Average Error: 0.6 → 0.6
Time: 3.0s
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 r733253 = re;
        double r733254 = r733253 * r733253;
        double r733255 = im;
        double r733256 = r733255 * r733255;
        double r733257 = r733254 + r733256;
        double r733258 = sqrt(r733257);
        return r733258;
}
↓
double f(double re, double im) {
        double r733259 = re;
        double r733260 = r733259 * r733259;
        double r733261 = im;
        double r733262 = r733261 * r733261;
        double r733263 = r733260 + r733262;
        double r733264 = sqrt(r733263);
        return r733264;
}

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))))