Average Error: 37.9 → 13.6
Time: 19.0s
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 r527858 = 0.5;
        double r527859 = 2.0;
        double r527860 = re;
        double r527861 = r527860 * r527860;
        double r527862 = im;
        double r527863 = r527862 * r527862;
        double r527864 = r527861 + r527863;
        double r527865 = sqrt(r527864);
        double r527866 = r527865 - r527860;
        double r527867 = r527859 * r527866;
        double r527868 = sqrt(r527867);
        double r527869 = r527858 * r527868;
        return r527869;
}
↓
double f(double re, double im) {
        double r527870 = re;
        double r527871 = im;
        double r527872 = hypot(r527870, r527871);
        double r527873 = r527872 - r527870;
        double r527874 = 2.0;
        double r527875 = r527873 * r527874;
        double r527876 = sqrt(r527875);
        double r527877 = 0.5;
        double r527878 = r527876 * r527877;
        return r527878;
}

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 37.9

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

  2. Simplified13.6

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

  3. Final simplification13.6

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

Reproduce

herbie shell --seed 2019129 +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)))))