Average Error: 38.6 → 13.4
Time: 24.0s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
0.5 \cdot \sqrt{\left(re + \mathsf{hypot}\left(re, im\right)\right) \cdot 2}
double f(double re, double im) {
        double r114869 = 0.5;
        double r114870 = 2.0;
        double r114871 = re;
        double r114872 = r114871 * r114871;
        double r114873 = im;
        double r114874 = r114873 * r114873;
        double r114875 = r114872 + r114874;
        double r114876 = sqrt(r114875);
        double r114877 = r114876 + r114871;
        double r114878 = r114870 * r114877;
        double r114879 = sqrt(r114878);
        double r114880 = r114869 * r114879;
        return r114880;
}
↓
double f(double re, double im) {
        double r114881 = 0.5;
        double r114882 = re;
        double r114883 = im;
        double r114884 = hypot(r114882, r114883);
        double r114885 = r114882 + r114884;
        double r114886 = 2.0;
        double r114887 = r114885 * r114886;
        double r114888 = sqrt(r114887);
        double r114889 = r114881 * r114888;
        return r114889;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original38.6
Target33.7
Herbie13.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Initial program 38.6

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

  2. Simplified13.4

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

  4. Applied sqrt-prod13.7

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

  6. Applied add-sqr-sqrt13.7

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

  7. Applied sqrt-prod13.8

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

  8. Applied associate-*r*13.8

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

  10. Applied sqrt-unprod13.6

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

  11. Applied sqrt-unprod13.8

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

  12. Simplified13.4

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

  13. Final simplification13.4

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

Reproduce

herbie shell --seed 2019323 +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)))))