tore 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 FN, unsigned long int N)
-- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
int N)
`mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
`mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
These functions are designed for calculating isolated Fibonacci
numbers. When a sequence of values is wanted it's best to start
with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
similar.
-- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
-- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned
long int N)
`mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
`mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
These functions are designed for calculating isolated Lucas
numbers. When a sequence of values is wanted it's best to start
with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
or similar.
The Fibonacci numbers and Lucas numbers are related sequences, so
it's never necessary to call both `mpz_fib2_ui' and
`mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
can be found in *Note Lucas Numbers Algorithm::, the reverse is
straightforward too.
File: gmp.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions
5.10 Comparison Functions
=========================
-- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
-- Function: int mpz_cmp_d (mpz_t OP1, double OP2)
-- Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
-- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, or a negative value if OP1 < OP2.
`mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their
arguments more than once. `mpz_cmp_d' can be called with an
infinity, but results are undefined for a NaN.
-- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
-- Function: int mpz_cmpabs_d (mpz_t OP1, double OP2)
-- Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
Compare the absolute values of OP1 and OP2. Return a positive
value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
negative value if abs(OP1) < abs(OP2).
`mpz_cmpabs_d' can be called with an infinity, but results are
undefined for a NaN.
-- Macro: int mpz_sgn (mpz_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates
its argument multiple times.
File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
5.11 Logical and Bit Manipulation Functions
===========================================
These functions behave as if twos complement arithmetic were used
(although sign-magnitude is the actual implementation). The least
significant bit is number 0.
-- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 bitwise-and OP2.
-- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 bitwise inclusive-or OP2.
-- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 bitwise exclusive-or OP2.
-- Function: void mpz_com (mpz_t ROP, mpz_t OP)
Set ROP to the one's complement of OP.
-- Function: mp_bitcnt_t mpz_popcount (mpz_t OP)
If OP>=0, return the population count of OP, which is the number
of 1 bits in the binary representation. If OP<0, the number of 1s
is infinite, and the return value is the largest possible
`mp_bitcnt_t'.
-- Function: mp_bitcnt_t mpz_hamdist (mpz_t OP1, mpz_t OP2)
If OP1 and OP2 are both >=0 or both <0, return the hamming
distance between the two operands, which is the number of bit
positions where OP1 and OP2 have different bit values. If one
operand is >=0 and the other <0 then the number of bits different
is infinite, and the return value is the largest possible
`mp_bitcnt_t'.
-- Function: mp_bitcnt_t mpz_scan0 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
-- Function: mp_bitcnt_t mpz_scan1 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
Scan OP, starting from bit STARTING_BIT, towards more significant
bits, until the first 0 or 1 bit (respectively) is found. Return
the index of the found bit.
If the bit at STARTING_BIT is already what's sought, then
STARTING_BIT is returned.
If there's no bit found, then the largest possible `mp_bitcnt_t' is
returned. This will happen in `mpz_scan0' past the end of a
negative number, or `mpz_scan1' past the end of a nonnegative
number.
-- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
Set bit BIT_INDEX in ROP.
-- Function: void mpz_clrbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
Clear bit BIT_INDEX in ROP.
-- Function: void mpz_combit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
Complement bit BIT_INDEX in ROP.
-- Function: int mpz_tstbit (mpz_t OP, mp_bitcnt_t BIT_INDEX)
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
File: gmp.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions
5.12 Input and Output Functions
===============================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream. Passing a `NULL' pointer for a STREAM
argument to any of these functions will make them read from `stdin' and
write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
-- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
Output OP on stdio stream STREAM, as a string of digits in base
BASE. The base argument may vary from 2 to 62 or from -2 to -36.
For BASE in the range 2..36, digits and lower-case letters are
used; for -2..-36, digits and upper-case letters are used; for
37..62, digits, upper-case letters, and lower-case letters (in
that significance order) are used.
Return the number of bytes written, or if an error occurred,
return 0.
-- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
Input a possibly white-space preceded string in base BASE from
stdio stream STREAM, and put the read integer in ROP.
The BASE may vary from 2 to 62, or if BASE is 0, then the leading
characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
for binary, `0' for octal, or decimal otherwise.
For bases up to 36, case is ignored; upper-case and lower-case
letters have the same value. For bases 37 to 62, upper-case
letter represent the usual 10..35 while lower-case letter
represent 36..61.
Return the number of bytes read, or if an error occurred, return 0.
-- Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
Output OP on stdio stream STREAM, in raw binary format. The
integer is written in a portable format, with 4 bytes of size
information, and that many bytes of limbs. Both the size and the
limbs are written in decreasing significance order (i.e., in
big-endian).
The output can be read with `mpz_inp_raw'.
Return the number of bytes written, or if an error occurred,
return 0.
The output of this can not be read by `mpz_inp_raw' from GMP 1,
because of changes necessary for compatibility between 32-bit and
64-bit machines.
-- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
Input from stdio stream STREAM in the format written by
`mpz_out_raw', and put the result in ROP. Return the number of
bytes read, or if an error occurred, return 0.
This routine can read the output from `mpz_out_raw' also from GMP
1, in spite of changes necessary for compatibility between 32-bit
and 64-bit machines.
File: gmp.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions
5.13 Random Number Functions
============================
The random number functions of GMP come in two groups; older function
that rely on a global state, and newer functions that accept a state
parameter that is read and modified. Please see the *Note Random
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,
mp_bitcnt_t N)
Generate a uniformly distributed random integer in the range 0 to
2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
-- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
mpz_t N)
Generate a uniform random integer in the range 0 to N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
-- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
mp_bitcnt_t N)
Generate a random integer with long strings of zeros and ones in
the binary representation. Useful for testing functions and
algorithms, since this kind of random numbers have proven to be
more likely to trigger corner-case bugs. The random number will
be in the range 0 to 2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
-- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs. The generated
random number doesn't satisfy any particular requirements of
randomness. Negative random numbers are generated when MAX_SIZE
is negative.
This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
instead.
-- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs, with long
strings of zeros and ones in the binary representation. Useful
for testing functions and algorithms, since this kind of random
numbers have proven to be more likely to trigger corner-case bugs.
Negative random numbers are generated when MAX_SIZE is negative.
This function is obsolete. Use `mpz_rrandomb' instead.
File: gmp.info, Node: Integer Import and Export, Next: Miscellaneous Integer Functions, Prev: Integer Random Numbers, Up: Integer Functions
5.14 Integer Import and Export
==============================
`mpz_t' variables can be converted to and from arbitrary words of binary
data with the following functions.
-- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER,
size_t SIZE, int ENDIAN, size_t NAILS, const void *OP)
Set ROP from an array of word data at OP.
The parameters specify the format of the data. COUNT many words
are read, each SIZE bytes. ORDER can be 1 for most significant
word first or -1 for least significant first. Within each word
ENDIAN can be 1 for most significant byte first, -1 for least
significant first, or 0 for the native endianness of the host CPU.
The most significant NAILS bits of each word are skipped, this
can be 0 to use the full words.
There is no sign taken from the data, ROP will simply be a positive
integer. An application can handle any sign itself, and apply it
for instance with `mpz_neg'.
There are no data alignment restrictions on OP, any address is
allowed.
Here's an example converting an array of `unsigned long' data, most
significant element first, and host byte order within each value.
unsigned long a[20];
/* Initialize Z and A */
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
This example assumes the full `sizeof' bytes are used for data in
the given type, which is usually true, and certainly true for
`unsigned long' everywhere we know of. However on Cray vector
systems it may be noted that `short' and `int' are always stored
in 8 bytes (and with `sizeof' indicating that) but use only 32 or
46 bits. The NAILS feature can account for this, by passing for
instance `8*sizeof(int)-INT_BIT'.
-- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
size_t SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
Fill ROP with word data from OP.
The parameters specify the format of the data produced. Each word
will be SIZE bytes and ORDER can be 1 for most significant word
first or -1 for least significant first. Within each word ENDIAN
can be 1 for most significant byte first, -1 for least significant
first, or 0 for the native endianness of the host CPU. The most
significant NAILS bits of each word are unused and set to zero,
this can be 0 to produce full words.
The number of words produced is written to `*COUNTP', or COUNTP
can be `NULL' to discard the count. ROP must have enough space
for the data, or if ROP is `NULL' then a result array of the
necessary size is allocated using the current GMP allocation
function (*note Custom Allocation::). In either case the return
value is the destination used, either ROP or the allocated block.
If OP is non-zero then the most significant word produced will be
non-zero. If OP is zero then the count returned will be zero and
nothing written to ROP. If ROP is `NULL' in this case, no block
is allocated, just `NULL' is returned.
The sign of OP is ignored, just the absolute value is exported. An
application can use `mpz_sgn' to get the sign and handle it as
desired. (*note Integer Comparisons::)
There are no data alignment restrictions on ROP, any address is
allowed.
When an application is allocating space itself the required size
can be determined with a calculation like the following. Since
`mpz_sizeinbase' always returns at least 1, `count' here will be
at least one, which avoids any portability problems with
`malloc(0)', though if `z' is zero no space at all is actually
needed (or written).
numb = 8*size - nail;
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
p = malloc (count * size);
File: gmp.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
5.15 Miscellaneous Functions
============================
-- Function: int mpz_fits_ulong_p (mpz_t OP)
-- Function: int mpz_fits_slong_p (mpz_t OP)
-- 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)
-- Macro: int mpz_even_p (mpz_t OP)
Determine whether OP is odd or even, respectively. Return
non-zero if yes, zero if no. These macros evaluate their argument
more than once.
-- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
Return the size of OP measured in number of digits in the given
BASE. BASE can vary from 2 to 62. The sign of OP is ignored,
just the absolute value is used. The result will be either exact
or 1 too big. If BASE is a power of 2, the result is always
exact. If OP is zero the return value is always 1.
