Average Error: 38.1 → 13.1
Time: 18.6s
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 r27459 = 0.5;
        double r27460 = 2.0;
        double r27461 = re;
        double r27462 = r27461 * r27461;
        double r27463 = im;
        double r27464 = r27463 * r27463;
        double r27465 = r27462 + r27464;
        double r27466 = sqrt(r27465);
        double r27467 = r27466 - r27461;
        double r27468 = r27460 * r27467;
        double r27469 = sqrt(r27468);
        double r27470 = r27459 * r27469;
        return r27470;
}
↓
double f(double re, double im) {
        double r27471 = 0.5;
        double r27472 = re;
        double r27473 = im;
        double r27474 = hypot(r27472, r27473);
        double r27475 = r27474 - r27472;
        double r27476 = 2.0;
        double r27477 = r27475 * r27476;
        double r27478 = sqrt(r27477);
        double r27479 = r27471 * r27478;
        return r27479;
}

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 38.1

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

  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 2019303 +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)))))