Average Error: 0.0 → 0.0
Time: 807.0ms
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)\]
x.re \cdot y.im + x.im \cdot y.re
↓
\mathsf{fma}\left(y.re, x.im, y.im \cdot x.re\right)
double f(double x_re, double x_im, double y_re, double y_im) {
        double r100964 = x_re;
        double r100965 = y_im;
        double r100966 = r100964 * r100965;
        double r100967 = x_im;
        double r100968 = y_re;
        double r100969 = r100967 * r100968;
        double r100970 = r100966 + r100969;
        return r100970;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r100971 = y_re;
        double r100972 = x_im;
        double r100973 = y_im;
        double r100974 = x_re;
        double r100975 = r100973 * r100974;
        double r100976 = fma(r100971, r100972, r100975);
        return r100976;
}

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. Taylor expanded around inf 0.0

    \[\leadsto \color{blue}{y.re \cdot x.im + y.im \cdot x.re}\]

  3. Simplified0.0

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

  4. Final simplification0.0

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

Reproduce

herbie shell --seed 2020064 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  :precision binary64
  (+ (* x.re y.im) (* x.im y.re)))