Average Error: 39.0 → 26.5
Time: 5.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 4.674718513276777827095282791297829776848 \cdot 10^{-152}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 1.777219208852638055427851320427769465928 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 2.388925724674624549087913716226938535243 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 4.674718513276777827095282791297829776848 \cdot 10^{-152}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\

\mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 1.777219208852638055427851320427769465928 \cdot 10^{-80}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 2.388925724674624549087913716226938535243 \cdot 10^{76}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\

\end{array}
double f(double re, double im) {
        double r23810 = 0.5;
        double r23811 = 2.0;
        double r23812 = re;
        double r23813 = r23812 * r23812;
        double r23814 = im;
        double r23815 = r23814 * r23814;
        double r23816 = r23813 + r23815;
        double r23817 = sqrt(r23816);
        double r23818 = r23817 - r23812;
        double r23819 = r23811 * r23818;
        double r23820 = sqrt(r23819);
        double r23821 = r23810 * r23820;
        return r23821;
}
↓
double f(double re, double im) {
        double r23822 = 2.0;
        double r23823 = re;
        double r23824 = r23823 * r23823;
        double r23825 = im;
        double r23826 = r23825 * r23825;
        double r23827 = r23824 + r23826;
        double r23828 = sqrt(r23827);
        double r23829 = r23828 - r23823;
        double r23830 = r23822 * r23829;
        double r23831 = sqrt(r23830);
        double r23832 = 4.674718513276778e-152;
        bool r23833 = r23831 <= r23832;
        double r23834 = 0.5;
        double r23835 = 0.0;
        double r23836 = r23826 + r23835;
        double r23837 = r23823 + r23828;
        double r23838 = r23836 / r23837;
        double r23839 = r23822 * r23838;
        double r23840 = sqrt(r23839);
        double r23841 = r23834 * r23840;
        double r23842 = 1.777219208852638e-80;
        bool r23843 = r23831 <= r23842;
        double r23844 = r23825 - r23823;
        double r23845 = r23822 * r23844;
        double r23846 = sqrt(r23845);
        double r23847 = r23834 * r23846;
        double r23848 = 2.3889257246746245e+76;
        bool r23849 = r23831 <= r23848;
        double r23850 = r23834 * r23831;
        double r23851 = r23849 ? r23850 : r23847;
        double r23852 = r23843 ? r23847 : r23851;
        double r23853 = r23833 ? r23841 : r23852;
        return r23853;
}

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. Split input into 3 regimes

  2. if (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 4.674718513276778e-152

    1. Initial program 57.3

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

    3. Applied add-sqr-sqrt57.3

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

    4. Applied sqrt-prod59.8

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

    6. Applied flip--59.8

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

    7. Simplified29.3

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{\color{blue}{im \cdot im + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}} + re}}\]

    8. Simplified29.2

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{\color{blue}{re + \sqrt{re \cdot re + im \cdot im}}}}\]

    if 4.674718513276778e-152 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 1.777219208852638e-80 or 2.3889257246746245e+76 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))

    1. Initial program 62.8

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

    2. Taylor expanded around 0 44.9

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

    if 1.777219208852638e-80 < (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re))) < 2.3889257246746245e+76

    1. Initial program 0.4

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 4.674718513276777827095282791297829776848 \cdot 10^{-152}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{im \cdot im + 0}{re + \sqrt{re \cdot re + im \cdot im}}}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 1.777219208852638055427851320427769465928 \cdot 10^{-80}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \mathbf{elif}\;\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)} \le 2.388925724674624549087913716226938535243 \cdot 10^{76}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(im - re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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)))))