Average Error: 38.2 → 13.4
Time: 20.6s
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 r973907 = 0.5;
        double r973908 = 2.0;
        double r973909 = re;
        double r973910 = r973909 * r973909;
        double r973911 = im;
        double r973912 = r973911 * r973911;
        double r973913 = r973910 + r973912;
        double r973914 = sqrt(r973913);
        double r973915 = r973914 - r973909;
        double r973916 = r973908 * r973915;
        double r973917 = sqrt(r973916);
        double r973918 = r973907 * r973917;
        return r973918;
}
↓
double f(double re, double im) {
        double r973919 = 0.5;
        double r973920 = re;
        double r973921 = im;
        double r973922 = hypot(r973920, r973921);
        double r973923 = r973922 - r973920;
        double r973924 = 2.0;
        double r973925 = r973923 * r973924;
        double r973926 = sqrt(r973925);
        double r973927 = r973919 * r973926;
        return r973927;
}

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 38.2

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

  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 2019174 +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)))))