Average Error: 37.4 → 12.6
Time: 23.1s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5
double f(double re, double im) {
        double r571128 = 0.5;
        double r571129 = 2.0;
        double r571130 = re;
        double r571131 = r571130 * r571130;
        double r571132 = im;
        double r571133 = r571132 * r571132;
        double r571134 = r571131 + r571133;
        double r571135 = sqrt(r571134);
        double r571136 = r571135 - r571130;
        double r571137 = r571129 * r571136;
        double r571138 = sqrt(r571137);
        double r571139 = r571128 * r571138;
        return r571139;
}
↓
double f(double re, double im) {
        double r571140 = re;
        double r571141 = im;
        double r571142 = hypot(r571140, r571141);
        double r571143 = r571142 - r571140;
        double r571144 = 2.0;
        double r571145 = r571143 * r571144;
        double r571146 = sqrt(r571145);
        double r571147 = 0.5;
        double r571148 = r571146 * r571147;
        return r571148;
}

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 37.4

    \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]

  2. Simplified12.6

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0}}\]

  3. Final simplification12.6

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019134 +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)))))