### 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 − n y, 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.