Average Error: 37.5 → 13.2
Time: 20.0s
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 r503315 = 0.5;
        double r503316 = 2.0;
        double r503317 = re;
        double r503318 = r503317 * r503317;
        double r503319 = im;
        double r503320 = r503319 * r503319;
        double r503321 = r503318 + r503320;
        double r503322 = sqrt(r503321);
        double r503323 = r503322 - r503317;
        double r503324 = r503316 * r503323;
        double r503325 = sqrt(r503324);
        double r503326 = r503315 * r503325;
        return r503326;
}
↓
double f(double re, double im) {
        double r503327 = re;
        double r503328 = im;
        double r503329 = hypot(r503327, r503328);
        double r503330 = r503329 - r503327;
        double r503331 = 2.0;
        double r503332 = r503330 * r503331;
        double r503333 = sqrt(r503332);
        double r503334 = 0.5;
        double r503335 = r503333 * r503334;
        return r503335;
}

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 37.5

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

  2. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2019149 +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)))))