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Constant gmp_mpfr_sys::C::MPFR::MPFR_Interface

source · [−]
pub const MPFR_Interface: ();
Expand description

This constant is a place-holder for documentation; do not use it in code.

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5 MPFR Interface

The floating-point functions expect arguments of type mpfr_t.

The MPFR floating-point functions have an interface that is similar to the GNU MP functions. The function prefix for floating-point operations is mpfr_.

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 cost of that computation should not depend on the precision of variables used as input (on average).

The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with “infinite accuracy”), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system.

MPFR does not keep track of the accuracy of a computation. This is left to the user or to a higher layer (for example the MPFI library for interval arithmetic). As a consequence, if two variables are used to store only a few significant bits, and their product is stored in a variable with large precision, then MPFR will still compute the result with full precision.

The value of the standard C macro errno may be set to non-zero after calling any MPFR function or macro, whether or not there is an error. Except when documented, MPFR will not set errno, but functions called by the MPFR code (libc functions, memory allocator, etc.) may do so.

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5.1 Initialization Functions

An mpfr_t object must be initialized before storing the first value in it. The functions mpfr_init and mpfr_init2 are used for that purpose.

Function: void mpfr_init2 (mpfr_t x, mpfr_prec_t prec)

Initialize x, set its precision to be exactly prec bits and its value to NaN. (Warning: the corresponding MPF function initializes to zero instead.)

Normally, a variable should be initialized once only or at least be cleared, using mpfr_clear, between initializations. To change the precision of a variable that has already been initialized, use mpfr_set_prec or mpfr_prec_round; note that if the precision is decreased, the unused memory will not be freed, so that it may be wise to choose a large enough initial precision in order to avoid reallocations. The precision prec must be an integer between MPFR_PREC_MIN and MPFR_PREC_MAX (otherwise the behavior is undefined).

Function: void mpfr_inits2 (mpfr_prec_t prec, mpfr_t x, ...)

Initialize all the mpfr_t variables of the given variable argument va_list, set their precision to be exactly prec bits and their value to NaN. See mpfr_init2 for more details. The va_list is assumed to be composed only of type mpfr_t (or equivalently mpfr_ptr). It begins from x, and ends when it encounters a null pointer (whose type must also be mpfr_ptr).

Function: void mpfr_clear (mpfr_t x)

Free the space occupied by the significand of x. Make sure to call this function for all mpfr_t variables when you are done with them.

Function: void mpfr_clears (mpfr_t x, ...)

Free the space occupied by all the mpfr_t variables of the given va_list. See mpfr_clear for more details. The va_list is assumed to be composed only of type mpfr_t (or equivalently mpfr_ptr). It begins from x, and ends when it encounters a null pointer (whose type must also be mpfr_ptr).

Here is an example of how to use multiple initialization functions (since NULL is not necessarily defined in this context, we use (mpfr_ptr) 0 instead, but (mpfr_ptr) NULL is also correct). 

{
  mpfr_t x, y, z, t;
  mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0);
  …
  mpfr_clears (x, y, z, t, (mpfr_ptr) 0);
}
Function: void mpfr_init (mpfr_t x)

Initialize x, set its precision to the default precision, and set its value to NaN. The default precision can be changed by a call to mpfr_set_default_prec.

Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use mpfr_init2.

Function: void mpfr_inits (mpfr_t x, ...)

Initialize all the mpfr_t variables of the given va_list, set their precision to the default precision and their value to NaN. See mpfr_init for more details. The va_list is assumed to be composed only of type mpfr_t (or equivalently mpfr_ptr). It begins from x, and ends when it encounters a null pointer (whose type must also be mpfr_ptr).

Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use mpfr_inits2.

Macro: MPFR_DECL_INIT (name, prec)

This macro declares name as an automatic variable of type mpfr_t, initializes it and sets its precision to be exactly prec bits and its value to NaN. name must be a valid identifier. You must use this macro in the declaration section. This macro is much faster than using mpfr_init2 but has some drawbacks:

  • You must not call mpfr_clear with variables created with this macro (the storage is allocated at the point of declaration and deallocated when the brace-level is exited).
  • You cannot change their precision.
  • You should not create variables with huge precision with this macro.
  • Your compiler must support ‘Non-Constant Initializers’ (standard in C++ and ISO C99) and ‘Token Pasting’ (standard in ISO C89). If prec is not a constant expression, your compiler must support ‘variable-length automatic arrays’ (standard in ISO C99). GCC 2.95.3 and above supports all these features. If you compile your program with GCC in C89 mode and with ‘-pedantic’, you may want to define the MPFR_USE_EXTENSION macro to avoid warnings due to the MPFR_DECL_INIT implementation.
Function: void mpfr_set_default_prec (mpfr_prec_t prec)

Set the default precision to be exactly prec bits, where prec can be any integer between MPFR_PREC_MIN and MPFR_PREC_MAX. The precision of a variable means the number of bits used to store its significand. All subsequent calls to mpfr_init or mpfr_inits will use this precision, but previously initialized variables are unaffected. The default precision is set to 53 bits initially.

Note: when MPFR is built with the ‘--enable-thread-safe’ configure option, the default precision is local to each thread. See Memory Handling, for more information.

Function: mpfr_prec_t mpfr_get_default_prec (void)

Return the current default MPFR precision in bits. See the documentation of mpfr_set_default_prec.

Here is an example on how to initialize floating-point variables: 

{
  mpfr_t x, y;
  mpfr_init (x);                /* use default precision */
  mpfr_init2 (y, 256);          /* precision exactly 256 bits */
  …
  /* When the program is about to exit, do ... */
  mpfr_clear (x);
  mpfr_clear (y);
  mpfr_free_cache ();           /* free the cache for constants like pi */
}

The following 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: void mpfr_set_prec (mpfr_t x, mpfr_prec_t prec)

Set the precision of x to be exactly prec bits, and set its value to NaN. The previous value stored in x is lost. It is equivalent to a call to mpfr_clear(x) followed by a call to mpfr_init2(x, prec), but more efficient as no allocation is done in case the current allocated space for the significand of x is enough. The precision prec can be any integer between MPFR_PREC_MIN and MPFR_PREC_MAX. In case you want to keep the previous value stored in x, use mpfr_prec_round instead.

Warning! You must not use this function if x was initialized with MPFR_DECL_INIT or with mpfr_custom_init_set (see Custom Interface).

Function: mpfr_prec_t mpfr_get_prec (mpfr_t x)

Return the precision of x, i.e., the number of bits used to store its significand.

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5.2 Assignment Functions

These functions assign new values to already initialized floats (see Initialization Functions).

Function: int mpfr_set (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_set_ui (mpfr_t rop, unsigned long int op, mpfr_rnd_t rnd)
Function: int mpfr_set_si (mpfr_t rop, long int op, mpfr_rnd_t rnd)
Function: int mpfr_set_uj (mpfr_t rop, uintmax_t op, mpfr_rnd_t rnd)
Function: int mpfr_set_sj (mpfr_t rop, intmax_t op, mpfr_rnd_t rnd)
Function: int mpfr_set_flt (mpfr_t rop, float op, mpfr_rnd_t rnd)
Function: int mpfr_set_d (mpfr_t rop, double op, mpfr_rnd_t rnd)
Function: int mpfr_set_ld (mpfr_t rop, long double op, mpfr_rnd_t rnd)
Function: int mpfr_set_float128 (mpfr_t rop, _Float128 op, mpfr_rnd_t rnd)
Function: int mpfr_set_decimal64 (mpfr_t rop, _Decimal64 op, mpfr_rnd_t rnd)
Function: int mpfr_set_decimal128 (mpfr_t rop, _Decimal128 op, mpfr_rnd_t rnd)
Function: int mpfr_set_z (mpfr_t rop, mpz_t op, mpfr_rnd_t rnd)
Function: int mpfr_set_q (mpfr_t rop, mpq_t op, mpfr_rnd_t rnd)
Function: int mpfr_set_f (mpfr_t rop, mpf_t op, mpfr_rnd_t rnd)

Set the value of rop from op, rounded toward the given direction rnd. Note that the input 0 is converted to +0 by mpfr_set_ui, mpfr_set_si, mpfr_set_uj, mpfr_set_sj, The mpfr_set_float128 function is built only with the configure option ‘--enable-float128’, which requires the compiler or system provides the ‘_Float128’ data type (GCC 4.3 or later supports this data type); to use mpfr_set_float128, one should define the macro MPFR_WANT_FLOAT128 before including mpfr.h. mpfr_set_z, mpfr_set_q and mpfr_set_f, regardless of the rounding mode. If the system does not support the IEEE 754 standard, mpfr_set_flt, mpfr_set_d, mpfr_set_ld, mpfr_set_decimal64 and mpfr_set_decimal128 might not preserve the signed zeros. The mpfr_set_decimal64 and mpfr_set_decimal128 functions are built only with the configure option ‘--enable-decimal-float’, and when the compiler or system provides the ‘_Decimal64’ and ‘_Decimal128’ data type; to use those functions, one should define the macro MPFR_WANT_DECIMAL_FLOATS before including mpfr.h. mpfr_set_q might fail if the numerator (or the denominator) cannot be represented as a mpfr_t.

For mpfr_set, the sign of a NaN is propagated in order to mimic the IEEE 754 copy operation. But contrary to IEEE 754, the NaN flag is set as usual.

Note: If you want to store a floating-point constant to a mpfr_t, you should use mpfr_set_str (or one of the MPFR constant functions, such as mpfr_const_pi for Pi) instead of mpfr_set_flt, mpfr_set_d, mpfr_set_ld, mpfr_set_decimal64 or mpfr_set_decimal128. Otherwise the floating-point constant will be first converted into a reduced-precision (e.g., 53-bit) binary (or decimal, for mpfr_set_decimal64 and mpfr_set_decimal128) number before MPFR can work with it.

Function: int mpfr_set_ui_2exp (mpfr_t rop, unsigned long int op, mpfr_exp_t e, mpfr_rnd_t rnd)
Function: int mpfr_set_si_2exp (mpfr_t rop, long int op, mpfr_exp_t e, mpfr_rnd_t rnd)
Function: int mpfr_set_uj_2exp (mpfr_t rop, uintmax_t op, intmax_t e, mpfr_rnd_t rnd)
Function: int mpfr_set_sj_2exp (mpfr_t rop, intmax_t op, intmax_t e, mpfr_rnd_t rnd)
Function: int mpfr_set_z_2exp (mpfr_t rop, mpz_t op, mpfr_exp_t e, mpfr_rnd_t rnd)

Set the value of rop from op multiplied by two to the power e, rounded toward the given direction rnd. Note that the input 0 is converted to +0.

