## 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 a 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.

### 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 C90). 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 C90 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.

### 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, mpfr_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`, `mpfr_set_z`, `mpfr_set_q` and `mpfr_set_f`, regardless of the rounding mode. To use the `mpfr_set_float128` function, see Nomenclature and Types. 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 (and in any case they don’t preserve the sign bit of NaN). 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 base (with its special value 0) and 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 with the rules of the C locale. An ‘`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 positive infinity (+Inf) or positive zero (+0) iff sign is non-negative; 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).

### 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`.

### 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: `mpfr_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 NaN (either quiet or signaling) or the result of 0.0/0.0 is returned (the sign bit is not preserved). 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 int` mpfr_get_si `(mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `unsigned long int` 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 int`, an `unsigned long int`, 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 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 times 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.

  Note: It is allowed to modify the string allocated by `mpfr_get_str`; but when `mpfr_free_str` is called, its length must be the same as the original one (unless it is guaranteed that the unallocation function ignores its size parameter).

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 int`, `long int`, `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`.

### 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 as 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 as for `mpfr_add_d` apply to `mpfr_mul_d`. Note: when op1 and op2 are equal, use `mpfr_sqr` instead of `mpfr_mul` for better efficiency.

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 as 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 (thus this differs from `mpfr_rootn_ui` and `mpfr_rootn_si` with n = 2). 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 standard (Section 9.2.1), which is −Inf instead of +Inf. However, `mpfr_rec_sqrt` is equivalent to `mpfr_rootn_si` with n = −2.

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)` ¶
Function: `int` mpfr_rootn_si `(mpfr_t rop, mpfr_t op, 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 operation of the IEEE 754 standard.

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 IEEE 754-2008, where rootn was introduced, and behaves differently from the IEEE 754 rootn operation. 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 (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.

### 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 int 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, 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.

### 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 (and in particular, that potentially exact cases are known and can be detected efficiently).

Function: `int` mpfr_log `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_log_ui `(mpfr_t rop, unsigned long int 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 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)` ¶
Function: `int` mpfr_log2p1 `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_log10p1 `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶

  Set rop to the logarithm of one plus op (in radix two for `mpfr_log2p1`, and in radix ten for `mpfr_log10p1`), 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)` ¶
Function: `int` mpfr_exp2m1 `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_exp10m1 `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶

  Set rop to the exponential of op followed by a subtraction by one (resp. 2 power of op followed by a subtraction by one, and 10 power 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_powr `(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_uj `(mpfr_t rop, mpfr_t op1, uintmax_t op2, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_pow_sj `(mpfr_t rop, mpfr_t op1, intmax_t op2, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_pown `(mpfr_t rop, mpfr_t op1, intmax_t 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. The `mpfr_powr` function corresponds to the `powr` function from IEEE 754, i.e., it computes the exponential of op2 multiplied by the logarithm of op1. The `mpfr_pown` function is just an alias for `mpfr_pow_sj` (defined with `#define mpfr_pown mpfr_pow_sj`), to follow the C2x function `pown`. Special values are handled as described in the ISO C99 and IEEE 754 standards for the `pow` function:

  - `pow(±0, y)` returns ±Inf for y a negative odd integer.
  - `pow(±0, y)` returns +Inf for y negative and not an odd integer.
  - `pow(±0, y)` returns ±0 for y a positive odd integer.
  - `pow(±0, y)` returns +0 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 +Inf for 0 < abs(x) < 1, and +0 for abs(x) > 1.
  - `pow(x, +Inf)` returns +0 for 0 < abs(x) < 1, and +Inf for abs(x) > 1.
  - `pow(-Inf, y)` returns −0 for y a negative odd integer.
  - `pow(-Inf, y)` returns +0 for y negative and not an odd integer.
  - `pow(-Inf, y)` returns −Inf for y a positive odd integer.
  - `pow(-Inf, y)` returns +Inf for y positive and not an odd integer.
  - `pow(+Inf, y)` returns +0 for y negative, and +Inf 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_compound_si `(mpfr_t rop, mpfr_t op, long int n, mpfr_rnd_t rnd)` ¶

  Set rop to the power n of one plus op, following IEEE 754 for the special cases and exceptions. In particular:

  - When op < −1, rop is set to NaN.
  - When n is zero and op is NaN (like any value greater or equal to −1), rop is set to 1.
  - When op = −1, rop is set to +Inf for n < 0, and to +0 for n > 0.

