:
   
 * Initializing Floats::  * Initializing Integers::
 * Assigning Floats::  * Assigning Integers::
 * Simultaneous Float Init & Assign::  * Simultaneous Integer Init & Assign::
 * Converting Floats::  * Converting Integers::
 * Float Arithmetic::  * Integer Arithmetic::
 * Float Comparison::  * Integer Division::
 * I/O of Floats::  * Integer Exponentiation::
 * Miscellaneous Float Functions::  * Integer Roots::
   * Number Theoretic Functions::
   * Integer Comparisons::
   * Integer Logic and Bit Fiddling::
   * I/O of Integers::
   * Integer Random Numbers::
   * Integer Import and Export::
   * Miscellaneous Integer Functions::
   
   
 File: gmp.info,  Node: Initializing Floats,  Next: Assigning Floats,  Prev: Floating-point Functions,  Up: Floating-point Functions  File: gmp.info,  Node: Initializing Integers,  Next: Assigning Integers,  Prev: Integer Functions,  Up: Integer Functions
   
 Initialization Functions  Initialization Functions
 ========================  ========================
   
  - Function: void mpf_set_default_prec (unsigned long int PREC)     The functions for integer arithmetic assume that all integer objects
      Set the default precision to be *at least* PREC bits.  All  are initialized.  You do that by calling the function `mpz_init'.  For
      subsequent calls to `mpf_init' will use this precision, but  example,
      previously initialized variables are unaffected.  
   
    An `mpf_t' object must be initialized before storing the first value  
 in it.  The functions `mpf_init' and `mpf_init2' are used for that  
 purpose.  
   
  - Function: void mpf_init (mpf_t X)  
      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'.  
   
  - Function: void mpf_init2 (mpf_t X, unsigned long int PREC)  
      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.  
   
  - Function: void mpf_clear (mpf_t X)  
      Free the space occupied by X.  Make sure to call this function for  
      all `mpf_t' variables when you are done with them.  
   
    Here is an example on how to initialize floating-point variables:  
      {       {
        mpf_t x, y;         mpz_t integ;
        mpf_init (x);                    /* use default precision */         mpz_init (integ);
        mpf_init2 (y, 256);              /* precision _at least_ 256 bits */  
        ...         ...
          mpz_add (integ, ...);
          ...
          mpz_sub (integ, ...);
   
        /* Unless the program is about to exit, do ... */         /* Unless the program is about to exit, do ... */
        mpf_clear (x);         mpz_clear (integ);
        mpf_clear (y);  
      }       }
   
    The following three functions are useful for changing the precision     As you can see, you can store new values any number of times, once an
 during a calculation.  A typical use would be for adjusting the  object is initialized.
 precision gradually in iterative algorithms like Newton-Raphson, making  
 the computation precision closely match the actual accurate part of the  
 numbers.  
   
  - Function: void mpf_set_prec (mpf_t ROP, unsigned long int PREC)   - Function: void mpz_init (mpz_t INTEGER)
      Set the precision of ROP to be *at least* PREC bits.  Since       Initialize INTEGER, and set its value to 0.
      changing the precision involves calls to `realloc', this routine  
      should not be called in a tight loop.  
   
  - Function: unsigned long int mpf_get_prec (mpf_t OP)   - Function: void mpz_init2 (mpz_t INTEGER, unsigned long N)
      Return the precision actually used for assignments of OP.       Initialize INTEGER, with space for N bits, and set its value to 0.
   
  - Function: void mpf_set_prec_raw (mpf_t ROP, unsigned long int PREC)       N is only the initial space, INTEGER will grow automatically in
      Set the precision of ROP to be *at least* PREC bits.  This is a       the normal way, if necessary, for subsequent values stored.
      low-level function that does not change the allocation.  The PREC       `mpz_init2' makes it possible to avoid such reallocations if a
      argument must not be larger that the precision previously returned       maximum size is known in advance.
      by `mpf_get_prec'.  It is crucial that the precision of ROP is  
      ultimately reset to exactly the value returned by `mpf_get_prec'  
      before the first call to `mpf_set_prec_raw'.  
   
