Average Error: 38.5 → 13.0
Time: 22.6s
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 r30750 = 0.5;
        double r30751 = 2.0;
        double r30752 = re;
        double r30753 = r30752 * r30752;
        double r30754 = im;
        double r30755 = r30754 * r30754;
        double r30756 = r30753 + r30755;
        double r30757 = sqrt(r30756);
        double r30758 = r30757 - r30752;
        double r30759 = r30751 * r30758;
        double r30760 = sqrt(r30759);
        double r30761 = r30750 * r30760;
        return r30761;
}
↓
double f(double re, double im) {
        double r30762 = 0.5;
        double r30763 = re;
        double r30764 = im;
        double r30765 = hypot(r30763, r30764);
        double r30766 = r30765 - r30763;
        double r30767 = 2.0;
        double r30768 = r30766 * r30767;
        double r30769 = sqrt(r30768);
        double r30770 = r30762 * r30769;
        return r30770;
}

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 38.5

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

  2. Simplified13.0

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

  3. Final simplification13.0

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

Reproduce

herbie shell --seed 2019325 +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)))))