Average Error: 38.4 → 13.2
Time: 12.2s
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 r609043 = 0.5;
        double r609044 = 2.0;
        double r609045 = re;
        double r609046 = r609045 * r609045;
        double r609047 = im;
        double r609048 = r609047 * r609047;
        double r609049 = r609046 + r609048;
        double r609050 = sqrt(r609049);
        double r609051 = r609050 - r609045;
        double r609052 = r609044 * r609051;
        double r609053 = sqrt(r609052);
        double r609054 = r609043 * r609053;
        return r609054;
}
↓
double f(double re, double im) {
        double r609055 = re;
        double r609056 = im;
        double r609057 = hypot(r609055, r609056);
        double r609058 = r609057 - r609055;
        double r609059 = 2.0;
        double r609060 = r609058 * r609059;
        double r609061 = sqrt(r609060);
        double r609062 = 0.5;
        double r609063 = r609061 * r609062;
        return r609063;
}

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 38.4

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019172 +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)))))