Average Error: 0.0 → 0.0
Time: 7.8s
Precision: 64
\[x.re \cdot y.re - x.im \cdot y.im\]
↓
\[\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)\]
x.re \cdot y.re - x.im \cdot y.im
↓
\mathsf{fma}\left(x.re, y.re, \left(-x.im \cdot y.im\right)\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1274934 = x_re;
        double r1274935 = y_re;
        double r1274936 = r1274934 * r1274935;
        double r1274937 = x_im;
        double r1274938 = y_im;
        double r1274939 = r1274937 * r1274938;
        double r1274940 = r1274936 - r1274939;
        return r1274940;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1274941 = x_re;
        double r1274942 = y_re;
        double r1274943 = x_im;
        double r1274944 = y_im;
        double r1274945 = r1274943 * r1274944;
        double r1274946 = -r1274945;
        double r1274947 = fma(r1274941, r1274942, r1274946);
        return r1274947;
}

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

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2019130 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, real part"
  (- (* x.re y.re) (* x.im y.im)))