Average Error: 0.6 → 0.6
Time: 2.4s
Precision: 64
\[\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\]
↓
\[\sqrt{re \cdot re + im \cdot im}\]
\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}
↓
\sqrt{re \cdot re + im \cdot im}
double f(double re, double im) {
        double r793911 = re;
        double r793912 = r793911 * r793911;
        double r793913 = im;
        double r793914 = r793913 * r793913;
        double r793915 = r793912 + r793914;
        double r793916 = sqrt(r793915);
        return r793916;
}
↓
double f(double re, double im) {
        double r793917 = re;
        double r793918 = r793917 * r793917;
        double r793919 = im;
        double r793920 = r793919 * r793919;
        double r793921 = r793918 + r793920;
        double r793922 = sqrt(r793921);
        return r793922;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.6

    \[\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\]

  2. Final simplification0.6

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

Reproduce

herbie shell --seed 2019104 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))