Average Error: 38.2 → 13.4
Time: 19.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2} \cdot 0.5
double f(double re, double im) {
        double r821601 = 0.5;
        double r821602 = 2.0;
        double r821603 = re;
        double r821604 = r821603 * r821603;
        double r821605 = im;
        double r821606 = r821605 * r821605;
        double r821607 = r821604 + r821606;
        double r821608 = sqrt(r821607);
        double r821609 = r821608 - r821603;
        double r821610 = r821602 * r821609;
        double r821611 = sqrt(r821610);
        double r821612 = r821601 * r821611;
        return r821612;
}
↓
double f(double re, double im) {
        double r821613 = re;
        double r821614 = im;
        double r821615 = hypot(r821613, r821614);
        double r821616 = r821615 - r821613;
        double r821617 = 2.0;
        double r821618 = r821616 * r821617;
        double r821619 = sqrt(r821618);
        double r821620 = 0.5;
        double r821621 = r821619 * r821620;
        return r821621;
}

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. Simplified13.4

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

  3. Final simplification13.4

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

Reproduce

herbie shell --seed 2019174 +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)))))