std::fma, std::fmaf, std::fmal

< cpp‎ | numeric‎ | math
Common mathematical functions
Basic operations
Exponential functions
Power functions
Trigonometric and hyperbolic functions
Error and gamma functions
Nearest integer floating point operations
Floating point manipulation functions
Macro constants
Defined in header <cmath>
float       fma ( float x, float y, float z );
float       fmaf( float x, float y, float z );
(1) (since C++11)
double      fma ( double x, double y, double z );
(2) (since C++11)
long double fma ( long double x, long double y, long double z );
long double fmal( long double x, long double y, long double z );
(3) (since C++11)
Promoted    fma ( Arithmetic1 x, Arithmetic2 y, Arithmetic3 z );
(4) (since C++11)
#define FP_FAST_FMA  /* implementation-defined */
(5) (since C++11)
#define FP_FAST_FMAF /* implementation-defined */
(6) (since C++11)
#define FP_FAST_FMAL /* implementation-defined */
(7) (since C++11)
1-3) Computes (x*y) + z as if to infinite precision and rounded only once to fit the result type.
4) A set of overloads or a function template for all combinations of arguments of arithmetic type not covered by 1-3). If any argument has integral type, it is cast to double. If any other argument is long double, then the return type is long double, otherwise it is double.
5-7) If the macro constants FP_FAST_FMAF, FP_FAST_FMA, 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.


x, y, z - values of floating-point or integral types

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)


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 additionally specifies that the situations specified to return FE_INVALID are domain errors.

Due to its infinite intermediate precision, 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.


#include <iostream>
#include <iomanip>
#include <cmath>
#include <cfenv>
int main()
    // demo the difference between fma and built-in operators
    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";
    double expr_result = 0.1 * 10 - 1;
    double fma_result = 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
    double high = 0.1 * 10;
    double low = fma(0.1, 10, -high);
    std::cout << "in double-double arithmetic, 0.1 * 10 is representable as "
              << high << " + " << low << "\n\n";
    // error handling 
    std::cout << "fma(+Inf, 10, -Inf) = " << std::fma(INFINITY, 10, -INFINITY) << '\n';
        std::cout << "    FE_INVALID raised\n";

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
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

See also

signed remainder of the division operation
signed remainder as well as the three last bits of the division operation