Average Error: 0.6 → 0.6
Time: 3.2s
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 r1040650 = re;
        double r1040651 = r1040650 * r1040650;
        double r1040652 = im;
        double r1040653 = r1040652 * r1040652;
        double r1040654 = r1040651 + r1040653;
        double r1040655 = sqrt(r1040654);
        return r1040655;
}
↓
double f(double re, double im) {
        double r1040656 = re;
        double r1040657 = r1040656 * r1040656;
        double r1040658 = im;
        double r1040659 = r1040658 * r1040658;
        double r1040660 = r1040657 + r1040659;
        double r1040661 = sqrt(r1040660);
        return r1040661;
}

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 2019134 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))