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 r1002901 = re;
        double r1002902 = r1002901 * r1002901;
        double r1002903 = im;
        double r1002904 = r1002903 * r1002903;
        double r1002905 = r1002902 + r1002904;
        double r1002906 = sqrt(r1002905);
        return r1002906;
}
↓
double f(double re, double im) {
        double r1002907 = re;
        double r1002908 = r1002907 * r1002907;
        double r1002909 = im;
        double r1002910 = r1002909 * r1002909;
        double r1002911 = r1002908 + r1002910;
        double r1002912 = sqrt(r1002911);
        return r1002912;
}

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))))