Average Error: 39.1 → 11.4
Time: 4.2s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -5229117268898383872:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{\frac{1}{2}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -5229117268898383872:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot {\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{\frac{1}{2}}\\

\end{array}
double f(double re, double im) {
        double r166731 = 0.5;
        double r166732 = 2.0;
        double r166733 = re;
        double r166734 = r166733 * r166733;
        double r166735 = im;
        double r166736 = r166735 * r166735;
        double r166737 = r166734 + r166736;
        double r166738 = sqrt(r166737);
        double r166739 = r166738 + r166733;
        double r166740 = r166732 * r166739;
        double r166741 = sqrt(r166740);
        double r166742 = r166731 * r166741;
        return r166742;
}
↓
double f(double re, double im) {
        double r166743 = re;
        double r166744 = -5.229117268898384e+18;
        bool r166745 = r166743 <= r166744;
        double r166746 = 0.5;
        double r166747 = 2.0;
        double r166748 = im;
        double r166749 = r166748 * r166748;
        double r166750 = hypot(r166743, r166748);
        double r166751 = r166750 - r166743;
        double r166752 = r166749 / r166751;
        double r166753 = r166747 * r166752;
        double r166754 = sqrt(r166753);
        double r166755 = r166746 * r166754;
        double r166756 = r166743 + r166750;
        double r166757 = r166747 * r166756;
        double r166758 = 0.5;
        double r166759 = pow(r166757, r166758);
        double r166760 = r166746 * r166759;
        double r166761 = r166745 ? r166755 : r166760;
        return r166761;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target34.1
Herbie11.4
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if re < -5.229117268898384e+18

    1. Initial program 58.6

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

    3. Applied flip-+58.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Simplified41.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im}}{\sqrt{re \cdot re + im \cdot im} - re}}\]

    5. Simplified31.5

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

    if -5.229117268898384e+18 < re

    1. Initial program 33.0

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

    3. Applied *-un-lft-identity33.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + \color{blue}{1 \cdot re}\right)}\]

    4. Applied *-un-lft-identity33.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{1 \cdot \sqrt{re \cdot re + im \cdot im}} + 1 \cdot re\right)}\]

    5. Applied distribute-lft-out33.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\left(1 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)\right)}}\]

    6. Simplified5.0

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(1 \cdot \color{blue}{\left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}\]
    7. Using strategy rm

    8. Applied sqrt-prod5.4

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{1 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)}\right)}\]

    9. Simplified5.4

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

    11. Applied add-sqr-sqrt5.4

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

    12. Applied sqrt-prod5.5

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

    13. Applied associate-*l*5.4

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

    15. Applied pow15.4

      \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \sqrt{\color{blue}{{\left(re + \mathsf{hypot}\left(re, im\right)\right)}^{1}}}\right)\right)\]

    16. Applied sqrt-pow15.4

      \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot \color{blue}{{\left(re + \mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{2}\right)}}\right)\right)\]

    17. Applied pow15.4

      \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\sqrt{\color{blue}{{\left(\sqrt{2}\right)}^{1}}} \cdot {\left(re + \mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right)\]

    18. Applied sqrt-pow15.4

      \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \left(\color{blue}{{\left(\sqrt{2}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(re + \mathsf{hypot}\left(re, im\right)\right)}^{\left(\frac{1}{2}\right)}\right)\right)\]

    19. Applied pow-prod-down5.3

      \[\leadsto 0.5 \cdot \left(\sqrt{\sqrt{2}} \cdot \color{blue}{{\left(\sqrt{2} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{\left(\frac{1}{2}\right)}}\right)\]

    20. Applied pow15.3

      \[\leadsto 0.5 \cdot \left(\sqrt{\color{blue}{{\left(\sqrt{2}\right)}^{1}}} \cdot {\left(\sqrt{2} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{\left(\frac{1}{2}\right)}\right)\]

    21. Applied sqrt-pow15.3

      \[\leadsto 0.5 \cdot \left(\color{blue}{{\left(\sqrt{2}\right)}^{\left(\frac{1}{2}\right)}} \cdot {\left(\sqrt{2} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{\left(\frac{1}{2}\right)}\right)\]

    22. Applied pow-prod-down5.5

      \[\leadsto 0.5 \cdot \color{blue}{{\left(\sqrt{2} \cdot \left(\sqrt{2} \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)\right)}^{\left(\frac{1}{2}\right)}}\]

    23. Simplified5.0

      \[\leadsto 0.5 \cdot {\color{blue}{\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}}^{\left(\frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5229117268898383872:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im}{\mathsf{hypot}\left(re, im\right) - re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot {\left(2 \cdot \left(re + \mathsf{hypot}\left(re, im\right)\right)\right)}^{\frac{1}{2}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019344 +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)))))