    - Function: void mpz_clear (mpz_t INTEGER)
        Free the space occupied by INTEGER.  Call this function for all
        `mpz_t' variables when you are done with them.
   
    - Function: void mpz_realloc2 (mpz_t INTEGER, unsigned long N)
        Change the space allocated for INTEGER to N bits.  The value in
        INTEGER is preserved if it fits, or is set to 0 if not.
   
        This function can be used to increase the space for a variable in
        order to avoid repeated automatic reallocations, or to decrease it
        to give memory back to the heap.
   
    - Function: void mpz_array_init (mpz_t INTEGER_ARRAY[], size_t
             ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
        This is a special type of initialization.  *Fixed* space of
        FIXED_NUM_BITS bits is allocated to each of the ARRAY_SIZE
        integers in INTEGER_ARRAY.
   
        The space will not be automatically increased, unlike the normal
        `mpz_init', but instead an application must ensure it's sufficient
        for any value stored.  The following space requirements apply to
        various functions,
   
           * `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and
             `mpz_set_ui' need room for the value they store.
   
           * `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room
             for the larger of the two operands, plus an extra
             `mp_bits_per_limb'.
   
           * `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum
             of the number of bits in their operands, but each rounded up
             to a multiple of `mp_bits_per_limb'.
   
           * `mpz_swap' can be used between two array variables, but not
             between an array and a normal variable.
   
        For other functions, or if in doubt, the suggestion is to
        calculate in a regular `mpz_init' variable and copy the result to
        an array variable with `mpz_set'.
   
        `mpz_array_init' can reduce memory usage in algorithms that need
        large arrays of integers, since it avoids allocating and
        reallocating lots of small memory blocks.  There is no way to free
        the storage allocated by this function.  Don't call `mpz_clear'!
   
    - Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
        Change the space for INTEGER to NEW_ALLOC limbs.  The value in
        INTEGER is preserved if it fits, or is set to 0 if not.  The return
        value is not useful to applications and should be ignored.
   
        `mpz_realloc2' is the preferred way to accomplish allocation
        changes like this.  `mpz_realloc2' and `_mpz_realloc' are the same
        except that `_mpz_realloc' takes the new size in limbs.
   
   
 File: gmp.info,  Node: Assigning Floats,  Next: Simultaneous Float Init & Assign,  Prev: Initializing Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Assigning Integers,  Next: Simultaneous Integer Init & Assign,  Prev: Initializing Integers,  Up: Integer Functions
   
 Assignment Functions  Assignment Functions
 ====================  ====================
   
    These functions assign new values to already initialized floats     These functions assign new values to already initialized integers
 (*note Initializing Floats::).  (*note Initializing Integers::).
   
  - Function: void mpf_set (mpf_t ROP, mpf_t OP)   - Function: void mpz_set (mpz_t ROP, mpz_t OP)
  - Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)   - Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
  - Function: void mpf_set_si (mpf_t ROP, signed long int OP)   - Function: void mpz_set_si (mpz_t ROP, signed long int OP)
  - Function: void mpf_set_d (mpf_t ROP, double OP)   - Function: void mpz_set_d (mpz_t ROP, double OP)
  - Function: void mpf_set_z (mpf_t ROP, mpz_t OP)   - Function: void mpz_set_q (mpz_t ROP, mpq_t OP)
  - Function: void mpf_set_q (mpf_t ROP, mpq_t OP)   - Function: void mpz_set_f (mpz_t ROP, mpf_t OP)
      Set the value of ROP from OP.       Set the value of ROP from OP.
   
  - Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)       `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
      Set the value of ROP from the string in STR.  The string is of the       integer.
      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 argument BASE may be in the ranges 2 to 36, or -36 to -2.   - Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
      Negative values are used to specify that the exponent is in       Set the value of ROP from STR, a null-terminated C string in base
      decimal.       BASE.  White space is allowed in the string, and is simply
        ignored.  The base may vary from 2 to 36.  If BASE is 0, the
        actual base is determined from the leading characters: if the
        first two characters are "0x" or "0X", hexadecimal is assumed,
        otherwise if the first character is "0", octal is assumed,
        otherwise decimal is assumed.
   
      Unlike the corresponding `mpz' function, the base will not be       This function returns 0 if the entire string is a valid number in
      determined from the leading characters of the string if BASE is 0.       base BASE.  Otherwise it returns -1.
      This is so that numbers like `0.23' are not interpreted as octal.  
   
      White space is allowed in the string, and is simply ignored.       [It turns out that it is not entirely true that this function
      [This is not really true; white-space is ignored in the beginning       ignores white-space.  It does ignore it between digits, but not
      of the string and within the mantissa, but not in other places,       after a minus sign or within or after "0x".  We are considering
      such as after a minus sign or in the exponent.  We are considering  
      changing the definition of this function, making it fail when       changing the definition of this function, making it fail when
      there is any white-space in the input, since that makes a lot of       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       sense.  Send your opinion of this change to <bug-gmp@gnu.org>.  Do
      really want it to accept "3 14" as meaning 314 as it does now?]       you really want it to accept "3 14" as meaning 314 as it does now?]
   
      This function returns 0 if the entire string up to the '\0' is a   - Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
      valid number in base BASE.  Otherwise it returns -1.  
   
  - Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)  
      Swap the values ROP1 and ROP2 efficiently.       Swap the values ROP1 and ROP2 efficiently.
   
   
 File: gmp.info,  Node: Simultaneous Float Init & Assign,  Next: Converting Floats,  Prev: Assigning Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Simultaneous Integer Init & Assign,  Next: Converting Integers,  Prev: Assigning Integers,  Up: Integer Functions
   
 Combined Initialization and Assignment Functions  Combined Initialization and Assignment Functions
 ================================================  ================================================
   
    For convenience, GMP provides a parallel series of     For convenience, GMP provides a parallel series of
 initialize-and-set functions which initialize the output and then store  initialize-and-set functions which initialize the output and then store
 the value there.  These functions' names have the form `mpf_init_set...'  the value there.  These functions' names have the form `mpz_init_set...'
   
    Once the float has been initialized by any of the `mpf_init_set...'     Here is an example of using one:
 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!  
   
  - Function: void mpf_init_set (mpf_t ROP, mpf_t OP)       {
  - Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)         mpz_t pie;
  - Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)         mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
  - Function: void mpf_init_set_d (mpf_t ROP, double OP)         ...
      Initialize ROP and set its value from OP.         mpz_sub (pie, ...);
          ...
          mpz_clear (pie);
        }
   
      The precision of ROP will be taken from the active default  Once the integer has been initialized by any of the `mpz_init_set...'
      precision, as set by `mpf_set_default_prec'.  functions, it can be used as the source or destination operand for the
   ordinary integer functions.  Don't use an initialize-and-set function
   on a variable already initialized!
   
  - Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)   - Function: void mpz_init_set (mpz_t ROP, mpz_t OP)
      Initialize ROP and set its value from the string in STR.  See   - Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
      `mpf_set_str' above for details on the assignment operation.   - Function: void mpz_init_set_si (mpz_t ROP, signed long int OP)
    - Function: void mpz_init_set_d (mpz_t ROP, double OP)
        Initialize ROP with limb space and set the initial numeric value
        from OP.
   
      Note that ROP is initialized even if an error occurs.  (I.e., you   - Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE)
      have to call `mpf_clear' for it.)       Initialize ROP and set its value like `mpz_set_str' (see its
        documentation above for details).
   
      The precision of ROP will be taken from the active default       If the string is a correct base BASE number, the function returns
      precision, as set by `mpf_set_default_prec'.       0; if an error occurs it returns -1.  ROP is initialized even if
        an error occurs.  (I.e., you have to call `mpz_clear' for it.)
   
