Average Error: 0.6 → 0.6
Time: 2.6s
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 r1129918 = re;
        double r1129919 = r1129918 * r1129918;
        double r1129920 = im;
        double r1129921 = r1129920 * r1129920;
        double r1129922 = r1129919 + r1129921;
        double r1129923 = sqrt(r1129922);
        return r1129923;
}
↓
double f(double re, double im) {
        double r1129924 = re;
        double r1129925 = r1129924 * r1129924;
        double r1129926 = im;
        double r1129927 = r1129926 * r1129926;
        double r1129928 = r1129925 + r1129927;
        double r1129929 = sqrt(r1129928);
        return r1129929;
}

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 2019128 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))