Average Error: 38.8 → 13.3
Time: 9.5s
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 r18908 = 0.5;
        double r18909 = 2.0;
        double r18910 = re;
        double r18911 = r18910 * r18910;
        double r18912 = im;
        double r18913 = r18912 * r18912;
        double r18914 = r18911 + r18913;
        double r18915 = sqrt(r18914);
        double r18916 = r18915 - r18910;
        double r18917 = r18909 * r18916;
        double r18918 = sqrt(r18917);
        double r18919 = r18908 * r18918;
        return r18919;
}
↓
double f(double re, double im) {
        double r18920 = 0.5;
        double r18921 = re;
        double r18922 = im;
        double r18923 = hypot(r18921, r18922);
        double r18924 = r18923 - r18921;
        double r18925 = 2.0;
        double r18926 = r18924 * r18925;
        double r18927 = sqrt(r18926);
        double r18928 = r18920 * r18927;
        return r18928;
}

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 38.8

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2020042 +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)))))