Average Error: 38.4 → 13.5
Time: 25.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(\sqrt{\mathsf{hypot}\left(re, im\right) + re} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(\sqrt{\mathsf{hypot}\left(re, im\right) + re} \cdot \sqrt{\mathsf{hypot}\left(re, im\right) + re}\right) \cdot 2}
double f(double re, double im) {
        double r154533 = 0.5;
        double r154534 = 2.0;
        double r154535 = re;
        double r154536 = r154535 * r154535;
        double r154537 = im;
        double r154538 = r154537 * r154537;
        double r154539 = r154536 + r154538;
        double r154540 = sqrt(r154539);
        double r154541 = r154540 + r154535;
        double r154542 = r154534 * r154541;
        double r154543 = sqrt(r154542);
        double r154544 = r154533 * r154543;
        return r154544;
}
↓
double f(double re, double im) {
        double r154545 = 0.5;
        double r154546 = re;
        double r154547 = im;
        double r154548 = hypot(r154546, r154547);
        double r154549 = r154548 + r154546;
        double r154550 = sqrt(r154549);
        double r154551 = r154550 * r154550;
        double r154552 = 2.0;
        double r154553 = r154551 * r154552;
        double r154554 = sqrt(r154553);
        double r154555 = r154545 * r154554;
        return r154555;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.4
Target33.5
Herbie13.5
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.4

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

  2. Simplified13.3

    \[\leadsto \color{blue}{0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}}\]
  3. Using strategy rm

  4. Applied add-sqr-sqrt14.4

    \[\leadsto 0.5 \cdot \sqrt{\left(re + \color{blue}{\sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \sqrt{\mathsf{hypot}\left(re, im\right)}}\right) \cdot 2}\]
  5. Using strategy rm

  6. Applied add-sqr-sqrt14.4

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

  7. Simplified14.1

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

  8. Simplified13.5

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

  9. Final simplification13.5

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

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(FPCore (re im)
  :name "math.sqrt on complex, real part"
  :precision binary64

  :herbie-target
  (if (< re 0.0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2 (+ (sqrt (+ (* re re) (* im im))) re)))))