Average Error: 0.0 → 0.0
Time: 430.0ms
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 r710 = re;
        double r711 = im;
        double r712 = r710 * r711;
        double r713 = r711 * r710;
        double r714 = r712 + r713;
        return r714;
}
↓
double f(double re, double im) {
        double r715 = re;
        double r716 = im;
        double r717 = r716 * r715;
        double r718 = fma(r715, r716, r717);
        return r718;
}

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 2019347 +o rules:numerics
(FPCore (re im)
  :name "math.square on complex, imaginary part"
  :precision binary64
  (+ (* re im) (* im re)))