Average Error: 38.8 → 13.1
Time: 14.5s
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 r30611 = 0.5;
        double r30612 = 2.0;
        double r30613 = re;
        double r30614 = r30613 * r30613;
        double r30615 = im;
        double r30616 = r30615 * r30615;
        double r30617 = r30614 + r30616;
        double r30618 = sqrt(r30617);
        double r30619 = r30618 - r30613;
        double r30620 = r30612 * r30619;
        double r30621 = sqrt(r30620);
        double r30622 = r30611 * r30621;
        return r30622;
}
↓
double f(double re, double im) {
        double r30623 = 0.5;
        double r30624 = re;
        double r30625 = im;
        double r30626 = hypot(r30624, r30625);
        double r30627 = r30626 - r30624;
        double r30628 = 2.0;
        double r30629 = r30627 * r30628;
        double r30630 = sqrt(r30629);
        double r30631 = r30623 * r30630;
        return r30631;
}

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 38.8

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

  2. Simplified13.1

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

  3. Final simplification13.1

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

Reproduce

herbie shell --seed 2019208 +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)))))