Average Error: 38.7 → 13.2
Time: 16.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r870767 = 0.5;
        double r870768 = 2.0;
        double r870769 = re;
        double r870770 = r870769 * r870769;
        double r870771 = im;
        double r870772 = r870771 * r870771;
        double r870773 = r870770 + r870772;
        double r870774 = sqrt(r870773);
        double r870775 = r870774 - r870769;
        double r870776 = r870768 * r870775;
        double r870777 = sqrt(r870776);
        double r870778 = r870767 * r870777;
        return r870778;
}
↓
double f(double re, double im) {
        double r870779 = re;
        double r870780 = im;
        double r870781 = hypot(r870779, r870780);
        double r870782 = r870781 - r870779;
        double r870783 = 2.0;
        double r870784 = r870782 * r870783;
        double r870785 = sqrt(r870784);
        double r870786 = 0.5;
        double r870787 = r870785 * r870786;
        return r870787;
}

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 38.7

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019179 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))