Average Error: 37.7 → 13.3
Time: 10.9s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r65808 = 0.5;
        double r65809 = 2.0;
        double r65810 = re;
        double r65811 = r65810 * r65810;
        double r65812 = im;
        double r65813 = r65812 * r65812;
        double r65814 = r65811 + r65813;
        double r65815 = sqrt(r65814);
        double r65816 = r65815 - r65810;
        double r65817 = r65809 * r65816;
        double r65818 = sqrt(r65817);
        double r65819 = r65808 * r65818;
        return r65819;
}
↓
double f(double re, double im) {
        double r65820 = re;
        double r65821 = im;
        double r65822 = hypot(r65820, r65821);
        double r65823 = r65822 - r65820;
        double r65824 = 2.0;
        double r65825 = r65823 * r65824;
        double r65826 = sqrt(r65825);
        double r65827 = 0.5;
        double r65828 = r65826 * r65827;
        return r65828;
}

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 37.7

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0}}\]

  3. Final simplification13.3

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019107 +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)))))