# fma, fmaf, fmal

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< c‎ | numeric‎ | math

Defined in header `<math.h>` |  |
float fmaf( float x, float y, float z ); | (1) | (since C99)
double fma( double x, double y, double z ); | (2) | (since C99)
long double fmal( long double x, long double y, long double z ); | (3) | (since C99)
#define FP_FAST_FMA /* implementation-defined */ | (4) | (since C99)
#define FP_FAST_FMAF /* implementation-defined */ | (5) | (since C99)
#define FP_FAST_FMAL /* implementation-defined */ | (6) | (since C99)
Defined in header `<tgmath.h>` |  |
#define fma( x, y, z ) | (7) | (since C99)

1-3) Computes (x*y) + z as if to infinite precision and rounded only once to fit the result type.
4-6) If the macro constants `FP_FAST_FMAF`, `FP_FAST_FMA`, or `FP_FAST_FMAL` are defined, the corresponding function `fmaf`, `fma`, or `fmal` evaluates faster (in addition to being more precise) than the expression x*y+z for float, double, and long double arguments, respectively. If defined, these macros evaluate to integer 1.
7) Type-generic macro: If any argument has type long double, `fmal` is called. Otherwise, if any argument has integer type or has type double, `fma` is called. Otherwise, `fmaf` is called.

### Parameters

x, y, z | - | floating point values

### Return value

If successful, returns the value of (x*y) + z as if calculated to infinite precision and rounded once to fit the result type (or, alternatively, calculated as a single ternary floating-point operation).

If a range error due to overflow occurs, `±HUGE_VAL`, `±HUGE_VALF`, or `±HUGE_VALL` is returned.

If a range error due to underflow occurs, the correct value (after rounding) is returned.

### Error handling

Errors are reported as specified in math_errhandling.

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

- If x is zero and y is infinite or if x is infinite and y is zero, and z is not a NaN, then NaN is returned and FE_INVALID is raised
- If x is zero and y is infinite or if x is infinite and y is zero, and z is a NaN, then NaN is returned and FE_INVALID may be raised
- If x*y is an exact infinity and z is an infinity with the opposite sign, NaN is returned and FE_INVALID is raised
- If x or y are NaN, NaN is returned
- If z is NaN, and x*y aren't 0*Inf or Inf*0, then NaN is returned (without FE_INVALID)

### Notes

This operation is commonly implemented in hardware as fused multiply-add CPU instruction. If supported by hardware, the appropriate FP_FAST_FMA* macros are expected to be defined, but many implementations make use of the CPU instruction even when the macros are not defined.

POSIX specifies that the situation where the value x*y is invalid and z is a NaN is a domain error.

Due to its infinite intermediate precision, `fma` is a common building block of other correctly-rounded mathematical operations, such as sqrt or even the division (where not provided by the CPU, e.g. Itanium).

As with all floating-point expressions, the expression (x*y) + z may be compiled as a fused mutiply-add unless the #pragma STDC FP_CONTRACT is off.

### Example

Run this code

    #include <stdio.h>
    #include <math.h>
    #include <float.h>
    #include <fenv.h>
    #pragma STDC FENV_ACCESS ON
    int main(void)
    {
        // demo the difference between fma and built-in operators
        double in = 0.1;
        printf("0.1 double is %.23f (%a)\n", in, in);
        printf("0.1*10 is 1.0000000000000000555112 (0x8.0000000000002p-3),"
               " or 1.0 if rounded to double\n");
        double expr_result = 0.1 * 10 - 1;
        printf("0.1 * 10 - 1 = %g : 1 subtracted after "
               "intermediate rounding to 1.0\n", expr_result);
        double fma_result = fma(0.1, 10, -1);
        printf("fma(0.1, 10, -1) = %g (%a)\n", fma_result, fma_result);

        // fma use in double-double arithmetic
        printf("\nin double-double arithmetic, 0.1 * 10 is representable as ");
        double high = 0.1 * 10;
        double low = fma(0.1, 10, -high);
        printf("%g + %g\n\n", high, low);

        //error handling
        feclearexcept(FE_ALL_EXCEPT);
        printf("fma(+Inf, 10, -Inf) = %f\n", fma(INFINITY, 10, -INFINITY));
        if(fetestexcept(FE_INVALID)) puts("    FE_INVALID raised");
    }

Possible output:

    0.1 double is 0.10000000000000000555112 (0x1.999999999999ap-4)
    0.1*10 is 1.0000000000000000555112 (0x8.0000000000002p-3), or 1.0 if rounded to double
    0.1 * 10 - 1 = 0 : 1 subtracted after intermediate rounding to 1.0
    fma(0.1, 10, -1) = 5.55112e-17 (0x1p-54)

    in double-double arithmetic, 0.1 * 10 is representable as 1 + 5.55112e-17

    fma(+Inf, 10, -Inf) = -nan
        FE_INVALID raised

### References

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

  - 7.12.13.1 The fma functions (p: 258)

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

  - F.10.10.1 The fma functions (p: 530)

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

  - 7.12.13.1 The fma functions (p: 239)

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

  - F.9.10.1 The fma functions (p: 466)

### See also

remainder remainderf remainderl (C99)(C99)(C99) | computes signed remainder of the floating-point division operation (function)
remquo remquof remquol (C99)(C99)(C99) | computes signed remainder as well as the three last bits of the division operation (function)
C++ documentation for fma

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