Average Error: 0.6 → 0.6
Time: 3.1s
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 r228636 = re;
        double r228637 = r228636 * r228636;
        double r228638 = im;
        double r228639 = r228638 * r228638;
        double r228640 = r228637 + r228639;
        double r228641 = sqrt(r228640);
        return r228641;
}
↓
double f(double re, double im) {
        double r228642 = re;
        double r228643 = r228642 * r228642;
        double r228644 = im;
        double r228645 = r228644 * r228644;
        double r228646 = r228643 + r228645;
        double r228647 = sqrt(r228646);
        return r228647;
}

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 2019132 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))