Function: int mpfr_set_str (mpfr_t rop, const char *s, int base, mpfr_rnd_t rnd)

Set rop to the value of the string s in base base, rounded in the direction rnd. See the documentation of mpfr_strtofr for a detailed description of the valid string formats. Contrary to mpfr_strtofr, mpfr_set_str requires the whole string to represent a valid floating-point number.

The meaning of the return value differs from other MPFR functions: it is 0 if the entire string up to the final null character is a valid number in base base; otherwise it is -1, and rop may have changed (users interested in the ternary value should use mpfr_strtofr instead).

Note: it is preferable to use mpfr_strtofr if one wants to distinguish between an infinite rop value coming from an infinite s or from an overflow.

Function: int mpfr_strtofr (mpfr_t rop, const char *nptr, char **endptr, int base, mpfr_rnd_t rnd)

Read a floating-point number from a string nptr in base base, rounded in the direction rnd; base must be either 0 (to detect the base, as described below) or a number from 2 to 62 (otherwise the behavior is undefined). If nptr starts with valid data, the result is stored in rop and *endptr points to the character just after the valid data (if endptr is not a null pointer); otherwise rop is set to zero (for consistency with strtod) and the value of nptr is stored in the location referenced by endptr (if endptr is not a null pointer). The usual ternary value is returned.

Parsing follows the standard C strtod function with some extensions. After optional leading whitespace, one has a subject sequence consisting of an optional sign (‘+’ or ‘-’), and either numeric data or special data. The subject sequence is defined as the longest initial subsequence of the input string, starting with the first non-whitespace character, that is of the expected form.

The form of numeric data is a non-empty sequence of significand digits with an optional decimal-point character, and an optional exponent consisting of an exponent prefix followed by an optional sign and a non-empty sequence of decimal digits. A significand digit is either a decimal digit or a Latin letter (62 possible characters), with ‘A’ = 10, ‘B’ = 11, …, ‘Z’ = 35; case is ignored in bases less than or equal to 36, in bases larger than 36, ‘a’ = 36, ‘b’ = 37, …, ‘z’ = 61. The value of a significand digit must be strictly less than the base. The decimal-point character can be either the one defined by the current locale or the period (the first one is accepted for consistency with the C standard and the practice, the second one is accepted to allow the programmer to provide MPFR numbers from strings in a way that does not depend on the current locale). The exponent prefix can be ‘e’ or ‘E’ for bases up to 10, or ‘@’ in any base; it indicates a multiplication by a power of the base. In bases 2 and 16, the exponent prefix can also be ‘p’ or ‘P’, in which case the exponent, called binary exponent, indicates a multiplication by a power of 2 instead of the base (there is a difference only for base 16); in base 16 for example ‘1p2’ represents 4 whereas ‘1@2’ represents 256. The value of an exponent is always written in base 10.

If the argument base is 0, then the base is automatically detected as follows. If the significand starts with ‘0b’ or ‘0B’, base 2 is assumed. If the significand starts with ‘0x’ or ‘0X’, base 16 is assumed. Otherwise base 10 is assumed.

Note: The exponent (if present) must contain at least a digit. Otherwise the possible exponent prefix and sign are not part of the number (which ends with the significand). Similarly, if ‘0b’, ‘0B’, ‘0x’ or ‘0X’ is not followed by a binary/hexadecimal digit, then the subject sequence stops at the character ‘0’, thus 0 is read.

Special data (for infinities and NaN) can be ‘@inf@’ or ‘@nan@(n-char-sequence-opt)’, and if base <= 16, it can also be ‘infinity’, ‘inf’, ‘nan’ or ‘nan(n-char-sequence-opt)’, all case insensitive. A ‘n-char-sequence-opt’ is a possibly empty string containing only digits, Latin letters and the underscore (0, 1, 2, …, 9, a, b, …, z, A, B, …, Z, _). Note: one has an optional sign for all data, even NaN. For example, ‘-@nAn@(This_Is_Not_17)’ is a valid representation for NaN in base 17.

Function: void mpfr_set_nan (mpfr_t x)
Function: void mpfr_set_inf (mpfr_t x, int sign)
Function: void mpfr_set_zero (mpfr_t x, int sign)

Set the variable x to NaN (Not-a-Number), infinity or zero respectively. In mpfr_set_inf or mpfr_set_zero, x is set to plus infinity or plus zero iff sign is nonnegative; in mpfr_set_nan, the sign bit of the result is unspecified.

Function: void mpfr_swap (mpfr_t x, mpfr_t y)

Swap the structures pointed to by x and y. In particular, the values are exchanged without rounding (this may be different from three mpfr_set calls using a third auxiliary variable).

Warning! Since the precisions are exchanged, this will affect future assignments. Moreover, since the significand pointers are also exchanged, you must not use this function if the allocation method used for x and/or y does not permit it. This is the case when x and/or y were declared and initialized with MPFR_DECL_INIT, and possibly with mpfr_custom_init_set (see Custom Interface).

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5.3 Combined Initialization and Assignment Functions

Macro: int mpfr_init_set (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Macro: int mpfr_init_set_ui (mpfr_t rop, unsigned long int op, mpfr_rnd_t rnd)
Macro: int mpfr_init_set_si (mpfr_t rop, long int op, mpfr_rnd_t rnd)
Macro: int mpfr_init_set_d (mpfr_t rop, double op, mpfr_rnd_t rnd)
Macro: int mpfr_init_set_ld (mpfr_t rop, long double op, mpfr_rnd_t rnd)
Macro: int mpfr_init_set_z (mpfr_t rop, mpz_t op, mpfr_rnd_t rnd)
Macro: int mpfr_init_set_q (mpfr_t rop, mpq_t op, mpfr_rnd_t rnd)
Macro: int mpfr_init_set_f (mpfr_t rop, mpf_t op, mpfr_rnd_t rnd)

Initialize rop and set its value from op, rounded in the direction rnd. The precision of rop will be taken from the active default precision, as set by mpfr_set_default_prec.

Function: int mpfr_init_set_str (mpfr_t x, const char *s, int base, mpfr_rnd_t rnd)

Initialize x and set its value from the string s in base base, rounded in the direction rnd. See mpfr_set_str.

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5.4 Conversion Functions

Function: float mpfr_get_flt (mpfr_t op, mpfr_rnd_t rnd)
Function: double mpfr_get_d (mpfr_t op, mpfr_rnd_t rnd)
Function: long double mpfr_get_ld (mpfr_t op, mpfr_rnd_t rnd)
Function: _Float128 mpfr_get_float128 (mpfr_t op, mpfr_rnd_t rnd)
Function: _Decimal64 mpfr_get_decimal64 (mpfr_t op, mpfr_rnd_t rnd)
Function: _Decimal128 mpfr_get_decimal128 (mpfr_t op, mpfr_rnd_t rnd)

Convert op to a float (respectively double, long double, _Decimal64, or _Decimal128) using the rounding mode rnd. If op is NaN, some fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is returned. If op is ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If op is zero, these functions return a zero, trying to preserve its sign, if possible. The mpfr_get_float128, mpfr_get_decimal64 and mpfr_get_decimal128 functions are built only under some conditions: see the documentation of mpfr_set_float128, mpfr_set_decimal64 and mpfr_set_decimal128 respectively.

Function: long mpfr_get_si (mpfr_t op, mpfr_rnd_t rnd)
Function: unsigned long mpfr_get_ui (mpfr_t op, mpfr_rnd_t rnd)
Function: intmax_t mpfr_get_sj (mpfr_t op, mpfr_rnd_t rnd)
Function: uintmax_t mpfr_get_uj (mpfr_t op, mpfr_rnd_t rnd)

Convert op to a long, an unsigned long, an intmax_t or an uintmax_t (respectively) after rounding it to an integer with respect to rnd. If op is NaN, 0 is returned and the erange flag is set. If op is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the erange flag is set too. When there is no such range error, if the return value differs from op, i.e., if op is not an integer, the inexact flag is set. See also mpfr_fits_slong_p, mpfr_fits_ulong_p, mpfr_fits_intmax_p and mpfr_fits_uintmax_p.

Function: double mpfr_get_d_2exp (long *exp, mpfr_t op, mpfr_rnd_t rnd)
Function: long double mpfr_get_ld_2exp (long *exp, mpfr_t op, mpfr_rnd_t rnd)

Return d and set exp (formally, the value pointed to by exp) such that 0.5<=abs(d)<1 and d times 2 raised to exp equals op rounded to double (resp. long double) precision, using the given rounding mode. If op is zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is returned, and exp is set to 0. If op is NaN or an infinity, then the corresponding double precision (resp. long-double precision) value is returned, and exp is undefined.

Function: int mpfr_frexp (mpfr_exp_t *exp, mpfr_t y, mpfr_t x, mpfr_rnd_t rnd)

Set exp (formally, the value pointed to by exp) and y such that 0.5<=abs(y)<1 and y times 2 raised to exp equals x rounded to the precision of y, using the given rounding mode. If x is zero, then y is set to a zero of the same sign and exp is set to 0. If x is NaN or an infinity, then y is set to the same value and exp is undefined.

Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t rop, mpfr_t op)

Put the scaled significand of op (regarded as an integer, with the precision of op) into rop, and return the exponent exp (which may be outside the current exponent range) such that op exactly equals rop times 2 raised to the power exp. If op is zero, the minimal exponent emin is returned. If op is NaN or an infinity, the erange flag is set, rop is set to 0, and the the minimal exponent emin is returned. The returned exponent may be less than the minimal exponent emin of MPFR numbers in the current exponent range; in case the exponent is not representable in the mpfr_exp_t type, the erange flag is set and the minimal value of the mpfr_exp_t type is returned.

Function: int mpfr_get_z (mpz_t rop, mpfr_t op, mpfr_rnd_t rnd)

Convert op to a mpz_t, after rounding it with respect to rnd. If op is NaN or an infinity, the erange flag is set, rop is set to 0, and 0 is returned. Otherwise the return value is zero when rop is equal to op (i.e., when op is an integer), positive when it is greater than op, and negative when it is smaller than op; moreover, if rop differs from op, i.e., if op is not an integer, the inexact flag is set.

Function: void mpfr_get_q (mpq_t rop, mpfr_t op)

Convert op to a mpq_t. If op is NaN or an infinity, the erange flag is set and rop is set to 0. Otherwise the conversion is always exact.

Function: int mpfr_get_f (mpf_t rop, mpfr_t op, mpfr_rnd_t rnd)

Convert op to a mpf_t, after rounding it with respect to rnd. The erange flag is set if op is NaN or an infinity, which do not exist in MPF. If op is NaN, then rop is undefined. If op is +Inf (resp. -Inf), then rop is set to the maximum (resp. minimum) value in the precision of the MPF number; if a future MPF version supports infinities, this behavior will be considered incorrect and will change (portable programs should assume that rop is set either to this finite number or to an infinite number). Note that since MPFR currently has the same exponent type as MPF (but not with the same radix), the range of values is much larger in MPF than in MPFR, so that an overflow or underflow is not possible.

