Average Error: 39.3 → 13.3
Time: 9.8s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}
double f(double re, double im) {
        double r23718 = 0.5;
        double r23719 = 2.0;
        double r23720 = re;
        double r23721 = r23720 * r23720;
        double r23722 = im;
        double r23723 = r23722 * r23722;
        double r23724 = r23721 + r23723;
        double r23725 = sqrt(r23724);
        double r23726 = r23725 - r23720;
        double r23727 = r23719 * r23726;
        double r23728 = sqrt(r23727);
        double r23729 = r23718 * r23728;
        return r23729;
}
↓
double f(double re, double im) {
        double r23730 = 0.5;
        double r23731 = re;
        double r23732 = im;
        double r23733 = hypot(r23731, r23732);
        double r23734 = r23733 - r23731;
        double r23735 = 2.0;
        double r23736 = r23734 * r23735;
        double r23737 = sqrt(r23736);
        double r23738 = r23730 * r23737;
        return r23738;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 39.3

    \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}}\]

  3. Final simplification13.3

    \[\leadsto 0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))