  The other special cases follow the usual rules.

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_cosu `(mpfr_t rop, mpfr_t op, unsigned long int u, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_sinu `(mpfr_t rop, mpfr_t op, unsigned long int u, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_tanu `(mpfr_t rop, mpfr_t op, unsigned long int u, mpfr_rnd_t rnd)` ¶

  Set rop to the cosine (resp. sine and tangent) of op multiplied by 2 Pi and divided by u. For example, if u equals 360, one gets the cosine (resp. sine and tangent) for op in degrees. For `mpfr_cosu`, when op multiplied by 2 and divided by u is a half-integer, the result is +0, following IEEE 754 (cosPi), so that the function is even. For `mpfr_sinu`, when op multiplied by 2 and divided by u is an integer, the result is zero with the same sign as op, following IEEE 754 (sinPi), so that the function is odd. Similarly, the function `mpfr_tanu` follows IEEE 754 (tanPi).

Function: `int` mpfr_cospi `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_sinpi `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_tanpi `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶

  Set rop to the cosine (resp. sine and tangent) of op multiplied by Pi. See the description of `mpfr_sinu`, `mpfr_cosu` and `mpfr_tanu` for special values.

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_acosu `(mpfr_t rop, mpfr_t op, unsigned long int u, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_asinu `(mpfr_t rop, mpfr_t op, unsigned long int u, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_atanu `(mpfr_t rop, mpfr_t op, unsigned long int u, mpfr_rnd_t rnd)` ¶

  Set rop to a multiplied by u and divided by 2 Pi, where a is the arc-cosine (resp. arc-sine and arc-tangent) of op. For example, if u equals 360, `mpfr_acosu` yields the arc-cosine in degrees.

Function: `int` mpfr_acospi `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_asinpi `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_atanpi `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶

  Set rop to `acos(op)` (resp. `asin(op)` and `atan(op)`) divided by Pi.

Function: `int` mpfr_atan2 `(mpfr_t rop, mpfr_t y, mpfr_t x, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_atan2u `(mpfr_t rop, mpfr_t y, mpfr_t x, unsigned long int u, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_atan2pi `(mpfr_t rop, mpfr_t y, mpfr_t x, mpfr_rnd_t rnd)` ¶

  For `mpfr_atan2`, set rop to the arc-tangent2 of y and x, rounded in the direction rnd: if x > 0, then `atan2(y, x)` returns atan(y/x); if x < 0, then `atan2(y, x)` returns the sign of y multiplied by 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. The function `mpfr_atan2u` behaves similarly, except the result is multiplied by u and divided by 2 Pi; and `mpfr_atan2pi` is the same as `mpfr_atan2u` with u = 2. For example, if u equals 360, `mpfr_atan2u` returns the arc-tangent in degrees, with values from −180 to 180.

  `atan2(y, 0)` does not raise any floating-point exception. Special values are handled as described in the ISO C99 and IEEE 754 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 the factorial of 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 non-positive integer, set rop to +Inf, following the general rules on special values. When −2k − 1 < op < −2k, k being a non-negative 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 non-positive 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 int 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 int 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 positive or negative 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 int 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`.

### 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.

  After skipping optional whitespace (as defined by `isspace`, which depends on the current locale), this function reads a word, defined as the longest sequence of non-whitespace characters, 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 (including the leading whitespace, if any), or if the string format is invalid or 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.

