Average Error: 0.6 → 0.6
Time: 2.7s
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 r733258 = re;
        double r733259 = r733258 * r733258;
        double r733260 = im;
        double r733261 = r733260 * r733260;
        double r733262 = r733259 + r733261;
        double r733263 = sqrt(r733262);
        return r733263;
}
↓
double f(double re, double im) {
        double r733264 = re;
        double r733265 = r733264 * r733264;
        double r733266 = im;
        double r733267 = r733266 * r733266;
        double r733268 = r733265 + r733267;
        double r733269 = sqrt(r733268);
        return r733269;
}

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 2019107 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))