Average Error: 0.6 → 0.6
Time: 3.6s
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 r1742254 = re;
        double r1742255 = r1742254 * r1742254;
        double r1742256 = im;
        double r1742257 = r1742256 * r1742256;
        double r1742258 = r1742255 + r1742257;
        double r1742259 = sqrt(r1742258);
        return r1742259;
}
↓
double f(double re, double im) {
        double r1742260 = re;
        double r1742261 = r1742260 * r1742260;
        double r1742262 = im;
        double r1742263 = r1742262 * r1742262;
        double r1742264 = r1742261 + r1742263;
        double r1742265 = sqrt(r1742264);
        return r1742265;
}

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