Fma

Defined in header <cmath>.

Description

Computes x * y + z as if to infinite precision and rounded only once to fit the result type. The library provides overloads of std::fma for all cv-unqualified floating-point types as the type of the parameters x, y and z.

If the macro constants FP_FAST_FMA, FP_FAST_FMAF, or FP_FAST_FMAL are defined, the function std::fma 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.

Declarations

C++23

// 1)
constexpr /* floating-point-type */
            fma ( /* floating-point-type */ x,
                  /* floating-point-type */ y,
                  /* floating-point-type */ z );
// 2)
constexpr float fmaf( float x, float y, float z );
// 3)
constexpr long double fmal( long double x, long double y, long double z );

Additional Overloads

// 4)
template< class Arithmetic1, class Arithmetic2, class Arithmetic3 >
constexpr /* common-floating-point-type */
     fma( Arithmetic1 x, Arithmetic2 y, Arithmetic3 z );

Macro Constants Definition

#define FP_FAST_FMA  /* implementation-defined */
#define FP_FAST_FMAF /* implementation-defined */
#define FP_FAST_FMAL /* implementation-defined */

C++11

// 1)
float fma ( float x, float y, float z );
// 2)
double fma ( double x, double y, double z );
// 3)
long double fma ( long double x, long double y, long double z );
// 4)
float fmaf( float x, float y, float z );
// 5)
long double fmal( long double x, long double y, long double z );

Additional Overloads

// 6)
template< class Arithmetic1, class Arithmetic2, class Arithmetic3 >
/* common-floating-point-type */
    fma( Arithmetic1 x, Arithmetic2 y, Arithmetic3 z );

Macro Constants Definition

#define FP_FAST_FMA  /* implementation-defined */
#define FP_FAST_FMAF /* implementation-defined */
#define FP_FAST_FMAL /* implementation-defined */

Parameters

x, y, z - floating-point or integer 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

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
if z is not a NaN, then NaN is returned and FE_INVALID is raised
if 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 is not 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 (fma, fmaf, fmal) additionally specifies that the situations specified to return FE_INVALID are domain errors.

Due to its infinite intermediate precision, std::fma is a common building block of other correctly-rounded mathematical operations, such as std::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 multiply-add unless the #pragma STDC FP_CONTRACT is off.

The additional overloads are not required to be provided exactly as in Addition Overloads. They only need to be sufficient to ensure that for their first argument num1, second argument num2 and third argument num3:

If num1, num2 or num3 has type long double, then
std::fma(num1, num2, num3) has the same effect as
​ std::fma(static_cast<long double>(num1), static_cast<long double>(num2), static_cast<long double>(num3)).

Otherwise, if num1, num2 and/or num3 has type double or an integer type, then
std::fma(num1, num2, num3) has the same effect as
​ std::fma(static_cast<double>(num1), static_cast<double>(num2), static_cast<double>(num3)).

Otherwise, if num1, num2 or num3 has type float, then
std::fma(num1, num2, num3) has the same effect as
​ std::fma(static_cast<float>(num1), static_cast<float>(num2), static_cast<float>(num3)).

If num1, num2 and num3 have arithmetic types, then
std::fma(num1, num2, num3) has the same effect as
std::fma(static_cast</* common-floating-point-type */>(num1), static_cast</* common-floating-point-type */>(num2), static_cast</* common-floating-point-type */>(num3))

where /* common-floating-point-type */ is the floating-point type with the greatest floating-point conversion rank and greatest floating-point conversion subrank among the types of num1, num2 and num3, arguments of integer type are considered to have the same floating-point conversion rank as double.

If no such floating-point type with the greatest rank and subrank exists, then overload resolution does not result in a usable candidate from the overloads provided.

Examples

#include <cfenv>
#include <cmath>
#include <iomanip>
#include <iostream>
 
#ifndef __GNUC__
#pragma STDC FENV_ACCESS ON
#endif
 
int main()
{
// demo the difference between fma and built-in operators
    const double in = 0.1;
    std::cout 
        << "0.1 double is " 
        << std::setprecision(23) << in
        << " (" << std::hexfloat << in 
        << std::defaultfloat << ")\n"
        << "0.1*10 is 1.0000000000000000555112 (0x8.0000000000002p-3), "
        << "or 1.0 if rounded to double\n";
 
    const double expr_result = 0.1 * 10 - 1;
    const double fma_result = std::fma(0.1, 10, -1);
    std::cout
        << "0.1 * 10 - 1 = " 
        << expr_result
        << " : 1 subtracted after intermediate rounding\n"
        << "fma(0.1, 10, -1) = " 
        << std::setprecision(6) 
        << fma_result << " ("
        << std::hexfloat 
        << fma_result 
        << std::defaultfloat << ")\n\n";
 
    // fma is used in double-double arithmetic
    const double high = 0.1 * 10;
    const double low = std::fma(0.1, 10, -high);
    std::cout 
        << "in double-double arithmetic, 0.1 * 10 is representable as "
        << high << " + " << low << "\n\n";
 
    // error handling 
    std::feclearexcept(FE_ALL_EXCEPT);
    std::cout
    << "fma(+Inf, 10, -Inf) = " 
    << std::fma(INFINITY, 10, -INFINITY) 
    << '\n';

    if (std::fetestexcept(FE_INVALID))
        std::cout 
            << "FE_INVALID raised\n";
}

Possible Result

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
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