Average Error: 0.6 → 0.6
Time: 2.6s
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 r1552160 = re;
        double r1552161 = r1552160 * r1552160;
        double r1552162 = im;
        double r1552163 = r1552162 * r1552162;
        double r1552164 = r1552161 + r1552163;
        double r1552165 = sqrt(r1552164);
        return r1552165;
}
↓
double f(double re, double im) {
        double r1552166 = re;
        double r1552167 = r1552166 * r1552166;
        double r1552168 = im;
        double r1552169 = r1552168 * r1552168;
        double r1552170 = r1552167 + r1552169;
        double r1552171 = sqrt(r1552170);
        return r1552171;
}

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 2019134 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))