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fma, fmaf, fmal

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_FMA, FP_FAST_FMAF, 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

#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

C17 standard (ISO/IEC 9899:2018):

7.12.13.1 The fma functions (p: 188-189)
7.25 Type-generic math <tgmath.h> (p: 272-273)
F.10.10.1 The fma functions (p: 386)

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)

remainderremainderfremainderl (C99)(C99)(C99)	computes signed remainder of the floating-point division operation (function)
remquoremquofremquol (C99)(C99)(C99)	computes signed remainder as well as the three last bits of the division operation (function)
C++ documentation for `fma`