Average Error: 0.5 → 0.5
Time: 3.5s
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 r1536100 = re;
        double r1536101 = r1536100 * r1536100;
        double r1536102 = im;
        double r1536103 = r1536102 * r1536102;
        double r1536104 = r1536101 + r1536103;
        double r1536105 = sqrt(r1536104);
        return r1536105;
}
↓
double f(double re, double im) {
        double r1536106 = re;
        double r1536107 = r1536106 * r1536106;
        double r1536108 = im;
        double r1536109 = r1536108 * r1536108;
        double r1536110 = r1536107 + r1536109;
        double r1536111 = sqrt(r1536110);
        return r1536111;
}

Error

Bits error versus re

Bits error versus im

Derivation

  1. Initial program 0.5

    \[\sqrt{\left(\frac{\left(re \cdot re\right)}{\left(im \cdot im\right)}\right)}\]

  2. Final simplification0.5

    \[\leadsto \sqrt{re \cdot re + im \cdot im}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))