Average Error: 39.5 → 19.7
Time: 11.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.971834220295259 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\\ \mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.971834220295259 \cdot 10^{153}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{-2 \cdot re}}\\

\mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\
\;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\\

\mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\

\mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)} + re\right)}\\

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

\end{array}
double f(double re, double im) {
        double r260782 = 0.5;
        double r260783 = 2.0;
        double r260784 = re;
        double r260785 = r260784 * r260784;
        double r260786 = im;
        double r260787 = r260786 * r260786;
        double r260788 = r260785 + r260787;
        double r260789 = sqrt(r260788);
        double r260790 = r260789 + r260784;
        double r260791 = r260783 * r260790;
        double r260792 = sqrt(r260791);
        double r260793 = r260782 * r260792;
        return r260793;
}
↓
double f(double re, double im) {
        double r260794 = re;
        double r260795 = -1.971834220295259e+153;
        bool r260796 = r260794 <= r260795;
        double r260797 = 0.5;
        double r260798 = im;
        double r260799 = 2.0;
        double r260800 = pow(r260798, r260799);
        double r260801 = 2.0;
        double r260802 = r260800 * r260801;
        double r260803 = sqrt(r260802);
        double r260804 = -2.0;
        double r260805 = r260804 * r260794;
        double r260806 = sqrt(r260805);
        double r260807 = r260803 / r260806;
        double r260808 = r260797 * r260807;
        double r260809 = -1.2504367945899628e-181;
        bool r260810 = r260794 <= r260809;
        double r260811 = sqrt(r260801);
        double r260812 = r260794 * r260794;
        double r260813 = r260798 * r260798;
        double r260814 = r260812 + r260813;
        double r260815 = sqrt(r260814);
        double r260816 = r260815 - r260794;
        double r260817 = sqrt(r260816);
        double r260818 = sqrt(r260817);
        double r260819 = r260811 / r260818;
        double r260820 = fabs(r260798);
        double r260821 = r260820 / r260818;
        double r260822 = r260819 * r260821;
        double r260823 = r260797 * r260822;
        double r260824 = 1.1547189018901299e-253;
        bool r260825 = r260794 <= r260824;
        double r260826 = r260794 + r260798;
        double r260827 = r260801 * r260826;
        double r260828 = sqrt(r260827);
        double r260829 = r260797 * r260828;
        double r260830 = 1.386148847085094e+97;
        bool r260831 = r260794 <= r260830;
        double r260832 = cbrt(r260814);
        double r260833 = r260832 * r260832;
        double r260834 = cbrt(r260833);
        double r260835 = cbrt(r260832);
        double r260836 = r260834 * r260835;
        double r260837 = r260833 * r260836;
        double r260838 = sqrt(r260837);
        double r260839 = r260838 + r260794;
        double r260840 = r260801 * r260839;
        double r260841 = sqrt(r260840);
        double r260842 = r260797 * r260841;
        double r260843 = r260794 + r260794;
        double r260844 = r260801 * r260843;
        double r260845 = sqrt(r260844);
        double r260846 = r260797 * r260845;
        double r260847 = r260831 ? r260842 : r260846;
        double r260848 = r260825 ? r260829 : r260847;
        double r260849 = r260810 ? r260823 : r260848;
        double r260850 = r260796 ? r260808 : r260849;
        return r260850;
}

Error

Bits error versus re

Bits error versus im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
Target34.4
Herbie19.7
\[\begin{array}{l} \mathbf{if}\;re \lt 0.0:\\ \;\;\;\;0.5 \cdot \left(\sqrt{2} \cdot \sqrt{\frac{im \cdot im}{\sqrt{re \cdot re + im \cdot im} - re}}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} + re\right)}\\ \end{array}\]

Derivation

  1. Split input into 5 regimes

  2. if re < -1.971834220295259e+153

    1. Initial program 64.0

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

    3. Applied flip-+64.0

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

    4. Applied associate-*r/64.0

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

    5. Applied sqrt-div64.0

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

    6. Simplified49.4

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]

    7. Taylor expanded around -inf 19.1

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}{\sqrt{\color{blue}{-2 \cdot re}}}\]

    if -1.971834220295259e+153 < re < -1.2504367945899628e-181

    1. Initial program 44.1

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

    3. Applied flip-+44.0

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

    4. Applied associate-*r/44.1

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

    5. Applied sqrt-div44.1

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

    6. Simplified30.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt30.2

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}{\sqrt{\color{blue}{\sqrt{\sqrt{re \cdot re + im \cdot im} - re} \cdot \sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\]

    9. Applied sqrt-prod30.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(0 + {im}^{2}\right)}}{\color{blue}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}} \cdot \sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}}\]

    10. Applied sqrt-prod30.3

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

    11. Applied times-frac30.3

      \[\leadsto 0.5 \cdot \color{blue}{\left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\sqrt{0 + {im}^{2}}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)}\]

    12. Simplified17.8

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

    if -1.2504367945899628e-181 < re < 1.1547189018901299e-253

    1. Initial program 32.7

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

    2. Taylor expanded around 0 33.9

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

    if 1.1547189018901299e-253 < re < 1.386148847085094e+97

    1. Initial program 20.3

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

    3. Applied add-cube-cbrt20.6

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

    5. Applied add-cube-cbrt20.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}}} + re\right)}\]

    6. Applied cbrt-prod20.6

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)}} + re\right)}\]

    if 1.386148847085094e+97 < re

    1. Initial program 51.2

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

    2. Taylor expanded around inf 10.7

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \left(\color{blue}{re} + re\right)}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification19.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.971834220295259 \cdot 10^{153}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{{im}^{2} \cdot 2}}{\sqrt{-2 \cdot re}}\\ \mathbf{elif}\;re \le -1.2504367945899628 \cdot 10^{-181}:\\ \;\;\;\;0.5 \cdot \left(\frac{\sqrt{2}}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}} \cdot \frac{\left|im\right|}{\sqrt{\sqrt{\sqrt{re \cdot re + im \cdot im} - re}}}\right)\\ \mathbf{elif}\;re \le 1.15471890189012987 \cdot 10^{-253}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + im\right)}\\ \mathbf{elif}\;re \le 1.3861488470850941 \cdot 10^{97}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \left(\sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt[3]{\sqrt[3]{re \cdot re + im \cdot im}}\right)} + re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(re + re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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)))))