This function can be used to determine the space required when
converting OP to a string. The right amount of allocation is
normally two more than the value returned by `mpz_sizeinbase', one
extra for a minus sign and one for the null-terminator.
It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate
the most significant 1 bit in OP, counting from 1. (Unlike the
bitwise functions which start from 0, *Note Logical and Bit
Manipulation Functions: Integer Logic and Bit Fiddling.)
File: gmp.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions
5.16 Special Functions
======================
The functions in this section are for various special purposes. Most
applications will not need them.
-- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, mp_size_t
ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
This is a special type of initialization. *Fixed* space of
FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in
INTEGER_ARRAY. There is no way to free the storage allocated by
this function. Don't call `mpz_clear'!
The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For
example,
mpz_t arr[20000];
mpz_array_init (arr[0], 20000, 512);
This function is only intended for programs that create a large
number of integers and need to reduce memory usage by avoiding the
overheads of allocating and reallocating lots of small blocks. In
normal programs this function is not recommended.
The space allocated to each integer by this function will not be
automatically increased, unlike the normal `mpz_init', so an
application must ensure it is sufficient for any value stored.
The following space requirements apply to various routines,
* `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'.
-- 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 its size in limbs.
-- 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'.
-- Function: size_t mpz_size (mpz_t OP)
Return the size of OP measured in number of limbs. If OP is zero,
the returned value will be zero.
File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
6 Rational Number Functions
***************************
This chapter describes the GMP functions for performing arithmetic on
rational numbers. These functions start with the prefix `mpq_'.
Rational numbers are stored in objects of type `mpq_t'.
All rational arithmetic functions assume operands have a canonical
form, and canonicalize their result. The canonical from means that the
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.
It is the responsibility of the user to canonicalize the assigned
variable before any arithmetic operations are performed on that
variable.
-- Function: void mpq_canonicalize (mpq_t OP)
Remove any factors that are common to the numerator and
denominator of OP, and make the denominator positive.
* Menu:
* Initializing Rationals::
* Rational Conversions::
* Rational Arithmetic::
* Comparing Rationals::
* Applying Integer Functions::
* I/O of Rationals::
File: gmp.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions
6.1 Initialization and Assignment Functions
===========================================
-- Function: void mpq_init (mpq_t X)
Initialize X and set it to 0/1. Each variable should normally
only be initialized once, or at least cleared out (using the
function `mpq_clear') between each initialization.
-- Function: void mpq_inits (mpq_t X, ...)
Initialize a NULL-terminated list of `mpq_t' variables, and set
their values to 0/1.
-- Function: void mpq_clear (mpq_t X)
Free the space occupied by X. Make sure to call this function for
all `mpq_t' variables when you are done with them.
-- Function: void mpq_clears (mpq_t X, ...)
Free the space occupied by a NULL-terminated list of `mpq_t'
variables.
-- Function: void mpq_set (mpq_t ROP, mpq_t OP)
-- Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
Assign ROP from OP.
-- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
unsigned long int OP2)
-- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
long int OP2)
Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have
common factors, ROP has to be passed to `mpq_canonicalize' before
any operations are performed on ROP.
-- Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE)
Set ROP from a null-terminated string STR in the given BASE.
The string can be an integer like "41" or a fraction like
"41/152". The fraction must be in canonical form (*note Rational
Number Functions::), or if not then `mpq_canonicalize' must be
called.
The numerator and optional denominator are parsed the same as in
`mpz_set_str' (*note Assigning Integers::). White space is
allowed in the string, and is simply ignored. The BASE can vary
from 2 to 62, or if BASE is 0 then the leading characters are
used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for
octal, or decimal otherwise. Note that this is done separately
for the numerator and denominator, so for instance `0xEF/100' is
239/100, whereas `0xEF/0x100' is 239/256.
The return value is 0 if the entire string is a valid number, or
-1 if not.
-- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
Swap the values ROP1 and ROP2 efficiently.
File: gmp.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions
6.2 Conversion Functions
========================
-- Function: double mpq_get_d (mpq_t OP)
Convert OP to a `double', truncating if necessary (ie. rounding
towards zero).
If the exponent from the conversion is too big or too small to fit
a `double' then the result is system dependent. For too big an
infinity is returned when available. For too small 0.0 is
normally returned. Hardware overflow, underflow and denorm traps
may or may not occur.
-- Function: void mpq_set_d (mpq_t ROP, double OP)
-- Function: void mpq_set_f (mpq_t ROP, mpf_t OP)
Set ROP to the value of OP. There is no rounding, this conversion
is exact.
-- Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP)
Convert OP to a string of digits in base BASE. The base may vary
from 2 to 36. The string will be of the form `num/den', or if the
denominator is 1 then just `num'.
