Average Error: 37.2 → 13.3
Time: 22.4s
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 r411458 = 0.5;
        double r411459 = 2.0;
        double r411460 = re;
        double r411461 = r411460 * r411460;
        double r411462 = im;
        double r411463 = r411462 * r411462;
        double r411464 = r411461 + r411463;
        double r411465 = sqrt(r411464);
        double r411466 = r411465 - r411460;
        double r411467 = r411459 * r411466;
        double r411468 = sqrt(r411467);
        double r411469 = r411458 * r411468;
        return r411469;
}
↓
double f(double re, double im) {
        double r411470 = re;
        double r411471 = im;
        double r411472 = hypot(r411470, r411471);
        double r411473 = r411472 - r411470;
        double r411474 = 2.0;
        double r411475 = r411473 * r411474;
        double r411476 = sqrt(r411475);
        double r411477 = 0.5;
        double r411478 = r411476 * r411477;
        return r411478;
}

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 37.2

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

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0}}\]

  3. Final simplification13.3

    \[\leadsto \sqrt{\left(\mathsf{hypot}\left(re, im\right) - re\right) \cdot 2.0} \cdot 0.5\]

Reproduce

herbie shell --seed 2019152 +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)))))