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 r649943 = re;
        double r649944 = r649943 * r649943;
        double r649945 = im;
        double r649946 = r649945 * r649945;
        double r649947 = r649944 + r649946;
        double r649948 = sqrt(r649947);
        return r649948;
}
↓
double f(double re, double im) {
        double r649949 = re;
        double r649950 = r649949 * r649949;
        double r649951 = im;
        double r649952 = r649951 * r649951;
        double r649953 = r649950 + r649952;
        double r649954 = sqrt(r649953);
        return r649954;
}

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 2019121 +o rules:numerics
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))