Average Error: 0.0 → 0.0
Time: 1.1s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r41957 = x_re;
        double r41958 = y_re;
        double r41959 = r41957 * r41958;
        double r41960 = x_im;
        double r41961 = y_im;
        double r41962 = r41960 * r41961;
        double r41963 = r41959 - r41962;
        return r41963;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r41964 = x_re;
        double r41965 = y_re;
        double r41966 = x_im;
        double r41967 = y_im;
        double r41968 = r41966 * r41967;
        double r41969 = -r41968;
        double r41970 = fma(r41964, r41965, r41969);
        return r41970;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Initial program 0.0

    \[x.re \cdot y.re - x.im \cdot y.im\]
  2. Using strategy rm

  3. Applied fma-neg0.0

    \[\leadsto \color{blue}{\mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)}\]

  4. Final simplification0.0

    \[\leadsto \mathsf{fma}\left(x.re, y.re, -x.im \cdot y.im\right)\]

Reproduce

herbie shell --seed 2019353 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  :precision binary64
  (- (* x.re y.re) (* x.im y.im)))