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 r1841646 = re;
        double r1841647 = r1841646 * r1841646;
        double r1841648 = im;
        double r1841649 = r1841648 * r1841648;
        double r1841650 = r1841647 + r1841649;
        double r1841651 = sqrt(r1841650);
        return r1841651;
}
↓
double f(double re, double im) {
        double r1841652 = re;
        double r1841653 = r1841652 * r1841652;
        double r1841654 = im;
        double r1841655 = r1841654 * r1841654;
        double r1841656 = r1841653 + r1841655;
        double r1841657 = sqrt(r1841656);
        return r1841657;
}

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 2019125 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))