Function: size_t mpfr_get_str_ndigits (int b, mpfr_prec_t p)

Return the minimal integer m such that any number of p bits, when output with m digits in radix b with rounding to nearest, can be recovered exactly when read again, still with rounding to nearest. More precisely, we have m = 1 + ceil(p*log(2)/log(b)), with p replaced by p-1 if b is a power of 2.

The argument b must be in the range 2 to 62; this is the range of bases supported by the mpfr_get_str function. Note that contrary to the base argument of this function, negative values are not accepted.

Function: char * mpfr_get_str (char *str, mpfr_exp_t *expptr, int base, size_t n, mpfr_t op, mpfr_rnd_t rnd)

Convert op to a string of digits in base abs(base), with rounding in the direction rnd, where n is either zero (see below) or the number of significant digits output in the string. The argument base may vary from 2 to 62 or from -2 to -36; otherwise the function does nothing and immediately returns a null pointer.

For base in the range 2 to 36, digits and lower-case letters are used; for -2 to -36, digits and upper-case letters are used; for 37 to 62, digits, upper-case letters, and lower-case letters, in that significance order, are used. Warning! This implies that for base > 10, the successor of the digit 9 depends on base. This choice has been done for compatibility with GMP’s mpf_get_str function. Users who wish a more consistent behavior should write a simple wrapper.

If the input is NaN, then the returned string is ‘@NaN@’ and the NaN flag is set. If the input is +Inf (resp. -Inf), then the returned string is ‘@Inf@’ (resp. ‘-@Inf@’).

If the input number is a finite number, the exponent is written through the pointer expptr (for input 0, the current minimal exponent is written); the type mpfr_exp_t is large enough to hold the exponent in all cases.

The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number -3.1416 would be returned as "-31416" in the string and 1 written at expptr. If rnd is to nearest, and op is exactly in the middle of two consecutive possible outputs, the one with an even significand is chosen, where both significands are considered with the exponent of op. Note that for an odd base, this may not correspond to an even last digit: for example with 2 digits in base 7, (14) and a half is rounded to (15) which is 12 in decimal, (16) and a half is rounded to (20) which is 14 in decimal, and (26) and a half is rounded to (26) which is 20 in decimal.

If n is zero, the number of digits of the significand is taken as mpfr_get_str_ndigits(base,p) where p is the precision of op (see mpfr_get_str_ndigits).

If str is a null pointer, space for the significand is allocated using the allocation function (see Memory Handling) and a pointer to the string is returned (unless the base is invalid). To free the returned string, you must use mpfr_free_str.

If str is not a null pointer, it should point to a block of storage large enough for the significand. A safe block size (sufficient for any value) is max(n + 2, 7) if n is not zero; if n is zero, replace it by mpfr_get_str_ndigits(base,p) where p is the precision of op, as mentioned above. The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for ‘-@Inf@’ plus the terminating null character. The pointer to the string str is returned (unless the base is invalid).

Like in usual functions, the inexact flag is set iff the result is inexact.

Function: void mpfr_free_str (char *str)

Free a string allocated by mpfr_get_str using the unallocation function (see Memory Handling). The block is assumed to be strlen(str)+1 bytes.

Function: int mpfr_fits_ulong_p (mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_fits_slong_p (mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_fits_uint_p (mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_fits_sint_p (mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_fits_ushort_p (mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_fits_sshort_p (mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_fits_uintmax_p (mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_fits_intmax_p (mpfr_t op, mpfr_rnd_t rnd)

Return non-zero if op would fit in the respective C data type, respectively unsigned long, long, unsigned int, int, unsigned short, short, uintmax_t, intmax_t, when rounded to an integer in the direction rnd. For instance, with the MPFR_RNDU rounding mode on -0.5, the result will be non-zero for all these functions. For MPFR_RNDF, those functions return non-zero when it is guaranteed that the corresponding conversion function (for example mpfr_get_ui for mpfr_fits_ulong_p), when called with faithful rounding, will always return a number that is representable in the corresponding type. As a consequence, for MPFR_RNDF, mpfr_fits_ulong_p will return non-zero for a non-negative number less than or equal to ULONG_MAX.

Next: , Previous: , Up: MPFR Interface   [Index]

5.5 Arithmetic Functions

Function: int mpfr_add (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_add_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_add_si (mpfr_t rop, mpfr_t op1, long int op2, mpfr_rnd_t rnd)
Function: int mpfr_add_d (mpfr_t rop, mpfr_t op1, double op2, mpfr_rnd_t rnd)
Function: int mpfr_add_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mpfr_rnd_t rnd)
Function: int mpfr_add_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mpfr_rnd_t rnd)

Set rop to op1 + op2 rounded in the direction rnd. The IEEE 754 rules are used, in particular for signed zeros. But for types having no signed zeros, 0 is considered unsigned (i.e., (+0) + 0 = (+0) and (-0) + 0 = (-0)). The mpfr_add_d function assumes that the radix of the double type is a power of 2, with a precision at most that declared by the C implementation (macro IEEE_DBL_MANT_DIG, and if not defined 53 bits).

Function: int mpfr_sub (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_ui_sub (mpfr_t rop, unsigned long int op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_sub_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_si_sub (mpfr_t rop, long int op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_sub_si (mpfr_t rop, mpfr_t op1, long int op2, mpfr_rnd_t rnd)
Function: int mpfr_d_sub (mpfr_t rop, double op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_sub_d (mpfr_t rop, mpfr_t op1, double op2, mpfr_rnd_t rnd)
Function: int mpfr_z_sub (mpfr_t rop, mpz_t op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_sub_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mpfr_rnd_t rnd)
Function: int mpfr_sub_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mpfr_rnd_t rnd)

Set rop to op1 - op2 rounded in the direction rnd. The IEEE 754 rules are used, in particular for signed zeros. But for types having no signed zeros, 0 is considered unsigned (i.e., (+0) - 0 = (+0), (-0) - 0 = (-0), 0 - (+0) = (-0) and 0 - (-0) = (+0)). The same restrictions than for mpfr_add_d apply to mpfr_d_sub and mpfr_sub_d.

Function: int mpfr_mul (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_mul_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_mul_si (mpfr_t rop, mpfr_t op1, long int op2, mpfr_rnd_t rnd)
Function: int mpfr_mul_d (mpfr_t rop, mpfr_t op1, double op2, mpfr_rnd_t rnd)
Function: int mpfr_mul_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mpfr_rnd_t rnd)
Function: int mpfr_mul_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mpfr_rnd_t rnd)

Set rop to op1 times op2 rounded in the direction rnd. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zeros, 0 is considered positive). The same restrictions than for mpfr_add_d apply to mpfr_mul_d.

Function: int mpfr_sqr (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the square of op rounded in the direction rnd.

Function: int mpfr_div (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_ui_div (mpfr_t rop, unsigned long int op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_div_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_si_div (mpfr_t rop, long int op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_div_si (mpfr_t rop, mpfr_t op1, long int op2, mpfr_rnd_t rnd)
Function: int mpfr_d_div (mpfr_t rop, double op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_div_d (mpfr_t rop, mpfr_t op1, double op2, mpfr_rnd_t rnd)
Function: int mpfr_div_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mpfr_rnd_t rnd)
Function: int mpfr_div_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mpfr_rnd_t rnd)

Set rop to op1/op2 rounded in the direction rnd. When a result is zero, its sign is the product of the signs of the operands. For types having no signed zeros, 0 is considered positive; but note that if op1 is non-zero and op2 is zero, the result might change from ±Inf to NaN in future MPFR versions if there is an opposite decision on the IEEE 754 side. The same restrictions than for mpfr_add_d apply to mpfr_d_div and mpfr_div_d.

Function: int mpfr_sqrt (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_sqrt_ui (mpfr_t rop, unsigned long int op, mpfr_rnd_t rnd)

Set rop to the square root of op rounded in the direction rnd. Set rop to -0 if op is -0, to be consistent with the IEEE 754 standard. Set rop to NaN if op is negative.

Function: int mpfr_rec_sqrt (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the reciprocal square root of op rounded in the direction rnd. Set rop to +Inf if op is ±0, +0 if op is +Inf, and NaN if op is negative. Warning! Therefore the result on -0 is different from the one of the rSqrt function recommended by the IEEE 754-2008 standard (Section 9.2.1), which is -Inf instead of +Inf.

Function: int mpfr_cbrt (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_rootn_ui (mpfr_t rop, mpfr_t op, unsigned long int n, mpfr_rnd_t rnd)

Set rop to the nth root (with n = 3, the cubic root, for mpfr_cbrt) of op rounded in the direction rnd. For n = 0, set rop to NaN. For n odd (resp. even) and op negative (including -Inf), set rop to a negative number (resp. NaN). If op is zero, set rop to zero with the sign obtained by the usual limit rules, i.e., the same sign as op if n is odd, and positive if n is even.

These functions agree with the rootn function of the IEEE 754-2008 standard and the P754/D2.41 draft of the next standard (Section 9.2). Note that it is here restricted to n >= 0. Functions allowing a negative n may be implemented in the future.

Function: int mpfr_root (mpfr_t rop, mpfr_t op, unsigned long int n, mpfr_rnd_t rnd)

This function is the same as mpfr_rootn_ui except when op is -0 and n is even: the result is -0 instead of +0 (the reason was to be consistent with mpfr_sqrt). Said otherwise, if op is zero, set rop to op.

This function predates the IEEE 754-2008 standard and behaves differently from its rootn function. It is marked as deprecated and will be removed in a future release.

Function: int mpfr_neg (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_abs (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to -op and the absolute value of op respectively, rounded in the direction rnd. Just changes or adjusts the sign if rop and op are the same variable, otherwise a rounding might occur if the precision of rop is less than that of op.

The sign rule also applies to NaN in order to mimic the IEEE 754 negate and abs operations, i.e., for mpfr_neg, the sign is reversed, and for mpfr_abs, the sign is set to positive. But contrary to IEEE 754, the NaN flag is set as usual.

Function: int mpfr_dim (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)

Set rop to the positive difference of op1 and op2, i.e., op1 - op2 rounded in the direction rnd if op1 > op2, +0 if op1 <= op2, and NaN if op1 or op2 is NaN.

Function: int mpfr_mul_2ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_mul_2si (mpfr_t rop, mpfr_t op1, long int op2, mpfr_rnd_t rnd)

Set rop to op1 times 2 raised to op2 rounded in the direction rnd. Just increases the exponent by op2 when rop and op1 are identical.

