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 r675309 = re;
        double r675310 = r675309 * r675309;
        double r675311 = im;
        double r675312 = r675311 * r675311;
        double r675313 = r675310 + r675312;
        double r675314 = sqrt(r675313);
        return r675314;
}
↓
double f(double re, double im) {
        double r675315 = re;
        double r675316 = r675315 * r675315;
        double r675317 = im;
        double r675318 = r675317 * r675317;
        double r675319 = r675316 + r675318;
        double r675320 = sqrt(r675319);
        return r675320;
}

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 2019124 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))