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 (mpq_numref(OP), BASE)
+ mpz_sizeinbase (mpq_denref(OP), BASE) + 3
The three extra bytes are for a possible minus sign, possible
slash, and the null-terminator.
A pointer to the result string is returned, being either the
allocated block, or the given STR.
File: gmp.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions
6.3 Arithmetic Functions
========================
-- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
Set SUM to ADDEND1 + ADDEND2.
-- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
SUBTRAHEND)
Set DIFFERENCE to MINUEND - SUBTRAHEND.
-- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
MULTIPLICAND)
Set PRODUCT to MULTIPLIER times MULTIPLICAND.
-- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
Set ROP to OP1 times 2 raised to OP2.
-- Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
DIVISOR)
Set QUOTIENT to DIVIDEND/DIVISOR.
-- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
Set ROP to OP1 divided by 2 raised to OP2.
-- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
Set NEGATED_OPERAND to -OPERAND.
-- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
Set ROP to the absolute value of OP.
-- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
this routine will divide by zero.
File: gmp.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions
6.4 Comparison Functions
========================
-- Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
To determine if two rationals are equal, `mpq_equal' is faster than
`mpq_cmp'.
-- Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
long int DEN2)
-- Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int
DEN2)
Compare OP1 and NUM2/DEN2. Return a positive value if OP1 >
NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
NUM2/DEN2.
NUM2 and DEN2 are allowed to have common factors.
These functions are implemented as a macros and evaluate their
arguments multiple times.
-- Macro: int mpq_sgn (mpq_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates its
arguments multiple times.
-- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
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.
File: gmp.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions
6.5 Applying Integer Functions to Rationals
===========================================
The set of `mpq' functions is quite small. In particular, there are few
functions for either input or output. The following functions give
direct access to the numerator and denominator of an `mpq_t'.
Note that if an assignment to the numerator and/or denominator could
take an `mpq_t' out of the canonical form described at the start of
this chapter (*note Rational Number Functions::) then
`mpq_canonicalize' must be called before any other `mpq' functions are
applied to that `mpq_t'.
-- Macro: mpz_t mpq_numref (mpq_t OP)
-- Macro: mpz_t mpq_denref (mpq_t OP)
Return a reference to the numerator and denominator of OP,
respectively. The `mpz' functions can be used on the result of
these macros.
-- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
-- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
-- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
-- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
Get or set the numerator or denominator of a rational. These
functions are equivalent to calling `mpz_set' with an appropriate
`mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or
`mpq_denref' is recommended instead of these functions.
File: gmp.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions
6.6 Input and Output Functions
==============================
When using any of these functions, it's a good idea to include `stdio.h'
before `gmp.h', since that will allow `gmp.h' to define prototypes for
these functions.
Passing a `NULL' pointer for a STREAM argument to any of these
functions will make them read from `stdin' and write to `stdout',
respectively.
-- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
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,
return 0.
-- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
Read a string of digits from STREAM and convert them to a rational
in ROP. Any initial white-space characters are read and
discarded. Return the number of characters read (including white
space), or 0 if a rational could not be read.
The input can be a fraction like `17/63' or just an integer like
`123'. Reading stops at the first character not in this form, and
white space is not permitted within the string. If the input
might not be in canonical form, then `mpq_canonicalize' must be
called (*note Rational Number Functions::).
The BASE can be between 2 and 36, or can be 0 in which case the
leading characters of the string determine the base, `0x' or `0X'
for hexadecimal, `0' for octal, or decimal otherwise. The leading
characters are examined separately for the numerator and
denominator of a fraction, so for instance `0x10/11' is 16/11,
whereas `0x10/0x11' is 16/17.
File: gmp.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
7 Floating-point Functions
**************************
GMP floating point numbers are stored in objects of type `mpf_t' and
functions operating on them have an `mpf_' prefix.
The mantissa of each float has a user-selectable precision, limited
only by available memory. Each variable has its own precision, and
that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on
most systems. In the current implementation the exponent is a count of
limbs, so for example on a 32-bit system this means a range of roughly
2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
greater. Note however `mpf_get_str' can only return an exponent which
fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
bigger than a `long'.
Each variable keeps a size for the mantissa data actually in use.
This means that if a float is exactly represented in only a few bits
then only those bits will be used in a calculation, even if the
selected precision is high.
All calculations are performed to the precision of the destination
variable. Each function is defined to calculate with "infinite
precision" followed by a truncation to the destination precision, but
of course the work done is only what's needed to determine a result
under that definition.
The precision selected for a variable is a minimum value, GMP may
increase it a little to facilitate efficient calculation. Currently
this means rounding up to a whole limb, and then sometimes having a
further partial limb, depending on the high limb of the mantissa. But
applications shouldn't be concerned by such details.
The mantissa in stored in binary, as might be imagined from the fact
precisions are expressed in bits. One consequence of this is that
decimal fractions like 0.1 cannot be represented exactly. The same is
true of plain IEEE `double' floats. This makes both highly unsuitable
for calculations involving money or other values that should be exact
decimal fractions. (Suitably scaled integers, or perhaps rationals,
are better choices.)