Function: int mpfr_div_2ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_div_2si (mpfr_t rop, mpfr_t op1, long int op2, mpfr_rnd_t rnd)

Set rop to op1 divided by 2 raised to op2 rounded in the direction rnd. Just decreases the exponent by op2 when rop and op1 are identical.

Function: int mpfr_fac_ui (mpfr_t rop, unsigned long int op, mpfr_rnd_t rnd)

Set rop to the factorial of op, rounded in the direction rnd.

Function: int mpfr_fma (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_t op3, mpfr_rnd_t rnd)
Function: int mpfr_fms (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_t op3, mpfr_rnd_t rnd)

Set rop to (op1 times op2) + op3 (resp. (op1 times op2) - op3) rounded in the direction rnd. Concerning special values (signed zeros, infinities, NaN), these functions behave like a multiplication followed by a separate addition or subtraction. That is, the fused operation matters only for rounding.

Function: int mpfr_fmma (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_t op3, mpfr_t op4, mpfr_rnd_t rnd)
Function: int mpfr_fmms (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_t op3, mpfr_t op4, mpfr_rnd_t rnd)

Set rop to (op1 times op2) + (op3 times op4) (resp. (op1 times op2) - (op3 times op4)) rounded in the direction rnd. In case the computation of op1 times op2 overflows or underflows (or that of op3 times op4), the result rop is computed as if the two intermediate products were computed with rounding toward zero.

Function: int mpfr_hypot (mpfr_t rop, mpfr_t x, mpfr_t y, mpfr_rnd_t rnd)

Set rop to the Euclidean norm of x and y, i.e., the square root of the sum of the squares of x and y, rounded in the direction rnd. Special values are handled as described in the ISO C99 (Section F.9.4.3) and IEEE 754-2008 (Section 9.2.1) standards: If x or y is an infinity, then +Inf is returned in rop, even if the other number is NaN.

Function: int mpfr_sum (mpfr_t rop, const mpfr_ptr tab[], unsigned long int n, mpfr_rnd_t rnd)

Set rop to the sum of all elements of tab, whose size is n, correctly rounded in the direction rnd. Warning: for efficiency reasons, tab is an array of pointers to mpfr_t, not an array of mpfr_t. If n = 0, then the result is +0, and if n = 1, then the function is equivalent to mpfr_set. For the special exact cases, the result is the same as the one obtained with a succession of additions (mpfr_add) in infinite precision. In particular, if the result is an exact zero and n >= 1:

  • if all the inputs have the same sign (i.e., all +0 or all -0), then the result has the same sign as the inputs;
  • otherwise, either because all inputs are zeros with at least a +0 and a -0, or because some inputs are non-zero (but they globally cancel), the result is +0, except for the MPFR_RNDD rounding mode, where it is -0.
Function: int mpfr_dot (mpfr_t rop, const mpfr_ptr a[], const mpfr_ptr b[], unsigned long int n, mpfr_rnd_t rnd)

Set rop to the dot product of elements of a by those of b, whose common size is n, correctly rounded in the direction rnd. Warning: for efficiency reasons, a and b are arrays of pointers to mpfr_t. This function is experimental, and does not yet handle intermediate overflows and underflows.

For the power functions (with an integer exponent or not), see mpfr_pow in Transcendental Functions.

Next: , Previous: , Up: MPFR Interface   [Index]

5.6 Comparison Functions

Function: int mpfr_cmp (mpfr_t op1, mpfr_t op2)
Function: int mpfr_cmp_ui (mpfr_t op1, unsigned long int op2)
Function: int mpfr_cmp_si (mpfr_t op1, long int op2)
Function: int mpfr_cmp_d (mpfr_t op1, double op2)
Function: int mpfr_cmp_ld (mpfr_t op1, long double op2)
Function: int mpfr_cmp_z (mpfr_t op1, mpz_t op2)
Function: int mpfr_cmp_q (mpfr_t op1, mpq_t op2)
Function: int mpfr_cmp_f (mpfr_t op1, mpf_t op2)

Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2. Both op1 and op2 are considered to their full own precision, which may differ. If one of the operands is NaN, set the erange flag and return zero.

Note: These functions may be useful to distinguish the three possible cases. If you need to distinguish two cases only, it is recommended to use the predicate functions (e.g., mpfr_equal_p for the equality) described below; they behave like the IEEE 754 comparisons, in particular when one or both arguments are NaN. But only floating-point numbers can be compared (you may need to do a conversion first).

Function: int mpfr_cmp_ui_2exp (mpfr_t op1, unsigned long int op2, mpfr_exp_t e)
Function: int mpfr_cmp_si_2exp (mpfr_t op1, long int op2, mpfr_exp_t e)

Compare op1 and op2 multiplied by two to the power e. Similar as above.

Function: int mpfr_cmpabs (mpfr_t op1, mpfr_t op2)
Function: int mpfr_cmpabs_ui (mpfr_t op1, unsigned long op2)

Compare |op1| and |op2|. Return a positive value if |op1| > |op2|, zero if |op1| = |op2|, and a negative value if |op1| < |op2|. If one of the operands is NaN, set the erange flag and return zero.

Function: int mpfr_nan_p (mpfr_t op)
Function: int mpfr_inf_p (mpfr_t op)
Function: int mpfr_number_p (mpfr_t op)
Function: int mpfr_zero_p (mpfr_t op)
Function: int mpfr_regular_p (mpfr_t op)

Return non-zero if op is respectively NaN, an infinity, an ordinary number (i.e., neither NaN nor an infinity), zero, or a regular number (i.e., neither NaN, nor an infinity nor zero). Return zero otherwise.

Macro: int mpfr_sgn (mpfr_t op)

Return a positive value if op > 0, zero if op = 0, and a negative value if op < 0. If the operand is NaN, set the erange flag and return zero. This is equivalent to mpfr_cmp_ui (op, 0), but more efficient.

Function: int mpfr_greater_p (mpfr_t op1, mpfr_t op2)
Function: int mpfr_greaterequal_p (mpfr_t op1, mpfr_t op2)
Function: int mpfr_less_p (mpfr_t op1, mpfr_t op2)
Function: int mpfr_lessequal_p (mpfr_t op1, mpfr_t op2)
Function: int mpfr_equal_p (mpfr_t op1, mpfr_t op2)

Return non-zero if op1 > op2, op1 >= op2, op1 < op2, op1 <= op2, op1 = op2 respectively, and zero otherwise. Those functions return zero whenever op1 and/or op2 is NaN.

Function: int mpfr_lessgreater_p (mpfr_t op1, mpfr_t op2)

Return non-zero if op1 < op2 or op1 > op2 (i.e., neither op1, nor op2 is NaN, and op1 <> op2), zero otherwise (i.e., op1 and/or op2 is NaN, or op1 = op2).

Function: int mpfr_unordered_p (mpfr_t op1, mpfr_t op2)

Return non-zero if op1 or op2 is a NaN (i.e., they cannot be compared), zero otherwise.

Function: int mpfr_total_order_p (mpfr_t x, mpfr_t y)

This function implements the totalOrder predicate from IEEE 754-2008, where -NaN < -Inf < negative finite numbers < -0 < +0 < positive finite numbers < +Inf < +NaN. It returns a non-zero value (true) when x is smaller than or equal to y for this order relation, and zero (false) otherwise. Contrary to mpfr_cmp (x, y), which returns a ternary value, mpfr_total_order_p returns a binary value (zero or non-zero). In particular, mpfr_total_order_p (x, x) returns true, mpfr_total_order_p (-0, +0) returns true and mpfr_total_order_p (+0, -0) returns false. The sign bit of NaN also matters.

Next: , Previous: , Up: MPFR Interface   [Index]

5.7 Transcendental Functions

All those functions, except explicitly stated (for example mpfr_sin_cos), return a ternary value, i.e., zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise.

Important note: In some domains, computing transcendental functions (even more with correct rounding) is expensive, even in small precision, for example the trigonometric and Bessel functions with a large argument. For some functions, the algorithm complexity and memory usage does not depend only on the output precision: for instance, the memory usage of mpfr_rootn_ui is also linear in the argument k, and the memory usage of the incomplete Gamma function also depends on the precision of the input op. It is also theoretically possible that some functions on some particular inputs might be very hard to round (i.e. the Table Maker’s Dilemma occurs in much larger precisions than normally expected from the context), meaning that the internal precision needs to be increased even more; but it is conjectured that the needed precision has a reasonable bound.

Function: int mpfr_log (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_log_ui (mpfr_t rop, unsigned long op, mpfr_rnd_t rnd)
Function: int mpfr_log2 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_log10 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the natural logarithm of op, log2(op) or log10(op), respectively, rounded in the direction rnd. Set rop to +0 if op is 1 (in all rounding modes), for consistency with the ISO C99 and IEEE 754-2008 standards. Set rop to -Inf if op is ±0 (i.e., the sign of the zero has no influence on the result).

Function: int mpfr_log1p (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the logarithm of one plus op, rounded in the direction rnd. Set rop to -Inf if op is -1.

Function: int mpfr_exp (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_exp2 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_exp10 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the exponential of op, to 2 power of op or to 10 power of op, respectively, rounded in the direction rnd.

Function: int mpfr_expm1 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the exponential of op followed by a subtraction by one, rounded in the direction rnd.

Function: int mpfr_pow (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_pow_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_pow_si (mpfr_t rop, mpfr_t op1, long int op2, mpfr_rnd_t rnd)
Function: int mpfr_pow_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mpfr_rnd_t rnd)
Function: int mpfr_ui_pow_ui (mpfr_t rop, unsigned long int op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_ui_pow (mpfr_t rop, unsigned long int op1, mpfr_t op2, mpfr_rnd_t rnd)

Set rop to op1 raised to op2, rounded in the direction rnd. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the pow function:

  • pow(±0, y) returns plus or minus infinity for y a negative odd integer.
  • pow(±0, y) returns plus infinity for y negative and not an odd integer.
  • pow(±0, y) returns plus or minus zero for y a positive odd integer.
  • pow(±0, y) returns plus zero for y positive and not an odd integer.
  • pow(-1, ±Inf) returns 1.
  • pow(+1, y) returns 1 for any y, even a NaN.
  • pow(x, ±0) returns 1 for any x, even a NaN.
  • pow(x, y) returns NaN for finite negative x and finite non-integer y.
  • pow(x, -Inf) returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1.
  • pow(x, +Inf) returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1.
  • pow(-Inf, y) returns minus zero for y a negative odd integer.
  • pow(-Inf, y) returns plus zero for y negative and not an odd integer.
  • pow(-Inf, y) returns minus infinity for y a positive odd integer.
  • pow(-Inf, y) returns plus infinity for y positive and not an odd integer.
  • pow(+Inf, y) returns plus zero for y negative, and plus infinity for y positive.