### 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 `gmp_printf` one (itself, 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`’, which are 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_vasprintf` (thus, if GMP uses its `vsnprintf` replacement, the behavior may differ from the C library or from other GMP formatted output functions; for instance, in such a case, up to GMP 6.3.0, the conversion specifiers ‘`a`’/‘`A`’ will not work).

Thus the ‘`conv`’ specifier ‘`F`’ is not supported (due to the use of ‘`F`’ as the ‘`type`’ specifier for `mpf_t`), except for the ‘`type`’ specifier ‘`R`’ (i.e., for `mpfr_t` arguments).

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`’ value 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 %Pd 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 positive infinity
> ‘`D`’ | round toward negative 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.

Non-real values are always displayed as ‘`nan`’ / ‘`inf`’ for the ‘`a`’, ‘`b`’, ‘`e`’, ‘`f`’, and ‘`g`’ specifiers, and ‘`NAN`’ / ‘`INF`’ for ‘`A`’, ‘`E`’, ‘`F`’, and ‘`G`’ specifiers, possibly preceded by a sign or a space (the minus sign when the value has a negative sign, the plus sign when the value has a positive sign and the ‘`+`’ flag is used, a space when the value has a positive sign and the space flag is used).

The `mpfr_t` number is rounded to the given precision in the direction specified by the rounding mode (see below if the precision is missing). Similarly to the native C types, the precision is the number of digits output after the decimal-point character, except for the ‘`g`’ and ‘`G`’ conversion specifiers, where it is the number of significant digits (but trailing zeros of the fractional part are not output by default), or 1 if the precision is zero. If the precision is zero with rounding to nearest mode and one of the following conversion 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 is missing, it is chosen as follows, depending on the conversion specifier.

- With ‘`a`’, ‘`A`’, and ‘`b`’, it is chosen to have an exact representation with no trailing zeros.
- With ‘`e`’ and ‘`E`’, it is ceil(p times 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 a missing precision and reading it back will yield the original value.
- With ‘`f`’, ‘`F`’, ‘`g`’, and ‘`G`’, it is 6.

For a given number with ‘`a`’/‘`A`’, 4 exponent values allow the first hexadecimal digit to be non-zero as required, and the choice is unspecified in ISO C99 and later. In MPFR, the choice currently depends on the value of the precision field (zero or non-zero) and on the input value. This may change in the future in order to have a more consistent behavior. Details and discussion: https://sympa.inria.fr/sympa/arc/mpfr/2021-05/msg00002.html

#### 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 array 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 array 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 first n − 1 characters are written in buf and the n-th one 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 array 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`, but you must first make sure that it does not contain non-terminating null characters or that the unallocation function does not use its size argument. The return value is the number of characters written in the array, excluding the null-terminator, or a negative value if an error occurred, in which case the contents of str are undefined. Note that this return value may be used to know how many bytes may need to be set to a non-zero value (in case ‘`%c`’ was used with the value 0) before the call to `mpfr_free_str`.

### 5.10 Integer and Remainder Related Functions ¶

Function: `int` mpfr_rint `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_ceil `(mpfr_t rop, mpfr_t op)` ¶
Function: `int` mpfr_floor `(mpfr_t rop, mpfr_t op)` ¶
Function: `int` mpfr_round `(mpfr_t rop, mpfr_t op)` ¶
Function: `int` mpfr_roundeven `(mpfr_t rop, mpfr_t op)` ¶
Function: `int` mpfr_trunc `(mpfr_t rop, mpfr_t op)` ¶

  Set rop to op rounded to an integer. `mpfr_rint` rounds to the nearest representable integer in the given direction rnd, and the other five functions behave in a similar way with some fixed rounding mode:

  - `mpfr_ceil`: to the next higher or equal representable integer (like `mpfr_rint` with `MPFR_RNDU`);
  - `mpfr_floor` to the next lower or equal representable integer (like `mpfr_rint` with `MPFR_RNDD`);
  - `mpfr_round` to the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754);
  - `mpfr_roundeven` to the nearest representable integer, rounding halfway cases with the even-rounding rule (like `mpfr_rint` with `MPFR_RNDN`);
  - `mpfr_trunc` to the next representable integer toward zero (like `mpfr_rint` with `MPFR_RNDZ`).

