Average Error: 38.8 → 13.4
Time: 19.2s
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 r31926 = 0.5;
        double r31927 = 2.0;
        double r31928 = re;
        double r31929 = r31928 * r31928;
        double r31930 = im;
        double r31931 = r31930 * r31930;
        double r31932 = r31929 + r31931;
        double r31933 = sqrt(r31932);
        double r31934 = r31933 - r31928;
        double r31935 = r31927 * r31934;
        double r31936 = sqrt(r31935);
        double r31937 = r31926 * r31936;
        return r31937;
}
↓
double f(double re, double im) {
        double r31938 = 0.5;
        double r31939 = re;
        double r31940 = im;
        double r31941 = hypot(r31939, r31940);
        double r31942 = r31941 - r31939;
        double r31943 = 2.0;
        double r31944 = r31942 * r31943;
        double r31945 = sqrt(r31944);
        double r31946 = r31938 * r31945;
        return r31946;
}

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. Simplified13.4

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

  3. Final simplification13.4

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

Reproduce

herbie shell --seed 2019212 +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)))))