Average Error: 0.0 → 0.0
Time: 622.0ms
Precision: 64
\[re \cdot im + im \cdot re\]
↓
\[\mathsf{fma}\left(re, im, im \cdot re\right)\]
re \cdot im + im \cdot re
↓
\mathsf{fma}\left(re, im, im \cdot re\right)
double f(double re, double im) {
        double r651 = re;
        double r652 = im;
        double r653 = r651 * r652;
        double r654 = r652 * r651;
        double r655 = r653 + r654;
        return r655;
}
↓
double f(double re, double im) {
        double r656 = re;
        double r657 = im;
        double r658 = r657 * r656;
        double r659 = fma(r656, r657, r658);
        return r659;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.0

    \[re \cdot im + im \cdot re\]

  2. Simplified0.0

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

  3. Final simplification0.0

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

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))