  When op is a zero or an infinity, set rop to the same value (with the same sign).

  The return value is zero when the result is exact, positive when it is greater than the original value of op, and negative when it is smaller. More precisely, the return value is 0 when op is an integer representable in rop, 1 or −1 when op is an integer that is not representable in rop, 2 or −2 when op is not an integer.

  When op is NaN, the NaN flag is set as usual. In the other cases, the inexact flag is set when rop differs from op, following the ISO C99 rule for the `rint` function. If you want the behavior to be more like IEEE 754 / ISO TS 18661-1, i.e., the usual behavior where the round-to-integer function is regarded as any other mathematical function, you should use one of the `mpfr_rint_*` functions instead.

  Note that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by `mpfr_rint` with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first rounded to an integer, one would round 10.5 to 10 with even rounding, and then 10 would be rounded to 8 again with even rounding.)

Function: `int` mpfr_rint_ceil `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_rint_floor `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_rint_round `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_rint_roundeven `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_rint_trunc `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶

  Set rop to op rounded to an integer:

  - `mpfr_rint_ceil`: to the next higher or equal integer;
  - `mpfr_rint_floor`: to the next lower or equal integer;
  - `mpfr_rint_round`: to the nearest integer, rounding halfway cases away from zero;
  - `mpfr_rint_roundeven`: to the nearest integer, rounding halfway cases to the nearest even integer;
  - `mpfr_rint_trunc` to the next integer toward zero.

  If the result is not representable, it is rounded in the direction rnd. When op is a zero or an infinity, set rop to the same value (with the same sign). The return value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function).

  Contrary to `mpfr_rint`, those functions do perform a double rounding: first op is rounded to the nearest integer in the direction given by the function name, then this nearest integer (if not representable) is rounded in the given direction rnd. Thus these round-to-integer functions behave more like the other mathematical functions, i.e., the returned result is the correct rounding of the exact result of the function in the real numbers.

  For example, `mpfr_rint_round` with rounding to nearest and a precision of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7 is rounded to 8 by the round-even rule, despite the fact that 6 is also representable on two bits, and is closer to 6.5 than 8.

Function: `int` mpfr_frac `(mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd)` ¶

  Set rop to the fractional part of op, having the same sign as op, rounded in the direction rnd (unlike in `mpfr_rint`, rnd affects only how the exact fractional part is rounded, not how the fractional part is generated). When op is an integer or an infinity, set rop to zero with the same sign as op.

Function: `int` mpfr_modf `(mpfr_t iop, mpfr_t fop, mpfr_t op, mpfr_rnd_t rnd)` ¶

  Set simultaneously iop to the integral part of op and fop to the fractional part of op, rounded in the direction rnd with the corresponding precision of iop and fop (equivalent to `mpfr_trunc(iop, op, rnd)` and `mpfr_frac(fop, op, rnd)`). The variables iop and fop must be different. Return 0 iff both results are exact (see `mpfr_sin_cos` for a more detailed description of the return value).

Function: `int` mpfr_fmod `(mpfr_t r, mpfr_t x, mpfr_t y, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_fmod_ui `(mpfr_t r, mpfr_t x, unsigned long int y, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_fmodquo `(mpfr_t r, long int* q, mpfr_t x, mpfr_t y, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_remainder `(mpfr_t r, mpfr_t x, mpfr_t y, mpfr_rnd_t rnd)` ¶
Function: `int` mpfr_remquo `(mpfr_t r, long int* q, mpfr_t x, mpfr_t y, mpfr_rnd_t rnd)` ¶

  Set r to the value of x − ny, rounded according to the direction rnd, where n is the integer quotient of x divided by y, defined as follows: n is rounded toward zero for `mpfr_fmod`, `mpfr_fmod_ui` and `mpfr_fmodquo`, and to the nearest integer (ties rounded to even) for `mpfr_remainder` and `mpfr_remquo`.

