Average Error: 38.8 → 14.8
Time: 12.7s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(re + \sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \left(\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}} \cdot \sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right)\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(re + \sqrt{\mathsf{hypot}\left(re, im\right)} \cdot \left(\sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}} \cdot \sqrt{\sqrt{\mathsf{hypot}\left(re, im\right)}}\right)\right) \cdot 2}
double f(double re, double im) {
        double r284362 = 0.5;
        double r284363 = 2.0;
        double r284364 = re;
        double r284365 = r284364 * r284364;
        double r284366 = im;
        double r284367 = r284366 * r284366;
        double r284368 = r284365 + r284367;
        double r284369 = sqrt(r284368);
        double r284370 = r284369 + r284364;
        double r284371 = r284363 * r284370;
        double r284372 = sqrt(r284371);
        double r284373 = r284362 * r284372;
        return r284373;
}
↓
double f(double re, double im) {
        double r284374 = 0.5;
        double r284375 = re;
        double r284376 = im;
        double r284377 = hypot(r284375, r284376);
        double r284378 = sqrt(r284377);
        double r284379 = sqrt(r284378);
        double r284380 = r284379 * r284379;
        double r284381 = r284378 * r284380;
        double r284382 = r284375 + r284381;
        double r284383 = 2.0;
        double r284384 = r284382 * r284383;
        double r284385 = sqrt(r284384);
        double r284386 = r284374 * r284385;
        return r284386;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.8
Target33.8
Herbie14.8
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.8

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

  2. Simplified13.2

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

  4. Applied add-sqr-sqrt14.5

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

  6. Applied add-sqr-sqrt14.5

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

  7. Applied sqrt-prod14.8

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

  8. Final simplification14.8

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

Reproduce

herbie shell --seed 2020043 +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)))))