Average Error: 36.8 → 13.1
Time: 22.7s
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 r744223 = 0.5;
        double r744224 = 2.0;
        double r744225 = re;
        double r744226 = r744225 * r744225;
        double r744227 = im;
        double r744228 = r744227 * r744227;
        double r744229 = r744226 + r744228;
        double r744230 = sqrt(r744229);
        double r744231 = r744230 - r744225;
        double r744232 = r744224 * r744231;
        double r744233 = sqrt(r744232);
        double r744234 = r744223 * r744233;
        return r744234;
}
↓
double f(double re, double im) {
        double r744235 = re;
        double r744236 = im;
        double r744237 = hypot(r744235, r744236);
        double r744238 = r744237 - r744235;
        double r744239 = 2.0;
        double r744240 = r744238 * r744239;
        double r744241 = sqrt(r744240);
        double r744242 = 0.5;
        double r744243 = r744241 * r744242;
        return r744243;
}

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 36.8

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019158 +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)))))