  Special values are handled as described in Section F.9.7.1 of the ISO C99 standard: If x is infinite or y is zero, r is NaN. If y is infinite and x is finite, r is x rounded to the precision of r. If r is zero, it has the sign of x. The return value is the ternary value corresponding to r.

  Additionally, `mpfr_fmodquo` and `mpfr_remquo` store the low significant bits from the quotient n in *q (more precisely the number of bits in a `long int` minus one), with the sign of x divided by y (except if those low bits are all zero, in which case zero is returned). If the result is NaN, the value of *q is unspecified. Note that x may be so large in magnitude relative to y that an exact representation of the quotient is not practical. The `mpfr_remainder` and `mpfr_remquo` functions are useful for additive argument reduction.

Function: `int` mpfr_integer_p `(mpfr_t op)` ¶

  Return non-zero iff op is an integer.

### 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 EXP(b) − err where EXP(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 int`, 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 C90 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);

### 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.

  Note: Concerning the exceptions and the sign of 0, the behavior differs from the ISO C `nextafter` and `nexttoward` functions. It is similar to the nextUp and nextDown operations from IEEE 754 (introduced in its 2008 revision).

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

  Equivalent to `mpfr_nexttoward` where y is +Inf (resp. −Inf).

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). As usual, the NaN flag is set only when the result is NaN, i.e., when both op1 and op2 are NaN.

  Note: These functions correspond to the minimumNumber and maximumNumber operations of IEEE 754-2019 for the result. But in MPFR, the NaN flag is set only when both operands are NaN.

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 binary128 support (a.k.a. float128), return zero otherwise. See Nomenclature and Types for additional information.

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.

### 5.13 Exception Related Functions ¶

Function: `mpfr_exp_t` mpfr_get_emin `(void)` ¶
Function: `mpfr_exp_t` mpfr_get_emax `(void)` ¶

  Return the (current) smallest and largest exponents allowed for a floating-point variable. The smallest positive value of a floating-point variable is one half times 2 raised to the smallest exponent and the largest value has the form (1 − epsilon) times 2 raised to the largest exponent, where epsilon depends on the precision of the considered variable.

Function: `int` mpfr_set_emin `(mpfr_exp_t exp)` ¶
Function: `int` mpfr_set_emax `(mpfr_exp_t exp)` ¶

  Set the smallest and largest exponents allowed for a floating-point variable. Return a non-zero value when exp is not in the range accepted by the implementation (in that case the smallest or largest exponent is not changed), and zero otherwise.

  For the subsequent operations, it is the user’s responsibility to check that any floating-point value used as an input is in the new exponent range (for example using `mpfr_check_range`). If a floating-point value outside the new exponent range is used as an input, the default behavior is undefined, in the sense of the ISO C standard; the behavior may also be explicitly documented, such as for `mpfr_check_range`.

  Note: Caches may still have values outside the current exponent range. This is not an issue as the user cannot use these caches directly via the API (MPFR extends the exponent range internally when need be).

  If emin > emax and a floating-point value needs to be produced as output, the behavior is undefined (`mpfr_set_emin` and `mpfr_set_emax` do not check this condition as it might occur between successive calls to these two functions).

Function: `mpfr_exp_t` mpfr_get_emin_min `(void)` ¶
Function: `mpfr_exp_t` mpfr_get_emin_max `(void)` ¶
Function: `mpfr_exp_t` mpfr_get_emax_min `(void)` ¶
Function: `mpfr_exp_t` mpfr_get_emax_max `(void)` ¶

  Return the minimum and maximum of the exponents allowed for `mpfr_set_emin` and `mpfr_set_emax` respectively. These values are implementation dependent, thus a program using `mpfr_set_emax(mpfr_get_emax_max())` or `mpfr_set_emin(mpfr_get_emin_min())` may not be portable.

