Average Error: 0.0 → 0.0
Time: 11.2s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(x.re, y.im, x.im \cdot y.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1766954 = x_re;
        double r1766955 = y_im;
        double r1766956 = r1766954 * r1766955;
        double r1766957 = x_im;
        double r1766958 = y_re;
        double r1766959 = r1766957 * r1766958;
        double r1766960 = r1766956 + r1766959;
        return r1766960;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1766961 = x_re;
        double r1766962 = y_im;
        double r1766963 = x_im;
        double r1766964 = y_re;
        double r1766965 = r1766963 * r1766964;
        double r1766966 = fma(r1766961, r1766962, r1766965);
        return r1766966;
}

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

    \[x.re \cdot y.im + x.im \cdot y.re\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2019164 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))