`mpf' functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the
exponent or results will be unpredictable. This might change in a
future release.
Note that the `mpf' functions are _not_ intended as a smooth
extension to IEEE P754 arithmetic. In particular results obtained on
one computer often differ from the results on a computer with a
different word size.
* Menu:
* Initializing Floats::
* Assigning Floats::
* Simultaneous Float Init & Assign::
* Converting Floats::
* Float Arithmetic::
* Float Comparison::
* I/O of Floats::
* Miscellaneous Float Functions::
File: gmp.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions
7.1 Initialization Functions
============================
-- Function: void mpf_set_default_prec (mp_bitcnt_t PREC)
Set the default precision to be *at least* PREC bits. All
subsequent calls to `mpf_init' will use this precision, but
previously initialized variables are unaffected.
-- Function: mp_bitcnt_t mpf_get_default_prec (void)
Return the default precision actually used.
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, mp_bitcnt_t 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_inits (mpf_t X, ...)
Initialize a NULL-terminated list of `mpf_t' variables, and set
their values to 0. The precision of the initialized variables is
undefined unless a default precision has already been established
by a call to `mpf_set_default_prec'.
-- 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.
-- Function: void mpf_clears (mpf_t X, ...)
Free the space occupied by a NULL-terminated list of `mpf_t'
variables.
Here is an example on how to initialize floating-point variables:
{
mpf_t x, y;
mpf_init (x); /* use default precision */
mpf_init2 (y, 256); /* precision _at least_ 256 bits */
...
/* Unless the program is about to exit, do ... */
mpf_clear (x);
mpf_clear (y);
}
The following three functions are useful for changing the precision
during a calculation. A typical use would be for adjusting the
precision gradually in iterative algorithms like Newton-Raphson, making
the computation precision closely match the actual accurate part of the
numbers.
-- Function: mp_bitcnt_t mpf_get_prec (mpf_t OP)
Return the current precision of OP, in bits.
-- Function: void mpf_set_prec (mpf_t ROP, mp_bitcnt_t PREC)
Set the precision of ROP to be *at least* PREC bits. The value in
ROP will be truncated to the new precision.
This function requires a call to `realloc', and so should not be
used in a tight loop.
-- Function: void mpf_set_prec_raw (mpf_t ROP, mp_bitcnt_t PREC)
Set the precision of ROP to be *at least* PREC bits, without
changing the memory allocated.
PREC must be no more than the allocated precision for ROP, that
being the precision when ROP was initialized, or in the most recent
`mpf_set_prec'.
The value in ROP is unchanged, and in particular if it had a higher
precision than PREC it will retain that higher precision. New
values written to ROP will use the new PREC.
Before calling `mpf_clear' or the full `mpf_set_prec', another
`mpf_set_prec_raw' call must be made to restore ROP to its original
allocated precision. Failing to do so will have unpredictable
results.
`mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
original allocated precision. After `mpf_set_prec_raw' it
reflects the PREC value set.
`mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
at different precisions during a calculation, perhaps to gradually
increase precision in an iteration, or just to use various
different precisions for different purposes during a calculation.
File: gmp.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions
7.2 Assignment Functions
========================
These functions assign new values to already initialized floats (*note
Initializing Floats::).
-- Function: void mpf_set (mpf_t ROP, mpf_t OP)
-- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
-- Function: void mpf_set_si (mpf_t ROP, signed long int OP)
-- Function: void mpf_set_d (mpf_t ROP, double OP)
-- Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
-- Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
Set the value of ROP from OP.
-- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
Set the value of ROP from the string in STR. The string is of the
form `M@N' or, if the base is 10 or less, alternatively `MeN'.
`M' is the mantissa and `N' is the exponent. The mantissa is
always in the specified base. The exponent is either in the
specified base or, if BASE is negative, in decimal. The decimal
point expected is taken from the current locale, on systems
providing `localeconv'.
The argument BASE may be in the ranges 2 to 62, or -62 to -2.
Negative values are used to specify that the exponent is in
decimal.
For bases up to 36, case is ignored; upper-case and lower-case
letters have the same value; for bases 37 to 62, upper-case letter
represent the usual 10..35 while lower-case letter represent
36..61.
Unlike the corresponding `mpz' function, the base will not be
determined from the leading characters of the string if BASE is 0.
This is so that numbers like `0.23' are not interpreted as octal.
White space is allowed in the string, and is simply ignored.
[This is not really true; white-space is ignored in the beginning
of the string and within the mantissa, but not in other places,
such as after a minus sign or in the exponent. We are considering
changing the definition of this function, making it fail when
there is any white-space in the input, since that makes a lot of
sense. Please tell us your opinion about this change. Do you
really want it to accept "3 14" as meaning 314 as it does now?]
This function returns 0 if the entire string is a valid number in
base BASE. Otherwise it returns -1.
-- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
Swap ROP1 and ROP2 efficiently. Both the values and the
precisions of the two variables are swapped.
File: gmp.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
7.3 Combined Initialization and Assignment Functions
====================================================
For convenience, GMP provides a parallel series of initialize-and-set
functions which initialize the output and then store the value there.
These functions' names have the form `mpf_init_set...'
Once the float has been initialized by any of the `mpf_init_set...'
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)
-- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
-- Function: void mpf_init_set_d (mpf_t ROP, double OP)
Initialize ROP and set its value from OP.
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.
-- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
Initialize ROP and set its value from the string in STR. See
`mpf_set_str' above for details on the assignment operation.
Note that ROP is initialized even if an error occurs. (I.e., you
have to call `mpf_clear' for it.)
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.
File: gmp.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions
7.4 Conversion Functions
========================
-- Function: double mpf_get_d (mpf_t OP)
Convert OP to a `double', truncating if necessary (ie. rounding
towards zero).
If the exponent in OP is too big or too small to fit a `double'
then the result is system dependent. For too big an infinity is
returned when available. For too small 0.0 is normally returned.
Hardware overflow, underflow and denorm traps may or may not occur.
-- Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP)
Convert OP to a `double', truncating if necessary (ie. rounding
towards zero), and with an exponent returned separately.
The return value is in the range 0.5<=abs(D)<1 and the exponent is
stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
is zero, the return is 0.0 and 0 is stored to `*EXP'.
This is similar to the standard C `frexp' function (*note
Normalization Functions: (libc)Normalization Functions.).
-- Function: long mpf_get_si (mpf_t OP)
-- Function: unsigned long mpf_get_ui (mpf_t OP)
Convert OP to a `long' or `unsigned long', truncating any fraction
part. If OP is too big for the return type, the result is
undefined.
See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
Miscellaneous Float Functions::).
-- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int
BASE, size_t N_DIGITS, mpf_t OP)
Convert OP to a string of digits in base BASE. The base argument
may vary from 2 to 62 or from -2 to -36. Up to N_DIGITS digits
will be generated. Trailing zeros are not returned. No more
digits than can be accurately represented by OP are ever
generated. If N_DIGITS is 0 then that accurate maximum number of
digits are generated.
For BASE in the range 2..36, digits and lower-case letters are
used; for -2..-36, digits and upper-case letters are used; for
37..62, digits, upper-case letters, and lower-case letters (in
that significance order) are used.
If STR is `NULL', the result string is allocated using the current
allocation function (*note Custom Allocation::). The block will be
`strlen(str)+1' bytes, that being exactly enough for the string and
null-terminator.
If STR is not `NULL', it should point to a block of N_DIGITS + 2
bytes, that being enough for the mantissa, a possible minus sign,
and a null-terminator. When N_DIGITS is 0 to get all significant
digits, an application won't be able to know the space required,
and STR should be `NULL' in that case.
The generated string is a fraction, with an implicit radix point
immediately to the left of the first digit. The applicable
exponent is written through the EXPPTR pointer. For example, the
number 3.1416 would be returned as string "31416" and exponent 1.
When OP is zero, an empty string is produced and the exponent
returned is 0.
A pointer to the result string is returned, being either the
allocated block or the given STR.
File: gmp.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
7.5 Arithmetic Functions
========================
-- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 + OP2.
-- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t
OP2)
-- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 - OP2.
-- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 times OP2.
Division is undefined if the divisor is zero, and passing a zero
divisor to the divide functions will make these functions intentionally
divide by zero. This lets the user handle arithmetic exceptions in
these functions in the same manner as other arithmetic exceptions.
-- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
OP2)
-- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1/OP2.
-- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
-- 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)
Set ROP to OP1 raised to the power OP2.
-- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
Set ROP to -OP.
-- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
Set ROP to the absolute value of OP.
-- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
Set ROP to OP1 times 2 raised to OP2.
-- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
Set ROP to OP1 divided by 2 raised to OP2.
File: gmp.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
7.6 Comparison Functions
========================
-- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
-- Function: int mpf_cmp_d (mpf_t OP1, double OP2)
-- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
-- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
`mpf_cmp_d' can be called with an infinity, but results are
undefined for a NaN.
-- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, mp_bitcnt_t op3)
Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
zero otherwise. I.e., test if OP1 and OP2 are approximately equal.
Caution 1: All version of GMP up to version 4.2.4 compared just
whole limbs, meaning sometimes more than OP3 bits, sometimes fewer.
Caution 2: This function will consider XXX11...111 and XX100...000
different, even if ... is replaced by a semi-infinite number of
bits. Such numbers are really just one ulp off, and should be
considered equal.
-- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
Compute the relative difference between OP1 and OP2 and store the
result in ROP. This is abs(OP1-OP2)/OP1.
-- Macro: int mpf_sgn (mpf_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates
its arguments multiple times.