Function: `int` mpfr_check_range `(mpfr_t x, int t, mpfr_rnd_t rnd)` ¶

  This function assumes that x is the correctly rounded value of some real value y in the direction rnd and some extended exponent range, and that t is the corresponding ternary value. For example, one performed `t = mpfr_log (x, u, rnd)`, and y is the exact logarithm of u. Thus t is negative if x is smaller than y, positive if x is larger than y, and zero if x equals y. This function modifies x if needed to be in the current range of acceptable values: It generates an underflow or an overflow if the exponent of x is outside the current allowed range; the value of t may be used to avoid a double rounding. This function returns zero if the new value of x equals the exact one y, a positive value if that new value is larger than y, and a negative value if it is smaller than y. Note that unlike most functions, the new result x is compared to the (unknown) exact one y, not the input value x, i.e., the ternary value is propagated.

  Note: If x is an infinity and t is different from zero (i.e., if the rounded result is an inexact infinity), then the overflow flag is set. This is useful because `mpfr_check_range` is typically called (at least in MPFR functions) after restoring the flags that could have been set due to internal computations.

Function: `int` mpfr_subnormalize `(mpfr_t x, int t, mpfr_rnd_t rnd)` ¶

  This function rounds x emulating subnormal number arithmetic: if x is outside the subnormal exponent range of the emulated floating-point system, this function just propagates the ternary value t; otherwise, if EXP(x) denotes the exponent of x, it rounds x to precision EXP(x)−emin+1 according to rounding mode rnd and previous ternary value t, avoiding double rounding problems. More precisely in the subnormal domain, denoting by e the value of emin, x is rounded in fixed-point arithmetic to an integer multiple of two to the power e − 1; as a consequence, 1.5 multiplied by two to the power e − 1 when t is zero is rounded to two to the power e with rounding to nearest.

  The precision PREC(x) of x is not modified by this function. rnd and t must be the rounding mode and the returned ternary value used when computing x (as in `mpfr_check_range`). The subnormal exponent range is from emin to emin+PREC(x)−2. If the result cannot be represented in the current exponent range of MPFR (due to a too small emax), the behavior is undefined. Note that unlike most functions, the result is compared to the exact one, not the input value x, i.e., the ternary value is propagated.

  As usual, if the returned ternary value is non zero, the inexact flag is set. Moreover, if a second rounding occurred (because the input x was in the subnormal range), the underflow flag is set.

  Warning! If you change emin (with `mpfr_set_emin`) just before calling `mpfr_subnormalize`, you need to make sure that the value is in the current exponent range of MPFR. But it is better to change emin before any computation, if possible.

This is an example of how to emulate binary64 IEEE 754 arithmetic (a.k.a. double precision) using MPFR:

    {
      mpfr_t xa, xb; int i; volatile double a, b;
      mpfr_set_default_prec (53);
      mpfr_set_emin (-1073); mpfr_set_emax (1024);
      mpfr_init (xa); mpfr_init (xb);
      b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN);
      a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN);
      a /= b;
      i = mpfr_div (xa, xa, xb, MPFR_RNDN);
      i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */
      mpfr_clear (xa); mpfr_clear (xb);
    }

Note that `mpfr_set_emin` and `mpfr_set_emax` are called early enough in order to make sure that all computed values are in the current exponent range. Warning! This emulates a double IEEE 754 arithmetic with correct rounding in the subnormal range, which may not be the case for your hardware.

