Average Error: 37.9 → 25.6
Time: 24.5s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \le 0.0:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \le 7.424954488058662 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + im\right)} \cdot 0.5\\ \end{array}\]
double f(double re, double im) {
        double r40199792 = 0.5;
        double r40199793 = 2.0;
        double r40199794 = re;
        double r40199795 = r40199794 * r40199794;
        double r40199796 = im;
        double r40199797 = r40199796 * r40199796;
        double r40199798 = r40199795 + r40199797;
        double r40199799 = sqrt(r40199798);
        double r40199800 = r40199799 + r40199794;
        double r40199801 = r40199793 * r40199800;
        double r40199802 = sqrt(r40199801);
        double r40199803 = r40199792 * r40199802;
        return r40199803;
}
↓
double f(double re, double im) {
        double r40199804 = re;
        double r40199805 = r40199804 * r40199804;
        double r40199806 = im;
        double r40199807 = r40199806 * r40199806;
        double r40199808 = r40199805 + r40199807;
        double r40199809 = sqrt(r40199808);
        double r40199810 = r40199804 + r40199809;
        double r40199811 = 2.0;
        double r40199812 = r40199810 * r40199811;
        double r40199813 = sqrt(r40199812);
        double r40199814 = 0.0;
        bool r40199815 = r40199813 <= r40199814;
        double r40199816 = 0.5;
        double r40199817 = r40199807 * r40199811;
        double r40199818 = sqrt(r40199817);
        double r40199819 = r40199809 - r40199804;
        double r40199820 = sqrt(r40199819);
        double r40199821 = r40199818 / r40199820;
        double r40199822 = r40199816 * r40199821;
        double r40199823 = 7.424954488058662e+76;
        bool r40199824 = r40199813 <= r40199823;
        double r40199825 = r40199813 * r40199816;
        double r40199826 = r40199804 + r40199806;
        double r40199827 = r40199811 * r40199826;
        double r40199828 = sqrt(r40199827);
        double r40199829 = r40199828 * r40199816;
        double r40199830 = r40199824 ? r40199825 : r40199829;
        double r40199831 = r40199815 ? r40199822 : r40199830;
        return r40199831;
}


0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \le 0.0:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\

\mathbf{elif}\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \le 7.424954488058662 \cdot 10^{+76}:\\
\;\;\;\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \cdot 0.5\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(re + im\right)} \cdot 0.5\\

\end{array}

Error

Bits error versus re

Bits error versus im

Target

Original37.9
Target33.0
Herbie25.6
\[\begin{array}{l} \mathbf{if}\;re \lt 0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 3 regimes

  2. if (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 0.0

    1. Initial program 57.4

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied flip-+57.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\frac{\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    4. Applied associate-*r/57.4

      \[\leadsto 0.5 \cdot \sqrt{\color{blue}{\frac{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    5. Applied sqrt-div57.4

      \[\leadsto 0.5 \cdot \color{blue}{\frac{\sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} \cdot \sqrt{re \cdot re + im \cdot im} - re \cdot re\right)}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\]

    6. Simplified27.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    if 0.0 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re))) < 7.424954488058662e+76

    1. Initial program 4.6

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]

    if 7.424954488058662e+76 < (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))

    1. Initial program 61.4

      \[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
    2. Using strategy rm

    3. Applied add-sqr-sqrt61.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im}}} + re\right)}\]

    4. Taylor expanded around 0 43.3

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{im} + re\right)}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification25.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \le 0.0:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\\ \mathbf{elif}\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \le 7.424954488058662 \cdot 10^{+76}:\\ \;\;\;\;\sqrt{\left(re + \sqrt{re \cdot re + im \cdot im}\right) \cdot 2.0} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(re + im\right)} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 
(FPCore (re im)
  :name "math.sqrt on complex, real part"

  :herbie-target
  (if (< re 0) (* 0.5 (* (sqrt 2) (sqrt (/ (* im im) (- (sqrt (+ (* re re) (* im im))) re))))) (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))

  (* 0.5 (sqrt (* 2.0 (+ (sqrt (+ (* re re) (* im im))) re)))))