Average Error: 38.1 → 12.7
Time: 16.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r36908 = 0.5;
        double r36909 = 2.0;
        double r36910 = re;
        double r36911 = r36910 * r36910;
        double r36912 = im;
        double r36913 = r36912 * r36912;
        double r36914 = r36911 + r36913;
        double r36915 = sqrt(r36914);
        double r36916 = r36915 - r36910;
        double r36917 = r36909 * r36916;
        double r36918 = sqrt(r36917);
        double r36919 = r36908 * r36918;
        return r36919;
}
↓
double f(double re, double im) {
        double r36920 = re;
        double r36921 = im;
        double r36922 = hypot(r36920, r36921);
        double r36923 = r36922 - r36920;
        double r36924 = 2.0;
        double r36925 = r36923 * r36924;
        double r36926 = sqrt(r36925);
        double r36927 = 0.5;
        double r36928 = r36926 * r36927;
        return r36928;
}

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 38.1

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

  2. Simplified12.7

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

  3. Final simplification12.7

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

Reproduce

herbie shell --seed 2019195 +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)))))