Average Error: 0.0 → 0.0
Time: 1.7s
Precision: 64
\[x.re \cdot y.im + x.im \cdot y.re\]
↓
\[(y.re \cdot x.im + \left(y.im \cdot x.re\right))_*\]
x.re \cdot y.im + x.im \cdot y.re
↓
(y.re \cdot x.im + \left(y.im \cdot x.re\right))_*
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1097280 = x_re;
        double r1097281 = y_im;
        double r1097282 = r1097280 * r1097281;
        double r1097283 = x_im;
        double r1097284 = y_re;
        double r1097285 = r1097283 * r1097284;
        double r1097286 = r1097282 + r1097285;
        return r1097286;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1097287 = y_re;
        double r1097288 = x_im;
        double r1097289 = y_im;
        double r1097290 = x_re;
        double r1097291 = r1097289 * r1097290;
        double r1097292 = fma(r1097287, r1097288, r1097291);
        return r1097292;
}

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

    \[\leadsto \color{blue}{(x.re \cdot y.im + \left(x.im \cdot y.re\right))_*}\]

  3. Taylor expanded around inf 0.0

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

  4. Simplified0.0

    \[\leadsto \color{blue}{(y.re \cdot x.im + \left(y.im \cdot x.re\right))_*}\]

  5. Final simplification0.0

    \[\leadsto (y.re \cdot x.im + \left(y.im \cdot x.re\right))_*\]

Reproduce

herbie shell --seed 2019119 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_multiplyComplex, imaginary part"
  (+ (* x.re y.im) (* x.im y.re)))