Average Error: 0.6 → 0.6
Time: 2.1s
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 r513541 = re;
        double r513542 = r513541 * r513541;
        double r513543 = im;
        double r513544 = r513543 * r513543;
        double r513545 = r513542 + r513544;
        double r513546 = sqrt(r513545);
        return r513546;
}
↓
double f(double re, double im) {
        double r513547 = re;
        double r513548 = r513547 * r513547;
        double r513549 = im;
        double r513550 = r513549 * r513549;
        double r513551 = r513548 + r513550;
        double r513552 = sqrt(r513551);
        return r513552;
}

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 2019112 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))