Average Error: 38.2 → 13.2
Time: 9.4s
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 r21502 = 0.5;
        double r21503 = 2.0;
        double r21504 = re;
        double r21505 = r21504 * r21504;
        double r21506 = im;
        double r21507 = r21506 * r21506;
        double r21508 = r21505 + r21507;
        double r21509 = sqrt(r21508);
        double r21510 = r21509 - r21504;
        double r21511 = r21503 * r21510;
        double r21512 = sqrt(r21511);
        double r21513 = r21502 * r21512;
        return r21513;
}
↓
double f(double re, double im) {
        double r21514 = 0.5;
        double r21515 = re;
        double r21516 = im;
        double r21517 = hypot(r21515, r21516);
        double r21518 = r21517 - r21515;
        double r21519 = 2.0;
        double r21520 = r21518 * r21519;
        double r21521 = sqrt(r21520);
        double r21522 = r21514 * r21521;
        return r21522;
}

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. Simplified13.2

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

  3. Final simplification13.2

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

Reproduce

herbie shell --seed 2020045 +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)))))