File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
7.7 Input and Output Functions
==============================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream. Passing a `NULL' pointer for a STREAM
argument to any of these functions will make them read from `stdin' and
write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
-- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
N_DIGITS, mpf_t OP)
Print OP to STREAM, as a string of digits. Return the number of
bytes written, or if an error occurred, return 0.
The mantissa is prefixed with an `0.' and is in the given BASE,
which may vary from 2 to 62 or from -2 to -36. An exponent is
then printed, separated by an `e', or if the base is greater than
10 then by an `@'. The exponent is always in decimal. The
decimal point follows the current locale, on systems providing
`localeconv'.
For BASE in the range 2..36, digits and lower-case letters are
used; for -2..-36, digits and upper-case letters are used; for
37..62, digits, upper-case letters, and lower-case letters (in
that significance order) are used.
Up to N_DIGITS will be printed from the mantissa, except that no
more digits than are accurately representable by OP will be
printed. N_DIGITS can be 0 to select that accurate maximum.
-- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
Read a string in base BASE from STREAM, and put the read float in
ROP. The string is of the form `M@N' or, if the base is 10 or
less, alternatively `MeN'. `M' is the mantissa and `N' is the
exponent. The mantissa is always in the specified base. The
exponent is either in the specified base or, if BASE is negative,
in decimal. The decimal point expected is taken from the current
locale, on systems providing `localeconv'.
The argument BASE may be in the ranges 2 to 36, or -36 to -2.
Negative values are used to specify that the exponent is in
decimal.
Unlike the corresponding `mpz' function, the base will not be
determined from the leading characters of the string if BASE is 0.
This is so that numbers like `0.23' are not interpreted as octal.
Return the number of bytes read, or if an error occurred, return 0.
File: gmp.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
7.8 Miscellaneous Functions
===========================
-- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
-- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
-- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
to the integer towards zero.
-- Function: int mpf_integer_p (mpf_t OP)
Return non-zero if OP is an integer.
-- Function: int mpf_fits_ulong_p (mpf_t OP)
-- Function: int mpf_fits_slong_p (mpf_t OP)
-- Function: int mpf_fits_uint_p (mpf_t OP)
-- Function: int mpf_fits_sint_p (mpf_t OP)
-- Function: int mpf_fits_ushort_p (mpf_t OP)
-- Function: int mpf_fits_sshort_p (mpf_t OP)
Return non-zero if OP would fit in the respective C data type, when
truncated to an integer.
-- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
mp_bitcnt_t NBITS)
Generate a uniformly distributed random float in ROP, such that 0
<= ROP < 1, with NBITS significant bits in the mantissa.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*Note Random State Initialization::)
before invoking this function.
-- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
EXP)
Generate a random float of at most MAX_SIZE limbs, with long
strings of zeros and ones in the binary representation. The
exponent of the number is in the interval -EXP to EXP (in limbs).
This function is useful for testing functions and algorithms,
since these kind of random numbers have proven to be more likely
to trigger corner-case bugs. Negative random numbers are
generated when MAX_SIZE is negative.
File: gmp.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
8 Low-level Functions
*********************
This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.
These functions start with the prefix `mpn_'.
The `mpn' functions are designed to be as fast as possible, *not* to
provide a coherent calling interface. The different functions have
somewhat similar interfaces, but there are variations that make them
hard to use. These functions do as little as possible apart from the
real multiple precision computation, so that no time is spent on things
that not all callers need.
A source operand is specified by a pointer to the least significant
limb and a limb count. A destination operand is specified by just a
pointer. It is the responsibility of the caller to ensure that the
destination has enough space for storing the result.
With this way of specifying operands, it is possible to perform
computations on subranges of an argument, and store the result into a
subrange of a destination.
A common requirement for all functions is that each source area
needs at least one limb. No size argument may be zero. Unless
otherwise stated, in-place operations are allowed where source and
destination are the same, but not where they only partly overlap.
The `mpn' functions are the base for the implementation of the
`mpz_', `mpf_', and `mpq_' functions.
This example adds the number beginning at S1P and the number
beginning at S2P and writes the sum at DESTP. All areas have N limbs.
cy = mpn_add_n (destp, s1p, s2p, n)
It should be noted that the `mpn' functions make no attempt to
identify high or low zero limbs on their operands, or other special
forms. On random data such cases will be unlikely and it'd be wasteful
for every function to check every time. An application knowing
something about its data can take steps to trim or perhaps split its
calculations.
In the notation used below, a source operand is identified by the
pointer to the least significant limb, and the limb count in braces.
For example, {S1P, S1N}.
-- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Add {S1P, N} and {S2P, N}, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
This is the lowest-level function for addition. It is the
preferred function for addition, since it is written in assembly
for most CPUs. For addition of a variable to itself (i.e., S1P
equals S2P) use `mpn_lshift' with a count of 1 for optimal speed.
-- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Add {S1P, N} and S2LIMB, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
-- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
limbs of the result to RP. Return carry, either 0 or 1.