Below is another example showing how to emulate fixed-point arithmetic in a specific case. Here we compute the sine of the integers 1 to 17 with a result in a fixed-point arithmetic rounded at two to the power −42 (using the fact that the result is at most 1 in absolute value):

    {
      mpfr_t x; int i, inex;
      mpfr_set_emin (-41);
      mpfr_init2 (x, 42);
      for (i = 1; i <= 17; i++)
        {
          mpfr_set_ui (x, i, MPFR_RNDN);
          inex = mpfr_sin (x, x, MPFR_RNDZ);
          mpfr_subnormalize (x, inex, MPFR_RNDZ);
          mpfr_dump (x);
        }
      mpfr_clear (x);
    }

Function: `void` mpfr_clear_underflow `(void)` ¶
Function: `void` mpfr_clear_overflow `(void)` ¶
Function: `void` mpfr_clear_divby0 `(void)` ¶
Function: `void` mpfr_clear_nanflag `(void)` ¶
Function: `void` mpfr_clear_inexflag `(void)` ¶
Function: `void` mpfr_clear_erangeflag `(void)` ¶

  Clear (lower) the underflow, overflow, divide-by-zero, invalid, inexact and erange flags.

Function: `void` mpfr_clear_flags `(void)` ¶

  Clear (lower) all global flags (underflow, overflow, divide-by-zero, invalid, inexact, erange). Note: a group of flags can be cleared by using `mpfr_flags_clear`.

Function: `void` mpfr_set_underflow `(void)` ¶
Function: `void` mpfr_set_overflow `(void)` ¶
Function: `void` mpfr_set_divby0 `(void)` ¶
Function: `void` mpfr_set_nanflag `(void)` ¶
Function: `void` mpfr_set_inexflag `(void)` ¶
Function: `void` mpfr_set_erangeflag `(void)` ¶

  Set (raise) the underflow, overflow, divide-by-zero, invalid, inexact and erange flags.

Function: `int` mpfr_underflow_p `(void)` ¶
Function: `int` mpfr_overflow_p `(void)` ¶
Function: `int` mpfr_divby0_p `(void)` ¶
Function: `int` mpfr_nanflag_p `(void)` ¶
Function: `int` mpfr_inexflag_p `(void)` ¶
Function: `int` mpfr_erangeflag_p `(void)` ¶

  Return the corresponding (underflow, overflow, divide-by-zero, invalid, inexact, erange) flag, which is non-zero iff the flag is set.

The `mpfr_flags_` functions below that take an argument mask can operate on any subset of the exception flags: a flag is part of this subset (or group) if and only if the corresponding bit of the argument mask is set. The `MPFR_FLAGS_` macros will normally be used to build this argument. See Exceptions.

Function: `void` mpfr_flags_clear `(mpfr_flags_t mask)` ¶

  Clear (lower) the group of flags specified by mask.

Function: `void` mpfr_flags_set `(mpfr_flags_t mask)` ¶

  Set (raise) the group of flags specified by mask.

Function: `mpfr_flags_t` mpfr_flags_test `(mpfr_flags_t mask)` ¶

  Return the flags specified by mask. To test whether any flag from mask is set, compare the return value to 0. You can also test individual flags by AND’ing the result with `MPFR_FLAGS_` macros. Example:

      mpfr_flags_t t = mpfr_flags_test (MPFR_FLAGS_UNDERFLOW|
                                        MPFR_FLAGS_OVERFLOW)
      ...
      if (t)  /* underflow and/or overflow (unlikely) */
        {
          if (t & MPFR_FLAGS_UNDERFLOW)  { /* handle underflow */ }
          if (t & MPFR_FLAGS_OVERFLOW)   { /* handle overflow  */ }
        }

Function: `mpfr_flags_t` mpfr_flags_save `(void)` ¶

  Return all the flags. It is equivalent to `mpfr_flags_test(MPFR_FLAGS_ALL)`.

Function: `void` mpfr_flags_restore `(mpfr_flags_t flags, mpfr_flags_t mask)` ¶

  Restore the flags specified by mask to their state represented in flags.

### 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.

### 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.

### 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. The `mpfr_custom_init_set` macro is not usable 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.

### 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.