Note: When 0 is of integer type, it is regarded as +0 by these functions. We do not use the usual limit rules in this case, as these rules are not used for pow.

Function: int mpfr_cos (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_sin (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_tan (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the cosine of op, sine of op, tangent of op, rounded in the direction rnd.

Function: int mpfr_sin_cos (mpfr_t sop, mpfr_t cop, mpfr_t op, mpfr_rnd_t rnd)

Set simultaneously sop to the sine of op and cop to the cosine of op, rounded in the direction rnd with the corresponding precisions of sop and cop, which must be different variables. Return 0 iff both results are exact, more precisely it returns s+4c where s=0 if sop is exact, s=1 if sop is larger than the sine of op, s=2 if sop is smaller than the sine of op, and similarly for c and the cosine of op.

Function: int mpfr_sec (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_csc (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_cot (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.

Function: int mpfr_acos (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_asin (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_atan (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the arc-cosine, arc-sine or arc-tangent of op, rounded in the direction rnd. Note that since acos(-1) returns the floating-point number closest to Pi according to the given rounding mode, this number might not be in the output range 0 <= rop < Pi of the arc-cosine function; still, the result lies in the image of the output range by the rounding function. The same holds for asin(-1), asin(1), atan(-Inf), atan(+Inf) or for atan(op) with large op and small precision of rop.

Function: int mpfr_atan2 (mpfr_t rop, mpfr_t y, mpfr_t x, mpfr_rnd_t rnd)

Set rop to the arc-tangent2 of y and x, rounded in the direction rnd: if x > 0, atan2(y, x) = atan(y/x); if x < 0, atan2(y, x) = sign(y)*(Pi - atan(abs(y/x))), thus a number from -Pi to Pi. As for atan, in case the exact mathematical result is +Pi or -Pi, its rounded result might be outside the function output range.

atan2(y, 0) does not raise any floating-point exception. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the atan2 function:

  • atan2(+0, -0) returns +Pi.
  • atan2(-0, -0) returns -Pi.
  • atan2(+0, +0) returns +0.
  • atan2(-0, +0) returns -0.
  • atan2(+0, x) returns +Pi for x < 0.
  • atan2(-0, x) returns -Pi for x < 0.
  • atan2(+0, x) returns +0 for x > 0.
  • atan2(-0, x) returns -0 for x > 0.
  • atan2(y, 0) returns -Pi/2 for y < 0.
  • atan2(y, 0) returns +Pi/2 for y > 0.
  • atan2(+Inf, -Inf) returns +3*Pi/4.
  • atan2(-Inf, -Inf) returns -3*Pi/4.
  • atan2(+Inf, +Inf) returns +Pi/4.
  • atan2(-Inf, +Inf) returns -Pi/4.
  • atan2(+Inf, x) returns +Pi/2 for finite x.
  • atan2(-Inf, x) returns -Pi/2 for finite x.
  • atan2(y, -Inf) returns +Pi for finite y > 0.
  • atan2(y, -Inf) returns -Pi for finite y < 0.
  • atan2(y, +Inf) returns +0 for finite y > 0.
  • atan2(y, +Inf) returns -0 for finite y < 0.
Function: int mpfr_cosh (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_sinh (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_tanh (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the hyperbolic cosine, sine or tangent of op, rounded in the direction rnd.

Function: int mpfr_sinh_cosh (mpfr_t sop, mpfr_t cop, mpfr_t op, mpfr_rnd_t rnd)

Set simultaneously sop to the hyperbolic sine of op and cop to the hyperbolic cosine of op, rounded in the direction rnd with the corresponding precision of sop and cop, which must be different variables. Return 0 iff both results are exact (see mpfr_sin_cos for a more detailed description of the return value).

Function: int mpfr_sech (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_csch (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_coth (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the hyperbolic secant of op, cosecant of op, cotangent of op, rounded in the direction rnd.

Function: int mpfr_acosh (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_asinh (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_atanh (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the inverse hyperbolic cosine, sine or tangent of op, rounded in the direction rnd.

Function: int mpfr_eint (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the exponential integral of op, rounded in the direction rnd. This is the sum of Euler’s constant, of the logarithm of the absolute value of op, and of the sum for k from 1 to infinity of op to the power k, divided by k and factorial(k). For positive op, it corresponds to the Ei function at op (see formula 5.1.10 from the Handbook of Mathematical Functions from Abramowitz and Stegun), and for negative op, to the opposite of the E1 function (sometimes called eint1) at -op (formula 5.1.1 from the same reference).

Function: int mpfr_li2 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to real part of the dilogarithm of op, rounded in the direction rnd. MPFR defines the dilogarithm function as the integral of -log(1-t)/t from 0 to op.

Function: int mpfr_gamma (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_gamma_inc (mpfr_t rop, mpfr_t op, mpfr_t op2, mpfr_rnd_t rnd)

Set rop to the value of the Gamma function on op, resp. the incomplete Gamma function on op and op2, rounded in the direction rnd. (In the literature, mpfr_gamma_inc is called upper incomplete Gamma function, or sometimes complementary incomplete Gamma function.) For mpfr_gamma (and mpfr_gamma_inc when op2 is zero), when op is a negative integer, rop is set to NaN.

Note: the current implementation of mpfr_gamma_inc is slow for large values of rop or op, in which case some internal overflow might also occur.

Function: int mpfr_lngamma (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the value of the logarithm of the Gamma function on op, rounded in the direction rnd. When op is 1 or 2, set rop to +0 (in all rounding modes). When op is an infinity or a nonpositive integer, set rop to +Inf, following the general rules on special values. When -2k-1 < op < -2k, k being a nonnegative integer, set rop to NaN. See also mpfr_lgamma.

Function: int mpfr_lgamma (mpfr_t rop, int *signp, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the value of the logarithm of the absolute value of the Gamma function on op, rounded in the direction rnd. The sign (1 or -1) of Gamma(op) is returned in the object pointed to by signp. When op is 1 or 2, set rop to +0 (in all rounding modes). When op is an infinity or a nonpositive integer, set rop to +Inf. When op is NaN, -Inf or a negative integer, *signp is undefined, and when op is ±0, *signp is the sign of the zero.

Function: int mpfr_digamma (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the value of the Digamma (sometimes also called Psi) function on op, rounded in the direction rnd. When op is a negative integer, set rop to NaN.

Function: int mpfr_beta (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)

Set rop to the value of the Beta function at arguments op1 and op2. Note: the current code does not try to avoid internal overflow or underflow, and might use a huge internal precision in some cases.

Function: int mpfr_zeta (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_zeta_ui (mpfr_t rop, unsigned long op, mpfr_rnd_t rnd)

Set rop to the value of the Riemann Zeta function on op, rounded in the direction rnd.

Function: int mpfr_erf (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_erfc (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the value of the error function on op (resp. the complementary error function on op) rounded in the direction rnd.

Function: int mpfr_j0 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_j1 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_jn (mpfr_t rop, long n, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the value of the first kind Bessel function of order 0, (resp. 1 and n) on op, rounded in the direction rnd. When op is NaN, rop is always set to NaN. When op is plus or minus Infinity, rop is set to +0. When op is zero, and n is not zero, rop is set to +0 or -0 depending on the parity and sign of n, and the sign of op.

Function: int mpfr_y0 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_y1 (mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)
Function: int mpfr_yn (mpfr_t rop, long n, mpfr_t op, mpfr_rnd_t rnd)

Set rop to the value of the second kind Bessel function of order 0 (resp. 1 and n) on op, rounded in the direction rnd. When op is NaN or negative, rop is always set to NaN. When op is +Inf, rop is set to +0. When op is zero, rop is set to +Inf or -Inf depending on the parity and sign of n.

Function: int mpfr_agm (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)

Set rop to the arithmetic-geometric mean of op1 and op2, rounded in the direction rnd. The arithmetic-geometric mean is the common limit of the sequences u_n and v_n, where u_0=op1, v_0=op2, u_(n+1) is the arithmetic mean of u_n and v_n, and v_(n+1) is the geometric mean of u_n and v_n. If any operand is negative and the other one is not zero, set rop to NaN. If any operand is zero and the other one is finite (resp. infinite), set rop to +0 (resp. NaN).

Function: int mpfr_ai (mpfr_t rop, mpfr_t x, mpfr_rnd_t rnd)

Set rop to the value of the Airy function Ai on x, rounded in the direction rnd. When x is NaN, rop is always set to NaN. When x is +Inf or -Inf, rop is +0. The current implementation is not intended to be used with large arguments. It works with abs(x) typically smaller than 500. For larger arguments, other methods should be used and will be implemented in a future version.

Function: int mpfr_const_log2 (mpfr_t rop, mpfr_rnd_t rnd)
Function: int mpfr_const_pi (mpfr_t rop, mpfr_rnd_t rnd)
Function: int mpfr_const_euler (mpfr_t rop, mpfr_rnd_t rnd)
Function: int mpfr_const_catalan (mpfr_t rop, mpfr_rnd_t rnd)

Set rop to the logarithm of 2, the value of Pi, of Euler’s constant 0.577…, of Catalan’s constant 0.915…, respectively, rounded in the direction rnd. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. To free these caches, use mpfr_free_cache or mpfr_free_cache2.

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5.8 Input and Output Functions

This section describes functions that perform input from an input/output stream, and functions that output to an input/output stream. Passing a null pointer for a stream to any of these functions will make them read from stdin and write to stdout, respectively.

When using a function that takes a FILE * argument, you must include the <stdio.h> standard header before mpfr.h, to allow mpfr.h to define prototypes for these functions.

Function: size_t mpfr_out_str (FILE *stream, int base, size_t n, mpfr_t op, mpfr_rnd_t rnd)

Output op on stream stream as a text string in base abs(base), rounded in the direction rnd. The base may vary from 2 to 62 or from -2 to -36 (any other value yields undefined behavior). The argument n has the same meaning as in mpfr_get_str (see mpfr_get_str): Print n significant digits exactly, or if n is 0, the number mpfr_get_str_ndigits(base,p) where p is the precision of op (see mpfr_get_str_ndigits).

If the input is NaN, +Inf, -Inf, +0, or -0, then ‘@NaN@’, ‘@Inf@’, ‘-@Inf@’, ‘0’, or ‘-0’ is output, respectively.

For the regular numbers, the format of the output is the following: the most significant digit, then a decimal-point character (defined by the current locale), then the remaining n-1 digits (including trailing zeros), then the exponent prefix, then the exponent in decimal. The exponent prefix is ‘e’ when abs(base) <= 10, and ‘@’ when abs(base) > 10. See mpfr_get_str for information on the digits depending on the base.

Return the number of characters written, or if an error occurred, return 0.

