Average Error: 0.6 → 0.6
Time: 3.7s
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 r1589400 = re;
        double r1589401 = r1589400 * r1589400;
        double r1589402 = im;
        double r1589403 = r1589402 * r1589402;
        double r1589404 = r1589401 + r1589403;
        double r1589405 = sqrt(r1589404);
        return r1589405;
}
↓
double f(double re, double im) {
        double r1589406 = re;
        double r1589407 = r1589406 * r1589406;
        double r1589408 = im;
        double r1589409 = r1589408 * r1589408;
        double r1589410 = r1589407 + r1589409;
        double r1589411 = sqrt(r1589410);
        return r1589411;
}

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 2019130 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))