Average Error: 37.7 → 13.1
Time: 18.9s
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 r685531 = 0.5;
        double r685532 = 2.0;
        double r685533 = re;
        double r685534 = r685533 * r685533;
        double r685535 = im;
        double r685536 = r685535 * r685535;
        double r685537 = r685534 + r685536;
        double r685538 = sqrt(r685537);
        double r685539 = r685538 - r685533;
        double r685540 = r685532 * r685539;
        double r685541 = sqrt(r685540);
        double r685542 = r685531 * r685541;
        return r685542;
}
↓
double f(double re, double im) {
        double r685543 = re;
        double r685544 = im;
        double r685545 = hypot(r685543, r685544);
        double r685546 = r685545 - r685543;
        double r685547 = 2.0;
        double r685548 = r685546 * r685547;
        double r685549 = sqrt(r685548);
        double r685550 = 0.5;
        double r685551 = r685549 * r685550;
        return r685551;
}

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 37.7

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

  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 2019135 +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)))))