Function: size_t mpfr_inp_str (mpfr_t rop, FILE *stream, int base, mpfr_rnd_t rnd)

Input a string in base base from stream stream, rounded in the direction rnd, and put the read float in rop.

This function reads a word (defined as a sequence of characters between whitespace) and parses it using mpfr_set_str. See the documentation of mpfr_strtofr for a detailed description of the valid string formats.

Return the number of bytes read, or if an error occurred, return 0.

Function: int mpfr_fpif_export (FILE *stream, mpfr_t op)

Export the number op to the stream stream in a floating-point interchange format. In particular one can export on a 32-bit computer and import on a 64-bit computer, or export on a little-endian computer and import on a big-endian computer. The precision of op and the sign bit of a NaN are stored too. Return 0 iff the export was successful.

Note: this function is experimental and its interface might change in future versions.

Function: int mpfr_fpif_import (mpfr_t op, FILE *stream)

Import the number op from the stream stream in a floating-point interchange format (see mpfr_fpif_export). Note that the precision of op is set to the one read from the stream, and the sign bit is always retrieved (even for NaN). If the stored precision is zero or greater than MPFR_PREC_MAX, the function fails (it returns non-zero) and op is unchanged. If the function fails for another reason, op is set to NaN and it is unspecified whether the precision of op has changed to the one read from the file. Return 0 iff the import was successful.

Note: this function is experimental and its interface might change in future versions.

Function: void mpfr_dump (mpfr_t op)

Output op on stdout in some unspecified format, then a newline character. This function is mainly for debugging purpose. Thus invalid data may be supported. Everything that is not specified may change without breaking the ABI and may depend on the environment.

The current output format is the following: a minus sign if the sign bit is set (even for NaN); ‘@NaN@’, ‘@Inf@’ or ‘0’ if the argument is NaN, an infinity or zero, respectively; otherwise the remaining of the output is as follows: ‘0.’ then the p bits of the binary significand, where p is the precision of the number; if the trailing bits are not all zeros (which must not occur with valid data), they are output enclosed by square brackets; the character ‘E’ followed by the exponent written in base 10; in case of invalid data or out-of-range exponent, this function outputs three exclamation marks (‘!!!’), followed by flags, followed by three exclamation marks (‘!!!’) again. These flags are: ‘N’ if the most significant bit of the significand is 0 (i.e., the number is not normalized); ‘T’ if there are non-zero trailing bits; ‘U’ if this is an UBF number (internal use only); ‘<’ if the exponent is less than the current minimum exponent; ‘>’ if the exponent is greater than the current maximum exponent.

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5.9 Formatted Output Functions

5.9.1 Requirements

The class of mpfr_printf functions provides formatted output in a similar manner as the standard C printf. These functions are defined only if your system supports ISO C variadic functions and the corresponding argument access macros.

When using any of these functions, you must include the <stdio.h> standard header before mpfr.h, to allow mpfr.h to define prototypes for these functions.

5.9.2 Format String

The format specification accepted by mpfr_printf is an extension of the printf one. The conversion specification is of the form: 

% [flags] [width] [.[precision]] [type] [rounding] conv

‘flags’, ‘width’, and ‘precision’ have the same meaning as for the standard printf (in particular, notice that the ‘precision’ is related to the number of digits displayed in the base chosen by ‘conv’ and not related to the internal precision of the mpfr_t variable), but note that for ‘Re’, the default precision is not the same as the one for ‘e’. mpfr_printf accepts the same ‘type’ specifiers as GMP (except the non-standard and deprecated ‘q’, use ‘ll’ instead), namely the length modifiers defined in the C standard:

‘h’short
‘hh’char
‘j’intmax_t or uintmax_t
‘l’long or wchar_t
‘ll’long long
‘L’long double
‘t’ptrdiff_t
‘z’size_t

and the ‘type’ specifiers defined in GMP plus ‘R’ and ‘P’ specific to MPFR (the second column in the table below shows the type of the argument read in the argument list and the kind of ‘conv’ specifier to use after the ‘type’ specifier):

‘F’mpf_t, float conversions
‘Q’mpq_t, integer conversions
‘M’mp_limb_t, integer conversions
‘N’mp_limb_t array, integer conversions
‘Z’mpz_t, integer conversions
‘P’mpfr_prec_t, integer conversions
‘R’mpfr_t, float conversions

The ‘type’ specifiers have the same restrictions as those mentioned in the GMP documentation: see Section “Formatted Output Strings” in GNU MP. In particular, the ‘type’ specifiers (except ‘R’ and ‘P’) are supported only if they are supported by gmp_printf in your GMP build; this implies that the standard specifiers, such as ‘t’, must also be supported by your C library if you want to use them.

The ‘rounding’ field is specific to mpfr_t arguments and should not be used with other types.

With conversion specification not involving ‘P’ and ‘R’ types, mpfr_printf behaves exactly as gmp_printf.

The ‘P’ type specifies that a following ‘d’, ‘i’, ‘o’, ‘u’, ‘x’, or ‘X’ conversion specifier applies to a mpfr_prec_t argument. It is needed because the mpfr_prec_t type does not necessarily correspond to an int or any fixed standard type. The ‘precision’ field specifies the minimum number of digits to appear. The default ‘precision’ is 1. For example: 

mpfr_t x;
mpfr_prec_t p;
mpfr_init (x);
…
p = mpfr_get_prec (x);
mpfr_printf ("variable x with %Pu bits", p);

The ‘R’ type specifies that a following ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, ‘f’, ‘F’, ‘g’, ‘G’, or ‘n’ conversion specifier applies to a mpfr_t argument. The ‘R’ type can be followed by a ‘rounding’ specifier denoted by one of the following characters:

‘U’round toward plus infinity
‘D’round toward minus infinity
‘Y’round away from zero
‘Z’round toward zero
‘N’round to nearest (with ties to even)
‘*’rounding mode indicated by the mpfr_rnd_t argument just before the corresponding mpfr_t variable.

The default rounding mode is rounding to nearest. The following three examples are equivalent: 

mpfr_t x;
mpfr_init (x);
…
mpfr_printf ("%.128Rf", x);
mpfr_printf ("%.128RNf", x);
mpfr_printf ("%.128R*f", MPFR_RNDN, x);

Note that the rounding away from zero mode is specified with ‘Y’ because ISO C reserves the ‘A’ specifier for hexadecimal output (see below).

The output ‘conv’ specifiers allowed with mpfr_t parameter are:

‘a’ ‘A’hex float, C99 style
‘b’binary output
‘e’ ‘E’scientific-format float
‘f’ ‘F’fixed-point float
‘g’ ‘G’fixed-point or scientific float

The conversion specifier ‘b’ which displays the argument in binary is specific to mpfr_t arguments and should not be used with other types. Other conversion specifiers have the same meaning as for a double argument.

In case of non-decimal output, only the significand is written in the specified base, the exponent is always displayed in decimal. Special values are always displayed as nan, -inf, and inf for ‘a’, ‘b’, ‘e’, ‘f’, and ‘g’ specifiers and NAN, -INF, and INF for ‘A’, ‘E’, ‘F’, and ‘G’ specifiers.

The mpfr_t number is rounded to the given precision in the direction specified by the rounding mode (see below if the ‘precision’ field is empty). If the precision is zero with rounding to nearest mode and one of the following ‘conv’ specifiers: ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, tie case is rounded to even when it lies between two consecutive values at the wanted precision which have the same exponent, otherwise, it is rounded away from zero. For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the format specification "%.0RNe". This also applies when the ‘g’ (resp. ‘G’) conversion specifier uses the ‘e’ (resp. ‘E’) style. If the precision is set to a value greater than the maximum value for an int, it will be silently reduced down to INT_MAX.

If the ‘precision’ field is empty with ‘conv’ specifier ‘e’ and ‘E’ (as in %Re or %.RE), the chosen precision (i.e., the number of digits to be displayed after the initial digit and the decimal point) is ceil(p*log(2)/log(10)), where p is the precision of the input variable, matching the choice done for mpfr_get_str; thus, if rounding to nearest is used, outputting the value with an empty ‘precision’ field and reading it back will yield the original value. The chosen precision for an empty ‘precision’ field with ‘conv’ specifiers ‘f’, ‘F’, ‘g’, and ‘G’ is 6.

5.9.3 Functions

For all the following functions, if the number of characters that ought to be written exceeds the maximum limit INT_MAX for an int, nothing is written in the stream (resp. to stdout, to buf, to str), the function returns -1, sets the erange flag, and errno is set to EOVERFLOW if the EOVERFLOW macro is defined (such as on POSIX systems). Note, however, that errno might be changed to another value by some internal library call if another error occurs there (currently, this would come from the unallocation function).

Function: int mpfr_fprintf (FILE *stream, const char *template, …)
Function: int mpfr_vfprintf (FILE *stream, const char *template, va_list ap)

Print to the stream stream the optional arguments under the control of the template string template. Return the number of characters written or a negative value if an error occurred.

Function: int mpfr_printf (const char *template, …)
Function: int mpfr_vprintf (const char *template, va_list ap)

Print to stdout the optional arguments under the control of the template string template. Return the number of characters written or a negative value if an error occurred.

Function: int mpfr_sprintf (char *buf, const char *template, …)
Function: int mpfr_vsprintf (char *buf, const char *template, va_list ap)

Form a null-terminated string corresponding to the optional arguments under the control of the template string template, and print it in buf. No overlap is permitted between buf and the other arguments. Return the number of characters written in the array buf not counting the terminating null character or a negative value if an error occurred.

Function: int mpfr_snprintf (char *buf, size_t n, const char *template, …)
Function: int mpfr_vsnprintf (char *buf, size_t n, const char *template, va_list ap)

Form a null-terminated string corresponding to the optional arguments under the control of the template string template, and print it in buf. If n is zero, nothing is written and buf may be a null pointer, otherwise, the n-1 first characters are written in buf and the n-th is a null character. Return the number of characters that would have been written had n been sufficiently large, not counting the terminating null character, or a negative value if an error occurred.

Function: int mpfr_asprintf (char **str, const char *template, …)
Function: int mpfr_vasprintf (char **str, const char *template, va_list ap)

Write their output as a null terminated string in a block of memory allocated using the allocation function (see Memory Handling). A pointer to the block is stored in str. The block of memory must be freed using mpfr_free_str. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred, in which case the contents of str are undefined.

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5.11 Rounding-Related Functions

Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t rnd)

Set the default rounding mode to rnd. The default rounding mode is to nearest initially.

Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void)

Get the default rounding mode.

Function: int mpfr_prec_round (mpfr_t x, mpfr_prec_t prec, mpfr_rnd_t rnd)

Round x according to rnd with precision prec, which must be an integer between MPFR_PREC_MIN and MPFR_PREC_MAX (otherwise the behavior is undefined). If prec is greater than or equal to the precision of x, then new space is allocated for the significand, and it is filled with zeros. Otherwise, the significand is rounded to precision prec with the given direction; no memory reallocation to free the unused limbs is done. In both cases, the precision of x is changed to prec.