This function requires that S1N is greater than or equal to S2N.
-- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Subtract {S2P, N} from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
This is the lowest-level function for subtraction. It is the
preferred function for subtraction, since it is written in
assembly for most CPUs.
-- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Subtract S2LIMB from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
-- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
significant limbs of the result to RP. Return borrow, either 0 or
1.
This function requires that S1N is greater than or equal to S2N.
-- Function: void mpn_neg (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N)
Perform the negation of {SP, N}, and write the result to {RP, N}.
Return carry-out.
-- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
RP.
The destination has to have space for 2*N limbs, even if the
product's most significant limb is zero. No overlap is permitted
between the destination and either source.
If the two input operands are the same, use `mpn_sqr'.
-- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Multiply {S1P, S1N} and {S2P, S2N}, and write the (S1N+S2N)-limb
result to RP. Return the most significant limb of the result.
The destination has to have space for S1N + S2N limbs, even if the
product's most significant limb is zero. No overlap is permitted
between the destination and either source.
This function requires that S1N is greater than or equal to S2N.
-- Function: void mpn_sqr (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N)
Compute the square of {S1P, N} and write the 2*N-limb result to RP.
The destination has to have space for 2*N limbs, even if the
result's most significant limb is zero. No overlap is permitted
between the destination and the source.
-- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} by S2LIMB, and write the N least significant
limbs of the product to RP. Return the most significant limb of
the product. {S1P, N} and {RP, N} are allowed to overlap provided
RP <= S1P.
This is a low-level function that is a building block for general
multiplication as well as other operations in GMP. It is written
in assembly for most CPUs.
Don't call this function if S2LIMB is a power of 2; use
`mpn_lshift' with a count equal to the logarithm of S2LIMB
instead, for optimal speed.
-- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} and S2LIMB, and add the N least significant
limbs of the product to {RP, N} and write the result to RP.
Return the most significant limb of the product, plus carry-out
from the addition.
This is a low-level function that is a building block for general
multiplication as well as other operations in GMP. It is written
in assembly for most CPUs.
-- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} and S2LIMB, and subtract the N least significant
limbs of the product from {RP, N} and write the result to RP.
Return the most significant limb of the product, plus borrow-out
from the subtraction.
This is a low-level function that is a building block for general
multiplication and division as well as other operations in GMP.
It is written in assembly for most CPUs.
-- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
mp_size_t DN)
Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
and the remainder at {RP, DN}. The quotient is rounded towards 0.
No overlap is permitted between arguments, except that NP might
equal RP. The dividend size NN must be greater than or equal to
divisor size DN. The most significant limb of the divisor must be
non-zero. The QXN operand must be zero.
-- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
mp_size_t S3N)
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
best performance.]
Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
with the exception of the most significant limb, which is
returned. The remainder replaces the dividend at RS2P; it will be
S3N limbs long (i.e., as many limbs as the divisor).
In addition to an integer quotient, QXN fraction limbs are
developed, and stored after the integral limbs. For most usages,
QXN will be zero.
It is required that RS2N is greater than or equal to S3N. It is
required that the most significant bit of the divisor is set.
If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
that special case, no overlap between arguments is permitted.
Return the most significant limb of the quotient, either 0 or 1.
The area at R1P needs to be RS2N - S3N + QXN limbs large.
-- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
-- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
mp_size_t S2N, mp_limb_t S3LIMB)
Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
Return the remainder.
The integer quotient is written to {R1P+QXN, S2N} and in addition
QXN fraction limbs are developed and written to {R1P, QXN}.
Either or both S2N and QXN can be zero. For most usages, QXN will
be zero.
`mpn_divmod_1' exists for upward source compatibility and is
simply a macro calling `mpn_divrem_1' with a QXN of 0.
The areas at R1P and S2P have to be identical or completely
separate, not partially overlapping.
-- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
[This function is obsolete. Please call `mpn_tdiv_qr' instead for
best performance.]
-- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t N)
-- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t
*SP, mp_size_t N, mp_limb_t CARRY)
Divide {SP, N} by 3, expecting it to divide exactly, and writing
the result to {RP, N}. If 3 divides exactly, the return value is
zero and the result is the quotient. If not, the return value is
non-zero and the result won't be anything useful.
`mpn_divexact_by3c' takes an initial carry parameter, which can be
the return value from a previous call, so a large calculation can
be done piece by piece from low to high. `mpn_divexact_by3' is
simply a macro calling `mpn_divexact_by3c' with a 0 carry
parameter.
These routines use a multiply-by-inverse and will be faster than
`mpn_divrem_1' on CPUs with fast multiplication but slow division.
The source a, result q, size n, initial carry i, and return value
c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return
c is always 0, 1 or 2, and the initial carry i must also be 0, 1
or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3.
When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b
== 1 mod 3 (when `mp_bits_per_limb' is even, which is always so
currently).
-- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
mp_limb_t S2LIMB)
Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
zero.
-- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N, unsi