Average Error: 0.6 → 0.6
Time: 3.3s
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 r1742288 = re;
        double r1742289 = r1742288 * r1742288;
        double r1742290 = im;
        double r1742291 = r1742290 * r1742290;
        double r1742292 = r1742289 + r1742291;
        double r1742293 = sqrt(r1742292);
        return r1742293;
}
↓
double f(double re, double im) {
        double r1742294 = re;
        double r1742295 = r1742294 * r1742294;
        double r1742296 = im;
        double r1742297 = r1742296 * r1742296;
        double r1742298 = r1742295 + r1742297;
        double r1742299 = sqrt(r1742298);
        return r1742299;
}

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 2019135 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))