Here is an example of how to use mpfr_prec_round to implement Newton’s algorithm to compute the inverse of a, assuming x is already an approximation to n bits: 

  mpfr_set_prec (t, 2 * n);
  mpfr_set (t, a, MPFR_RNDN);         /* round a to 2n bits */
  mpfr_mul (t, t, x, MPFR_RNDN);      /* t is correct to 2n bits */
  mpfr_ui_sub (t, 1, t, MPFR_RNDN);   /* high n bits cancel with 1 */
  mpfr_prec_round (t, n, MPFR_RNDN);  /* t is correct to n bits */
  mpfr_mul (t, t, x, MPFR_RNDN);      /* t is correct to n bits */
  mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */
  mpfr_add (x, x, t, MPFR_RNDN);      /* x is correct to 2n bits */

Warning! You must not use this function if x was initialized with MPFR_DECL_INIT or with mpfr_custom_init_set (see Custom Interface).

Function: int mpfr_can_round (mpfr_t b, mpfr_exp_t err, mpfr_rnd_t rnd1, mpfr_rnd_t rnd2, mpfr_prec_t prec)

Assuming b is an approximation of an unknown number x in the direction rnd1 with error at most two to the power E(b)-err where E(b) is the exponent of b, return a non-zero value if one is able to round correctly x to precision prec with the direction rnd2 assuming an unbounded exponent range, and 0 otherwise (including for NaN and Inf). In other words, if the error on b is bounded by two to the power k ulps, and b has precision prec, you should give err=prec-k. This function does not modify its arguments.

If rnd1 is MPFR_RNDN or MPFR_RNDF, the error is considered to be either positive or negative, thus the possible range is twice as large as with a directed rounding for rnd1 (with the same value of err).

When rnd2 is MPFR_RNDF, let rnd3 be the opposite direction if rnd1 is a directed rounding, and MPFR_RNDN if rnd1 is MPFR_RNDN or MPFR_RNDF. The returned value of mpfr_can_round (b, err, rnd1, MPFR_RNDF, prec) is non-zero iff after the call mpfr_set (y, b, rnd3) with y of precision prec, y is guaranteed to be a faithful rounding of x.

Note: The ternary value cannot be determined in general with this function. However, if it is known that the exact value is not exactly representable in precision prec, then one can use the following trick to determine the (non-zero) ternary value in any rounding mode rnd2 (note that MPFR_RNDZ below can be replaced by any directed rounding mode): 

if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ,
                    prec + (rnd2 == MPFR_RNDN)))
  {
    /* round the approximation 'b' to the result 'r' of 'prec' bits
       with rounding mode 'rnd2' and get the ternary value 'inex' */
    inex = mpfr_set (r, b, rnd2);
  }

Indeed, if rnd2 is MPFR_RNDN, this will check if one can round to prec+1 bits with a directed rounding: if so, one can surely round to nearest to prec bits, and in addition one can determine the correct ternary value, which would not be the case when b is near from a value exactly representable on prec bits.

A detailed example is available in the examples subdirectory, file can_round.c.

Function: mpfr_prec_t mpfr_min_prec (mpfr_t x)

Return the minimal number of bits required to store the significand of x, and 0 for special values, including 0.

Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t rnd)

Return a string ("MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDU", "MPFR_RNDD", "MPFR_RNDA", "MPFR_RNDF") corresponding to the rounding mode rnd, or a null pointer if rnd is an invalid rounding mode.

Macro: int mpfr_round_nearest_away (int (foo)(mpfr_t, type1_t, ..., mpfr_rnd_t), mpfr_t rop, type1_t op, ...)

Given a function foo and one or more values op (which may be a mpfr_t, a long, a double, etc.), put in rop the round-to-nearest-away rounding of foo(op,...). This rounding is defined in the same way as round-to-nearest-even, except in case of tie, where the value away from zero is returned. The function foo takes as input, from second to penultimate argument(s), the argument list given after rop, a rounding mode as final argument, puts in its first argument the value foo(op,...) rounded according to this rounding mode, and returns the corresponding ternary value (which is expected to be correct, otherwise mpfr_round_nearest_away will not work as desired). Due to implementation constraints, this function must not be called when the minimal exponent emin is the smallest possible one. This macro has been made such that the compiler is able to detect mismatch between the argument list op and the function prototype of foo. Multiple input arguments op are supported only with C99 compilers. Otherwise, for C89 compilers, only one such argument is supported.

Note: this macro is experimental and its interface might change in future versions. 

unsigned long ul;
mpfr_t f, r;
/* Code that inits and sets r, f, and ul, and if needed sets emin */
int i = mpfr_round_nearest_away (mpfr_add_ui, r, f, ul);

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5.12 Miscellaneous Functions

Function: void mpfr_nexttoward (mpfr_t x, mpfr_t y)

If x or y is NaN, set x to NaN; note that the NaN flag is set as usual. If x and y are equal, x is unchanged. Otherwise, if x is different from y, replace x by the next floating-point number (with the precision of x and the current exponent range) in the direction of y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow, overflow, or inexact exception is raised.

Function: void mpfr_nextabove (mpfr_t x)
Function: void mpfr_nextbelow (mpfr_t x)

Equivalent to mpfr_nexttoward where y is plus infinity (resp. minus infinity).

Function: int mpfr_min (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)
Function: int mpfr_max (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)

Set rop to the minimum (resp. maximum) of op1 and op2. If op1 and op2 are both NaN, then rop is set to NaN. If op1 or op2 is NaN, then rop is set to the numeric value. If op1 and op2 are zeros of different signs, then rop is set to -0 (resp. +0).

Function: int mpfr_urandomb (mpfr_t rop, gmp_randstate_t state)

Generate a uniformly distributed random float in the interval 0 <= rop < 1. More precisely, the number can be seen as a float with a random non-normalized significand and exponent 0, which is then normalized (thus if e denotes the exponent after normalization, then the least -e significant bits of the significand are always 0).

Return 0, unless the exponent is not in the current exponent range, in which case rop is set to NaN and a non-zero value is returned (this should never happen in practice, except in very specific cases). The second argument is a gmp_randstate_t structure which should be created using the GMP gmp_randinit function (see the GMP manual).

Note: for a given version of MPFR, the returned value of rop and the new value of state (which controls further random values) do not depend on the machine word size.

Function: int mpfr_urandom (mpfr_t rop, gmp_randstate_t state, mpfr_rnd_t rnd)

Generate a uniformly distributed random float. The floating-point number rop can be seen as if a random real number is generated according to the continuous uniform distribution on the interval [0, 1] and then rounded in the direction rnd.

The second argument is a gmp_randstate_t structure which should be created using the GMP gmp_randinit function (see the GMP manual).

Note: the note for mpfr_urandomb holds too. Moreover, the exact number (the random value to be rounded) and the next random state do not depend on the current exponent range and the rounding mode. However, they depend on the target precision: from the same state of the random generator, if the precision of the destination is changed, then the value may be completely different (and the state of the random generator is different too).

Function: int mpfr_nrandom (mpfr_t rop1, gmp_randstate_t state, mpfr_rnd_t rnd)
Function: int mpfr_grandom (mpfr_t rop1, mpfr_t rop2, gmp_randstate_t state, mpfr_rnd_t rnd)

Generate one (possibly two for mpfr_grandom) random floating-point number according to a standard normal Gaussian distribution (with mean zero and variance one). For mpfr_grandom, if rop2 is a null pointer, then only one value is generated and stored in rop1.

The floating-point number rop1 (and rop2) can be seen as if a random real number were generated according to the standard normal Gaussian distribution and then rounded in the direction rnd.

The gmp_randstate_t argument should be created using the GMP gmp_randinit function (see the GMP manual).

For mpfr_grandom, the combination of the ternary values is returned like with mpfr_sin_cos. If rop2 is a null pointer, the second ternary value is assumed to be 0 (note that the encoding of the only ternary value is not the same as the usual encoding for functions that return only one result). Otherwise the ternary value of a random number is always non-zero.

Note: the note for mpfr_urandomb holds too. In addition, the exponent range and the rounding mode might have a side effect on the next random state.

Note: mpfr_nrandom is much more efficient than mpfr_grandom, especially for large precision. Thus mpfr_grandom is marked as deprecated and will be removed in a future release.

Function: int mpfr_erandom (mpfr_t rop1, gmp_randstate_t state, mpfr_rnd_t rnd)

Generate one random floating-point number according to an exponential distribution, with mean one. Other characteristics are identical to mpfr_nrandom.

Function: mpfr_exp_t mpfr_get_exp (mpfr_t x)

Return the exponent of x, assuming that x is a non-zero ordinary number and the significand is considered in [1/2,1). For this function, x is allowed to be outside of the current range of acceptable values. The behavior for NaN, infinity or zero is undefined.

Function: int mpfr_set_exp (mpfr_t x, mpfr_exp_t e)

Set the exponent of x to e if x is a non-zero ordinary number and e is in the current exponent range, and return 0; otherwise, return a non-zero value (x is not changed).

Function: int mpfr_signbit (mpfr_t op)

Return a non-zero value iff op has its sign bit set (i.e., if it is negative, -0, or a NaN whose representation has its sign bit set).

Function: int mpfr_setsign (mpfr_t rop, mpfr_t op, int s, mpfr_rnd_t rnd)

Set the value of rop from op, rounded toward the given direction rnd, then set (resp. clear) its sign bit if s is non-zero (resp. zero), even when op is a NaN.

Function: int mpfr_copysign (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)

Set the value of rop from op1, rounded toward the given direction rnd, then set its sign bit to that of op2 (even when op1 or op2 is a NaN). This function is equivalent to mpfr_setsign (rop, op1, mpfr_signbit (op2), rnd).

Function: const char * mpfr_get_version (void)

Return the MPFR version, as a null-terminated string.

Macro: MPFR_VERSION
Macro: MPFR_VERSION_MAJOR
Macro: MPFR_VERSION_MINOR
Macro: MPFR_VERSION_PATCHLEVEL
Macro: MPFR_VERSION_STRING

MPFR_VERSION is the version of MPFR as a preprocessing constant. MPFR_VERSION_MAJOR, MPFR_VERSION_MINOR and MPFR_VERSION_PATCHLEVEL are respectively the major, minor and patch level of MPFR version, as preprocessing constants. MPFR_VERSION_STRING is the version (with an optional suffix, used in development and pre-release versions) as a string constant, which can be compared to the result of mpfr_get_version to check at run time the header file and library used match: 

if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING))
  fprintf (stderr, "Warning: header and library do not match\n");

Note: Obtaining different strings is not necessarily an error, as in general, a program compiled with some old MPFR version can be dynamically linked with a newer MPFR library version (if allowed by the library versioning system).

