Average Error: 0.5 → 0.5
Time: 3.6s
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 r228619 = re;
        double r228620 = r228619 * r228619;
        double r228621 = im;
        double r228622 = r228621 * r228621;
        double r228623 = r228620 + r228622;
        double r228624 = sqrt(r228623);
        return r228624;
}
↓
double f(double re, double im) {
        double r228625 = re;
        double r228626 = r228625 * r228625;
        double r228627 = im;
        double r228628 = r228627 * r228627;
        double r228629 = r228626 + r228628;
        double r228630 = sqrt(r228629);
        return r228630;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.5

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

  2. Final simplification0.5

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

Reproduce

herbie shell --seed 2019133 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))