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 r843532 = re;
        double r843533 = r843532 * r843532;
        double r843534 = im;
        double r843535 = r843534 * r843534;
        double r843536 = r843533 + r843535;
        double r843537 = sqrt(r843536);
        return r843537;
}
↓
double f(double re, double im) {
        double r843538 = re;
        double r843539 = r843538 * r843538;
        double r843540 = im;
        double r843541 = r843540 * r843540;
        double r843542 = r843539 + r843541;
        double r843543 = sqrt(r843542);
        return r843543;
}

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 2019119 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))