   
 File: gmp.info,  Node: Converting Floats,  Next: Float Arithmetic,  Prev: Simultaneous Float Init & Assign,  Up: Floating-point Functions  File: gmp.info,  Node: Converting Integers,  Next: Integer Arithmetic,  Prev: Simultaneous Integer Init & Assign,  Up: Integer Functions
   
 Conversion Functions  Conversion Functions
 ====================  ====================
   
  - Function: double mpf_get_d (mpf_t OP)     This section describes functions for converting GMP integers to
      Convert OP to a double.  standard C types.  Functions for converting _to_ GMP integers are
   described in *Note Assigning Integers:: and *Note I/O of Integers::.
   
  - Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE,   - Function: unsigned long int mpz_get_ui (mpz_t OP)
           size_t N_DIGITS, mpf_t OP)       Return the value of OP as an `unsigned long'.
      Convert OP to a string of digits in base BASE.  The base may vary  
      from 2 to 36.  Generate at most N_DIGITS significant digits, or if  
      N_DIGITS is 0, the maximum number of digits accurately  
      representable by OP.  
   
      If STR is `NULL', space for the mantissa is allocated using the       If OP is too big to fit an `unsigned long' then just the least
      default allocation function.       significant bits that do fit are returned.  The sign of OP is
        ignored, only the absolute value is used.
   
      If STR is not `NULL', it should point to a block of storage enough   - Function: signed long int mpz_get_si (mpz_t OP)
      large for the mantissa, i.e., N_DIGITS + 2.  The two extra bytes       If OP fits into a `signed long int' return the value of OP.
      are for a possible minus sign, and for the terminating null       Otherwise return the least significant part of OP, with the same
      character.       sign as OP.
   
      The exponent is written through the pointer EXPPTR.       If OP is too big to fit in a `signed long int', the returned
        result is probably not very useful.  To find out if the value will
        fit, use the function `mpz_fits_slong_p'.
   
      If N_DIGITS is 0, the maximum number of digits meaningfully   - Function: double mpz_get_d (mpz_t OP)
      achievable from the precision of OP will be generated.  Note that       Convert OP to a `double'.
      the space requirements for STR in this case will be impossible for  
      the user to predetermine.  Therefore, you need to pass `NULL' for  
      the string argument whenever N_DIGITS is 0.  
   
      The generated string is a fraction, with an implicit radix point   - Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP)
      immediately to the left of the first digit.  For example, the       Find D and EXP such that D times 2 raised to EXP, with
      number 3.1416 would be returned as "31416" in the string and 1       0.5<=abs(D)<1, is a good approximation to OP.
      written at EXPPTR.  
   
      A pointer to the result string is returned.  This pointer will   - Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
      will either equal STR, or if that is `NULL', will point to the       Convert OP to a string of digits in base BASE.  The base may vary
      allocated storage.       from 2 to 36.
   
        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 storage large
        enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
        The two extra bytes are for a possible minus sign, and the
        null-terminator.
   
        A pointer to the result string is returned, being either the
        allocated block, or the given STR.
   
    - Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
        Return limb number N from OP.  The sign of OP is ignored, just the
        absolute value is used.  The least significant limb is number 0.
   
        `mpz_size' can be used to find how many limbs make up OP.
        `mpz_getlimbn' returns zero if N is outside the range 0 to
        `mpz_size(OP)-1'.
   
   
 File: gmp.info,  Node: Float Arithmetic,  Next: Float Comparison,  Prev: Converting Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Integer Arithmetic,  Next: Integer Division,  Prev: Converting Integers,  Up: Integer Functions
   
 Arithmetic Functions  Arithmetic Functions
 ====================  ====================
   
  - Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)   - Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
  - Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int   - Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int
           OP2)            OP2)
      Set ROP to OP1 + OP2.       Set ROP to OP1 + OP2.
   
  - Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)   - Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
  - Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t   - Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int
           OP2)            OP2)
  - Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int   - Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t
           OP2)            OP2)
      Set ROP to OP1 - OP2.       Set ROP to OP1 - OP2.
   
  - Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)   - Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
  - Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int   - Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2)
    - Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int
           OP2)            OP2)
      Set ROP to OP1 times OP2.       Set ROP to OP1 times OP2.
   
    Division is undefined if the divisor is zero, and passing a zero   - Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
 divisor to the divide functions will make these functions intentionally   - Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long
 divide by zero.  This lets the user handle arithmetic exceptions in            int OP2)
 these functions in the same manner as other arithmetic exceptions.       Set ROP to ROP + OP1 times OP2.
   
  - Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)   - Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
  - Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t   - Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long
           OP2)            int OP2)
  - Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int       Set ROP to ROP - OP1 times OP2.
           OP2)  
      Set ROP to OP1/OP2.  
   
  - Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)   - Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, unsigned long int
  - Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)  
      Set ROP to the square root of OP.  
   
  - Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int  
           OP2)            OP2)
      Set ROP to OP1 raised to the power OP2.       Set ROP to OP1 times 2 raised to OP2.  This operation can also be
        defined as a left shift by OP2 bits.
   
  - Function: void mpf_neg (mpf_t ROP, mpf_t OP)   - Function: void mpz_neg (mpz_t ROP, mpz_t OP)
      Set ROP to -OP.       Set ROP to -OP.
   
  - Function: void mpf_abs (mpf_t ROP, mpf_t OP)   - Function: void mpz_abs (mpz_t ROP, mpz_t OP)
      Set ROP to the absolute value of OP.       Set ROP to the absolute value of OP.
   
  - Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long int  
           OP2)  File: gmp.info,  Node: Integer Division,  Next: Integer Exponentiation,  Prev: Integer Arithmetic,  Up: Integer Functions
      Set ROP to OP1 times 2 raised to OP2.  
   
  - Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long int  Division Functions
           OP2)  ==================
      Set ROP to OP1 divided by 2 raised to OP2.  
   
      Division is undefined if the divisor is zero.  Passing a zero
 File: gmp.info,  Node: Float Comparison,  Next: I/O of Floats,  Prev: Float Arithmetic,  Up: Floating-point Functions  divisor to the division or modulo functions (including the modular
   powering functions `mpz_powm' and `mpz_powm_ui'), will cause an
   intentional division by zero.  This lets a program handle arithmetic
   exceptions in these functions the same way as for normal C `int'
   arithmetic.
   
 Comparison Functions   - Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
 ====================   - Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
    - Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
    - Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
             unsigned long int D)
    - Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
             unsigned long int D)
    - Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
             mpz_t N, unsigned long int D)
    - Function: unsigned long int mpz_cdiv_ui (mpz_t N,
             unsigned long int D)
    - Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N,
             unsigned long int B)
    - Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N,
             unsigned long int B)
   
  - Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)   - Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
  - Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)   - Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
  - Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)   - Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
      Compare OP1 and OP2.  Return a positive value if OP1 > OP2, zero   - Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
      if OP1 = OP2, and a negative value if OP1 < OP2.            unsigned long int D)
    - Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
             unsigned long int D)
    - Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
             mpz_t N, unsigned long int D)
    - Function: unsigned long int mpz_fdiv_ui (mpz_t N,
             unsigned long int D)
    - Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N,
             unsigned long int B)
    - Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N,
             unsigned long int B)
   
  - Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3)   - Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
      Return non-zero if the first OP3 bits of OP1 and OP2 are equal,   - Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
      zero otherwise.  I.e., test of OP1 and OP2 are approximately equal.   - Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
    - Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
             unsigned long int D)
    - Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
             unsigned long int D)
    - Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
             mpz_t N, unsigned long int D)
    - Function: unsigned long int mpz_tdiv_ui (mpz_t N,
             unsigned long int D)
    - Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N,
             unsigned long int B)
    - Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N,
             unsigned long int B)
   
  - Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)       Divide N by D, forming a quotient Q and/or remainder R.  For the
      Compute the relative difference between OP1 and OP2 and store the       `2exp' functions, D=2^B.  The rounding is in three styles, each
      result in ROP.       suiting different applications.
   
