Average Error: 38.2 → 13.7
Time: 16.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2}
double f(double re, double im) {
        double r30585 = 0.5;
        double r30586 = 2.0;
        double r30587 = re;
        double r30588 = r30587 * r30587;
        double r30589 = im;
        double r30590 = r30589 * r30589;
        double r30591 = r30588 + r30590;
        double r30592 = sqrt(r30591);
        double r30593 = r30592 - r30587;
        double r30594 = r30586 * r30593;
        double r30595 = sqrt(r30594);
        double r30596 = r30585 * r30595;
        return r30596;
}
↓
double f(double re, double im) {
        double r30597 = 0.5;
        double r30598 = re;
        double r30599 = im;
        double r30600 = hypot(r30598, r30599);
        double r30601 = r30600 - r30598;
        double r30602 = 2.0;
        double r30603 = r30601 * r30602;
        double r30604 = sqrt(r30603);
        double r30605 = r30597 * r30604;
        return r30605;
}

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 38.2

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

  2. Simplified13.7

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

  3. Final simplification13.7

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

Reproduce

herbie shell --seed 2019209 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  :precision binary64
  (* 0.5 (sqrt (* 2 (- (sqrt (+ (* re re) (* im im))) re)))))