# remquo, remquof, remquol

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< c‎ | numeric‎ | math

Defined in header `<math.h>` |  |
float remquof( float x, float y, int *quo ); | (1) | (since C99)
double remquo( double x, double y, int *quo ); | (2) | (since C99)
long double remquol( long double x, long double y, int *quo ); | (3) | (since C99)
Defined in header `<tgmath.h>` |  |
#define remquo( x, y, quo ) | (4) | (since C99)

1-3) Computes the floating-point remainder of the division operation x/y as the remainder() function does. Additionally, the sign and at least the three of the last bits of x/y will be stored in quo, sufficient to determine the octant of the result within a period.
4) Type-generic macro: If any non-pointer argument has type long double, `remquol` is called. Otherwise, if any non-pointer argument has integer type or has type double, `remquo` is called. Otherwise, `remquof` is called.

### Parameters

x, y | - | floating point values
quo | - | pointer to an integer value to store the sign and some bits of x/y

### Return value

If successful, returns the floating-point remainder of the division x/y as defined in remainder, and stores, in *quo, the sign and at least three of the least significant bits of x/y (formally, stores a value whose sign is the sign of x/y and whose magnitude is congruent modulo 2n
to the magnitude of the integral quotient of x/y, where n is an implementation-defined integer greater than or equal to 3).

If `y` is zero, the value stored in *quo is unspecified.

If a domain error occurs, an implementation-defined value is returned (NaN where supported).

If a range error occurs due to underflow, the correct result is returned if subnormals are supported.

If `y` is zero, but the domain error does not occur, zero is returned.

### Error handling

Errors are reported as specified in math_errhandling.

Domain error may occur if `y` is zero.

If the implementation supports IEEE floating-point arithmetic (IEC 60559),

- The current rounding mode has no effect.
- FE_INEXACT is never raised
- If `x` is ±∞ and `y` is not NaN, NaN is returned and FE_INVALID is raised
- If `y` is ±0 and `x` is not NaN, NaN is returned and FE_INVALID is raised
- If either `x` or `y` is NaN, NaN is returned

### Notes

POSIX requires that a domain error occurs if `x` is infinite or `y` is zero.

This function is useful when implementing periodic functions with the period exactly representable as a floating-point value: when calculating sin(πx) for a very large `x`, calling sin directly may result in a large error, but if the function argument is first reduced with `remquo`, he low-order bits of the quotient may be used to determine the sign and the octant of the result within the period, while the remainder may be used to calculate the value with high precision.

On some platforms this operation is supported by hardware (and, for example, on Intel CPU, `FPREM1` leaves exactly 3 bits of precision in the quotient)

### Example

Run this code

    #include <stdio.h>
    #include <math.h>
    #include <fenv.h>

    #pragma STDC FENV_ACCESS ON
    double cos_pi_x_naive(double x)
    {
        double pi = acos(-1);
        return cos(pi * x);
    }
    // the period is 2, values are (0;0.5) positive, (0.5;1.5) negative, (1.5,2) positive
    double cos_pi_x_smart(double x)
    {
        int quadrant;
        double rem = remquo(x, 1, &quadrant);
        quadrant = (unsigned)quadrant % 4; // keep 2 bits to determine quadrant

        double pi = acos(-1);
        switch(quadrant) {
            case 0: return cos(pi * rem);
            case 1: return -cos(pi * rem);
            case 2: return -cos(pi * rem);
            case 3: return cos(pi * rem);
        };
    }
    int main(void)
    {
        printf("cos(pi * 0.25) = %f\n", cos_pi_x_naive(0.25));
        printf("cos(pi * 1.25) = %f\n", cos_pi_x_naive(1.25));
        printf("cos(pi * 1000000000000.25) = %f\n", cos_pi_x_naive(1000000000000.25));
        printf("cos(pi * 1000000000001.25) = %f\n", cos_pi_x_naive(1000000000001.25));
        printf("cos(pi * 1000000000000.25) = %f\n", cos_pi_x_smart(1000000000000.25));
        printf("cos(pi * 1000000000001.25) = %f\n", cos_pi_x_smart(1000000000001.25));
        // error handling
        feclearexcept(FE_ALL_EXCEPT);
        int quo;
        printf("remquo(+Inf, 1) = %.1f\n", remquo(INFINITY, 1, &quo));
        if(fetestexcept(FE_INVALID)) puts("    FE_INVALID raised");
    }

Possible output:

    cos(pi * 0.25) = 0.707107
    cos(pi * 1.25) = -0.707107
    cos(pi * 1000000000000.25) = 0.707123
    cos(pi * 1000000000001.25) = -0.707117
    cos(pi * 1000000000000.25) = 0.707107
    cos(pi * 1000000000001.25) = -0.707107
    remquo(+Inf, 1) = -nan
        FE_INVALID raised

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.12.10.3 The remquo functions (p: 255)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.7.3 The remquo functions (p: 529)

- C99 standard (ISO/IEC 9899:1999):

  - 7.12.10.3 The remquo functions (p: 236)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - F.9.7.3 The remquo functions (p: 465)

### See also

div ldiv lldiv (C99) | computes quotient and remainder of integer division (function)
fmod fmodf fmodl (C99)(C99) | computes remainder of the floating-point division operation (function)
remainder remainderf remainderl (C99)(C99)(C99) | computes signed remainder of the floating-point division operation (function)

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