  - Macro: int mpf_sgn (mpf_t OP)          * `cdiv' rounds Q up towards +infinity, and R will have the
      Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.            opposite sign to D.  The `c' stands for "ceil".
   
      This function is actually implemented as a macro.  It evaluates its          * `fdiv' rounds Q down towards -infinity, and R will have the
      arguments multiple times.            same sign as D.  The `f' stands for "floor".
   
           * `tdiv' rounds Q towards zero, and R will have the same sign
 File: gmp.info,  Node: I/O of Floats,  Next: Miscellaneous Float Functions,  Prev: Float Comparison,  Up: Floating-point Functions            as N.  The `t' stands for "truncate".
   
 Input and Output Functions       In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
 ==========================       0<=abs(R)<abs(D).
   
    Functions that perform input from a stdio stream, and functions that       The `q' functions calculate only the quotient, the `r' functions
 output to a stdio stream.  Passing a `NULL' pointer for a STREAM       only the remainder, and the `qr' functions calculate both.  Note
 argument to any of these functions will make them read from `stdin' and       that for `qr' the same variable cannot be passed for both Q and R,
 write to `stdout', respectively.       or results will be unpredictable.
   
    When using any of these functions, it is a good idea to include       For the `ui' variants the return value is the remainder, and in
 `stdio.h' before `gmp.h', since that will allow `gmp.h' to define       fact returning the remainder is all the `div_ui' functions do.  For
 prototypes for these functions.       `tdiv' and `cdiv' the remainder can be negative, so for those the
        return value is the absolute value of the remainder.
   
  - Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t       The `2exp' functions are right shifts and bit masks, but of course
           N_DIGITS, mpf_t OP)       rounding the same as the other functions.  For positive N both
      Output OP on stdio stream STREAM, as a string of digits in base       `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are simple bitwise right
      BASE.  The base may vary from 2 to 36.  Print at most N_DIGITS       shifts.  For negative N, `mpz_fdiv_q_2exp' is effectively an
      significant digits, or if N_DIGITS is 0, the maximum number of       arithmetic right shift treating N as twos complement the same as
      digits accurately representable by OP.       the bitwise logical functions do, whereas `mpz_tdiv_q_2exp'
        effectively treats N as sign and magnitude.
   
      In addition to the significant digits, a leading `0.' and a   - Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D)
      trailing exponent, in the form `eNNN', are printed.  If BASE is   - Function: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N,
      greater than 10, `@' will be used instead of `e' as exponent            unsigned long int D)
      delimiter.       Set R to N `mod' D.  The sign of the divisor is ignored; the
        result is always non-negative.
   
      Return the number of bytes written, or if an error occurred,       `mpz_mod_ui' is identical to `mpz_fdiv_r_ui' above, returning the
      return 0.       remainder as well as setting R.  See `mpz_fdiv_ui' above if only
        the return value is wanted.
   
  - Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)   - Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D)
      Input a string in base BASE from stdio stream STREAM, and put the   - Function: void mpz_divexact_ui (mpz_t Q, mpz_t N, unsigned long D)
      read float in ROP.  The string is of the form `M@N' or, if the       Set Q to N/D.  These functions produce correct results only when
      base is 10 or less, alternatively `MeN'.  `M' is the mantissa and       it is known in advance that D divides N.
      `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 argument BASE may be in the ranges 2 to 36, or -36 to -2.       These routines are much faster than the other division functions,
      Negative values are used to specify that the exponent is in       and are the best choice when exact division is known to occur, for
      decimal.       example reducing a rational to lowest terms.
   
      Unlike the corresponding `mpz' function, the base will not be   - Function: int mpz_divisible_p (mpz_t N, mpz_t D)
      determined from the leading characters of the string if BASE is 0.   - Function: int mpz_divisible_ui_p (mpz_t N, unsigned long int D)
      This is so that numbers like `0.23' are not interpreted as octal.   - Function: int mpz_divisible_2exp_p (mpz_t N, unsigned long int B)
        Return non-zero if N is exactly divisible by D, or in the case of
        `mpz_divisible_2exp_p' by 2^B.
   
