Average Error: 29.1 → 0.0
Time: 1.3s
Precision: 64
\[\sqrt{re \cdot re + im \cdot im}\]
↓
\[\sqrt{re^2 + im^2}^*\]
double f(double re, double im) {
        double r490667 = re;
        double r490668 = r490667 * r490667;
        double r490669 = im;
        double r490670 = r490669 * r490669;
        double r490671 = r490668 + r490670;
        double r490672 = sqrt(r490671);
        return r490672;
}
↓
double f(double re, double im) {
        double r490673 = re;
        double r490674 = im;
        double r490675 = hypot(r490673, r490674);
        return r490675;
}


\sqrt{re \cdot re + im \cdot im}
↓
\sqrt{re^2 + im^2}^*

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 29.1

    \[\sqrt{re \cdot re + im \cdot im}\]

  2. Simplified0.0

    \[\leadsto \color{blue}{\sqrt{re^2 + im^2}^*}\]

  3. Final simplification0.0

    \[\leadsto \sqrt{re^2 + im^2}^*\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt (+ (* re re) (* im im))))