Average Error: 0.6 → 0.6
Time: 2.9s
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 r629690 = re;
        double r629691 = r629690 * r629690;
        double r629692 = im;
        double r629693 = r629692 * r629692;
        double r629694 = r629691 + r629693;
        double r629695 = sqrt(r629694);
        return r629695;
}
↓
double f(double re, double im) {
        double r629696 = re;
        double r629697 = r629696 * r629696;
        double r629698 = im;
        double r629699 = r629698 * r629698;
        double r629700 = r629697 + r629699;
        double r629701 = sqrt(r629700);
        return r629701;
}

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 2019104 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))