Average Error: 38.5 → 13.3
Time: 21.9s
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 r2294455 = 0.5;
        double r2294456 = 2.0;
        double r2294457 = re;
        double r2294458 = r2294457 * r2294457;
        double r2294459 = im;
        double r2294460 = r2294459 * r2294459;
        double r2294461 = r2294458 + r2294460;
        double r2294462 = sqrt(r2294461);
        double r2294463 = r2294462 - r2294457;
        double r2294464 = r2294456 * r2294463;
        double r2294465 = sqrt(r2294464);
        double r2294466 = r2294455 * r2294465;
        return r2294466;
}
↓
double f(double re, double im) {
        double r2294467 = 0.5;
        double r2294468 = re;
        double r2294469 = im;
        double r2294470 = hypot(r2294468, r2294469);
        double r2294471 = r2294470 - r2294468;
        double r2294472 = 2.0;
        double r2294473 = r2294471 * r2294472;
        double r2294474 = sqrt(r2294473);
        double r2294475 = r2294467 * r2294474;
        return r2294475;
}

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 38.5

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

  2. Simplified13.3

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

  3. Final simplification13.3

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

Reproduce

herbie shell --seed 2019173 +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)))))