      Return the number of bytes read, or if an error occurred, return 0.   - Function: int mpz_congruent_p (mpz_t N, mpz_t C, mpz_t D)
    - Function: int mpz_congruent_ui_p (mpz_t N, unsigned long int C,
             unsigned long int D)
    - Function: int mpz_congruent_2exp_p (mpz_t N, mpz_t C, unsigned long
             int B)
        Return non-zero if N is congruent to C modulo D, or in the case of
        `mpz_congruent_2exp_p' modulo 2^B.
   
   
 File: gmp.info,  Node: Miscellaneous Float Functions,  Prev: I/O of Floats,  Up: Floating-point Functions  File: gmp.info,  Node: Integer Exponentiation,  Next: Integer Roots,  Prev: Integer Division,  Up: Integer Functions
   
 Miscellaneous Functions  Exponentiation Functions
 =======================  ========================
   
  - Function: void mpf_ceil (mpf_t ROP, mpf_t OP)   - Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t MOD)
  - Function: void mpf_floor (mpf_t ROP, mpf_t OP)   - Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long int
  - Function: void mpf_trunc (mpf_t ROP, mpf_t OP)            EXP, mpz_t MOD)
      Set ROP to OP rounded to an integer.  `mpf_ceil' rounds to the       Set ROP to (BASE raised to EXP) modulo MOD.
      next higher integer, `mpf_floor' to the next lower, and  
      `mpf_trunc' to the integer towards zero.  
   
  - Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,       Negative EXP is supported if an inverse BASE^-1 mod MOD exists
           unsigned long int NBITS)       (see `mpz_invert' in *Note Number Theoretic Functions::).  If an
      Generate a uniformly distributed random float in ROP, such that 0       inverse doesn't exist then a divide by zero is raised.
      <= ROP < 1, with NBITS significant bits in the mantissa.  
   
      The variable STATE must be initialized by calling one of the   - Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
      `gmp_randinit' functions (*Note Random State Initialization::)            EXP)
      before invoking this function.   - Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
             unsigned long int EXP)
        Set ROP to BASE raised to EXP.  The case 0^0 yields 1.
   
  - Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t  
           MAX_EXP)  File: gmp.info,  Node: Integer Roots,  Next: Number Theoretic Functions,  Prev: Integer Exponentiation,  Up: Integer Functions
      Generate a random float of at most MAX_SIZE limbs, with long  
      strings of zeros and ones in the binary representation.  The  Root Extraction Functions
      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   - Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)
      trigger corner-case bugs.  Negative random numbers are generated       Set ROP to  the truncated integer part of the Nth root of OP.
      when MAX_SIZE is negative.       Return non-zero if the computation was exact, i.e., if OP is ROP
        to the Nth power.
   
    - Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
        Set ROP to  the truncated integer part of the square root of OP.
   
    - Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
        Set ROP1 to the truncated integer part of the square root of OP,
        like `mpz_sqrt'.  Set ROP2 to the remainder OP-ROP1*ROP1, which
        will be zero if OP is a perfect square.
   
        If ROP1 and ROP2 are the same variable, the results are undefined.
   
    - Function: int mpz_perfect_power_p (mpz_t OP)
        Return non-zero if OP is a perfect power, i.e., if there exist
        integers A and B, with B>1, such that OP equals A raised to the
        power B.
   
        Under this definition both 0 and 1 are considered to be perfect
        powers.  Negative values of OP are accepted, but of course can
        only be odd perfect powers.
   
    - Function: int mpz_perfect_square_p (mpz_t OP)
        Return non-zero if OP is a perfect square, i.e., if the square
        root of OP is an integer.  Under this definition both 0 and 1 are
        considered to be perfect squares.
   
Legend:
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changed lines
  Added in v.1.1.1.4
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