Average Error: 0.6 → 0.6
Time: 2.8s
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 r649974 = re;
        double r649975 = r649974 * r649974;
        double r649976 = im;
        double r649977 = r649976 * r649976;
        double r649978 = r649975 + r649977;
        double r649979 = sqrt(r649978);
        return r649979;
}
↓
double f(double re, double im) {
        double r649980 = re;
        double r649981 = r649980 * r649980;
        double r649982 = im;
        double r649983 = r649982 * r649982;
        double r649984 = r649981 + r649983;
        double r649985 = sqrt(r649984);
        return r649985;
}

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 2019119 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))