Average Error: 39.2 → 14.0
Time: 18.0s
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 r33852 = 0.5;
        double r33853 = 2.0;
        double r33854 = re;
        double r33855 = r33854 * r33854;
        double r33856 = im;
        double r33857 = r33856 * r33856;
        double r33858 = r33855 + r33857;
        double r33859 = sqrt(r33858);
        double r33860 = r33859 - r33854;
        double r33861 = r33853 * r33860;
        double r33862 = sqrt(r33861);
        double r33863 = r33852 * r33862;
        return r33863;
}
↓
double f(double re, double im) {
        double r33864 = 0.5;
        double r33865 = re;
        double r33866 = im;
        double r33867 = hypot(r33865, r33866);
        double r33868 = r33867 - r33865;
        double r33869 = 2.0;
        double r33870 = r33868 * r33869;
        double r33871 = sqrt(r33870);
        double r33872 = r33864 * r33871;
        return r33872;
}

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 39.2

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

  2. Simplified14.0

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

  3. Final simplification14.0

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

Reproduce

herbie shell --seed 2019235 +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)))))