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 r1330553 = re;
        double r1330554 = r1330553 * r1330553;
        double r1330555 = im;
        double r1330556 = r1330555 * r1330555;
        double r1330557 = r1330554 + r1330556;
        double r1330558 = sqrt(r1330557);
        return r1330558;
}
↓
double f(double re, double im) {
        double r1330559 = re;
        double r1330560 = r1330559 * r1330559;
        double r1330561 = im;
        double r1330562 = r1330561 * r1330561;
        double r1330563 = r1330560 + r1330562;
        double r1330564 = sqrt(r1330563);
        return r1330564;
}

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 2019128 
(FPCore (re im)
  :name "math.abs on complex"
  (sqrt.p16 (+.p16 (*.p16 re re) (*.p16 im im))))