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 r1801129 = re;
        double r1801130 = r1801129 * r1801129;
        double r1801131 = im;
        double r1801132 = r1801131 * r1801131;
        double r1801133 = r1801130 + r1801132;
        double r1801134 = sqrt(r1801133);
        return r1801134;
}
↓
double f(double re, double im) {
        double r1801135 = re;
        double r1801136 = r1801135 * r1801135;
        double r1801137 = im;
        double r1801138 = r1801137 * r1801137;
        double r1801139 = r1801136 + r1801138;
        double r1801140 = sqrt(r1801139);
        return r1801140;
}

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))))