Macro: long MPFR_VERSION_NUM (major, minor, patchlevel)

Create an integer in the same format as used by MPFR_VERSION from the given major, minor and patchlevel. Here is an example of how to check the MPFR version at compile time: 

#if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(3,0,0)))
# error "Wrong MPFR version."
#endif
Function: const char * mpfr_get_patches (void)

Return a null-terminated string containing the ids of the patches applied to the MPFR library (contents of the PATCHES file), separated by spaces. Note: If the program has been compiled with an older MPFR version and is dynamically linked with a new MPFR library version, the identifiers of the patches applied to the old (compile-time) MPFR version are not available (however this information should not have much interest in general).

Function: int mpfr_buildopt_tls_p (void)

Return a non-zero value if MPFR was compiled as thread safe using compiler-level Thread-Local Storage (that is, MPFR was built with the ‘--enable-thread-safe’ configure option, see INSTALL file), return zero otherwise.

Function: int mpfr_buildopt_float128_p (void)

Return a non-zero value if MPFR was compiled with ‘_Float128’ support (that is, MPFR was built with the ‘--enable-float128’ configure option), return zero otherwise.

Function: int mpfr_buildopt_decimal_p (void)

Return a non-zero value if MPFR was compiled with decimal float support (that is, MPFR was built with the ‘--enable-decimal-float’ configure option), return zero otherwise.

Function: int mpfr_buildopt_gmpinternals_p (void)

Return a non-zero value if MPFR was compiled with GMP internals (that is, MPFR was built with either ‘--with-gmp-build’ or ‘--enable-gmp-internals’ configure option), return zero otherwise.

Function: int mpfr_buildopt_sharedcache_p (void)

Return a non-zero value if MPFR was compiled so that all threads share the same cache for one MPFR constant, like mpfr_const_pi or mpfr_const_log2 (that is, MPFR was built with the ‘--enable-shared-cache’ configure option), return zero otherwise. If the return value is non-zero, MPFR applications may need to be compiled with the ‘-pthread’ option.

Function: const char * mpfr_buildopt_tune_case (void)

Return a string saying which thresholds file has been used at compile time. This file is normally selected from the processor type.

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5.14 Memory Handling Functions

These are general functions concerning memory handling (see Memory Handling, for more information).

Function: void mpfr_free_cache (void)

Free all caches and pools used by MPFR internally (those local to the current thread and those shared by all threads). You should call this function before terminating a thread, even if you did not call mpfr_const_* functions directly (they could have been called internally).

Function: void mpfr_free_cache2 (mpfr_free_cache_t way)

Free various caches and pools used by MPFR internally, as specified by way, which is a set of flags:

  • those local to the current thread if flag MPFR_FREE_LOCAL_CACHE is set;
  • those shared by all threads if flag MPFR_FREE_GLOBAL_CACHE is set.

The other bits of way are currently ignored and are reserved for future use; they should be zero.

Note: mpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE) is currently equivalent to mpfr_free_cache().

Function: void mpfr_free_pool (void)

Free the pools used by MPFR internally. Note: This function is automatically called after the thread-local caches are freed (with mpfr_free_cache or mpfr_free_cache2).

Function: int mpfr_mp_memory_cleanup (void)

This function should be called before calling mp_set_memory_functions. See Memory Handling, for more information. Zero is returned in case of success, non-zero in case of error. Errors are currently not possible, but checking the return value is recommended for future compatibility.

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5.15 Compatibility With MPF

A header file mpf2mpfr.h is included in the distribution of MPFR for compatibility with the GNU MP class MPF. By inserting the following two lines after the #include <gmp.h> line, 

#include <mpfr.h>
#include <mpf2mpfr.h>

many programs written for MPF can be compiled directly against MPFR without any changes. All operations are then performed with the default MPFR rounding mode, which can be reset with mpfr_set_default_rounding_mode.

Warning! There are some differences. In particular:

  • The precision is different: MPFR rounds to the exact number of bits (zeroing trailing bits in the internal representation). Users may need to increase the precision of their variables.
  • The exponent range is also different.
  • The formatted output functions (gmp_printf, etc.) will not work for arguments of arbitrary-precision floating-point type (mpf_t, which mpf2mpfr.h redefines as mpfr_t).
  • The output of mpf_out_str has a format slightly different from the one of mpfr_out_str (concerning the position of the decimal-point character, trailing zeros and the output of the value 0).
Function: void mpfr_set_prec_raw (mpfr_t x, mpfr_prec_t prec)

Reset the precision of x to be exactly prec bits. The only difference with mpfr_set_prec is that prec is assumed to be small enough so that the significand fits into the current allocated memory space for x. Otherwise the behavior is undefined.

Function: int mpfr_eq (mpfr_t op1, mpfr_t op2, unsigned long int op3)

Return non-zero if op1 and op2 are both non-zero ordinary numbers with the same exponent and the same first op3 bits, both zero, or both infinities of the same sign. Return zero otherwise. This function is defined for compatibility with MPF, we do not recommend to use it otherwise. Do not use it either if you want to know whether two numbers are close to each other; for instance, 1.011111 and 1.100000 are regarded as different for any value of op3 larger than 1.

Function: void mpfr_reldiff (mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd)

Compute the relative difference between op1 and op2 and store the result in rop. This function does not guarantee the correct rounding on the relative difference; it just computes |op1-op2|/op1, using the precision of rop and the rounding mode rnd for all operations.

Function: int mpfr_mul_2exp (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)
Function: int mpfr_div_2exp (mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd)

These functions are identical to mpfr_mul_2ui and mpfr_div_2ui respectively. These functions are only kept for compatibility with MPF, one should prefer mpfr_mul_2ui and mpfr_div_2ui otherwise.

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5.16 Custom Interface

Some applications use a stack to handle the memory and their objects. However, the MPFR memory design is not well suited for such a thing. So that such applications are able to use MPFR, an auxiliary memory interface has been created: the Custom Interface.

The following interface allows one to use MPFR in two ways:

  • Either directly store a floating-point number as a mpfr_t on the stack.
  • Either store its own representation on the stack and construct a new temporary mpfr_t each time it is needed.

Nothing has to be done to destroy the floating-point numbers except garbaging the used memory: all the memory management (allocating, destroying, garbaging) is left to the application.

Each function in this interface is also implemented as a macro for efficiency reasons: for example mpfr_custom_init (s, p) uses the macro, while (mpfr_custom_init) (s, p) uses the function. Note that the macro may evaluate arguments multiple times (or none). Moreover, macros implementing functions with the void return type may not be used in contexts where an expression is expected, e.g., inside for(...) or before a comma operator.

Note 1: MPFR functions may still initialize temporary floating-point numbers using mpfr_init and similar functions. See Custom Allocation (GNU MP).

Note 2: MPFR functions may use the cached functions (mpfr_const_pi for example), even if they are not explicitly called. You have to call mpfr_free_cache each time you garbage the memory iff mpfr_init, through GMP Custom Allocation, allocates its memory on the application stack.

Function: size_t mpfr_custom_get_size (mpfr_prec_t prec)

Return the needed size in bytes to store the significand of a floating-point number of precision prec.

Function: void mpfr_custom_init (void *significand, mpfr_prec_t prec)

Initialize a significand of precision prec, where significand must be an area of mpfr_custom_get_size (prec) bytes at least and be suitably aligned for an array of mp_limb_t (GMP type, see Internals).

Function: void mpfr_custom_init_set (mpfr_t x, int kind, mpfr_exp_t exp, mpfr_prec_t prec, void *significand)

Perform a dummy initialization of a mpfr_t and set it to:

  • if abs(kind) = MPFR_NAN_KIND, x is set to NaN;
  • if abs(kind) = MPFR_INF_KIND, x is set to the infinity of the same sign as kind;
  • if abs(kind) = MPFR_ZERO_KIND, x is set to the zero of the same sign as kind;
  • if abs(kind) = MPFR_REGULAR_KIND, x is set to the regular number whose sign is the one of kind, and whose exponent and significand are given by exp and significand.

In all cases, significand will be used directly for further computing involving x. This function does not allocate anything. A floating-point number initialized with this function cannot be resized using mpfr_set_prec or mpfr_prec_round, or cleared using mpfr_clear! The significand must have been initialized with mpfr_custom_init using the same precision prec.

Function: int mpfr_custom_get_kind (mpfr_t x)

Return the current kind of a mpfr_t as created by mpfr_custom_init_set. The behavior of this function for any mpfr_t not initialized with mpfr_custom_init_set is undefined.

Function: void * mpfr_custom_get_significand (mpfr_t x)

Return a pointer to the significand used by a mpfr_t initialized with mpfr_custom_init_set. The behavior of this function for any mpfr_t not initialized with mpfr_custom_init_set is undefined.

Function: mpfr_exp_t mpfr_custom_get_exp (mpfr_t x)

Return the exponent of x, assuming that x is a non-zero ordinary number and the significand is considered in [1/2,1). But if x is NaN, infinity or zero, contrary to mpfr_get_exp (where the behavior is undefined), the return value is here an unspecified, valid value of the mpfr_exp_t type. The behavior of this function for any mpfr_t not initialized with mpfr_custom_init_set is undefined.

Function: void mpfr_custom_move (mpfr_t x, void *new_position)

Inform MPFR that the significand of x has moved due to a garbage collect and update its new position to new_position. However the application has to move the significand and the mpfr_t itself. The behavior of this function for any mpfr_t not initialized with mpfr_custom_init_set is undefined.

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5.17 Internals

A limb means the part of a multi-precision number that fits in a single word. Usually a limb contains 32 or 64 bits. The C data type for a limb is mp_limb_t.

The mpfr_t type is internally defined as a one-element array of a structure, and mpfr_ptr is the C data type representing a pointer to this structure. The mpfr_t type consists of four fields:

  • The _mpfr_prec field is used to store the precision of the variable (in bits); this is not less than MPFR_PREC_MIN.
  • The _mpfr_sign field is used to store the sign of the variable.
  • The _mpfr_exp field stores the exponent. An exponent of 0 means a radix point just above the most significant limb. Non-zero values n are a multiplier 2^n relative to that point. A NaN, an infinity and a zero are indicated by special values of the exponent field.
  • Finally, the _mpfr_d field is a pointer to the limbs, least significant limbs stored first. The number of limbs in use is controlled by _mpfr_prec, namely ceil(_mpfr_prec/mp_bits_per_limb). Non-singular (i.e., different from NaN, Infinity or zero) values always have the most significant bit of the most significant limb set to 1. When the precision does not correspond to a whole number of limbs, the excess bits at the low end of the data are zeros.

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