Average Error: 38.2 → 13.4
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 r27929 = 0.5;
        double r27930 = 2.0;
        double r27931 = re;
        double r27932 = r27931 * r27931;
        double r27933 = im;
        double r27934 = r27933 * r27933;
        double r27935 = r27932 + r27934;
        double r27936 = sqrt(r27935);
        double r27937 = r27936 - r27931;
        double r27938 = r27930 * r27937;
        double r27939 = sqrt(r27938);
        double r27940 = r27929 * r27939;
        return r27940;
}
↓
double f(double re, double im) {
        double r27941 = re;
        double r27942 = im;
        double r27943 = hypot(r27941, r27942);
        double r27944 = r27943 - r27941;
        double r27945 = 2.0;
        double r27946 = r27944 * r27945;
        double r27947 = sqrt(r27946);
        double r27948 = 0.5;
        double r27949 = r27947 * r27948;
        return r27949;
}

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

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

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)))))