Average Error: 38.8 → 24.5
Time: 4.5s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 1.0984713903285393 \cdot 10^{132}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} \cdot \frac{1}{2}}{re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\

\mathbf{elif}\;re \le 1.0984713903285393 \cdot 10^{132}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} \cdot \frac{1}{2}}{re}}\\

\end{array}
double f(double re, double im) {
        double r17099 = 0.5;
        double r17100 = 2.0;
        double r17101 = re;
        double r17102 = r17101 * r17101;
        double r17103 = im;
        double r17104 = r17103 * r17103;
        double r17105 = r17102 + r17104;
        double r17106 = sqrt(r17105);
        double r17107 = r17106 - r17101;
        double r17108 = r17100 * r17107;
        double r17109 = sqrt(r17108);
        double r17110 = r17099 * r17109;
        return r17110;
}
↓
double f(double re, double im) {
        double r17111 = re;
        double r17112 = -2.2710920961423764e+59;
        bool r17113 = r17111 <= r17112;
        double r17114 = 0.5;
        double r17115 = 2.0;
        double r17116 = -1.0;
        double r17117 = r17116 * r17111;
        double r17118 = r17117 - r17111;
        double r17119 = r17115 * r17118;
        double r17120 = sqrt(r17119);
        double r17121 = r17114 * r17120;
        double r17122 = -1.612699741875562e-300;
        bool r17123 = r17111 <= r17122;
        double r17124 = r17111 * r17111;
        double r17125 = im;
        double r17126 = r17125 * r17125;
        double r17127 = r17124 + r17126;
        double r17128 = sqrt(r17127);
        double r17129 = r17128 - r17111;
        double r17130 = r17115 * r17129;
        double r17131 = sqrt(r17130);
        double r17132 = r17114 * r17131;
        double r17133 = 1.0984713903285393e+132;
        bool r17134 = r17111 <= r17133;
        double r17135 = 2.0;
        double r17136 = pow(r17125, r17135);
        double r17137 = r17128 + r17111;
        double r17138 = r17136 / r17137;
        double r17139 = r17115 * r17138;
        double r17140 = sqrt(r17139);
        double r17141 = r17114 * r17140;
        double r17142 = 0.5;
        double r17143 = r17136 * r17142;
        double r17144 = r17143 / r17111;
        double r17145 = r17115 * r17144;
        double r17146 = sqrt(r17145);
        double r17147 = r17114 * r17146;
        double r17148 = r17134 ? r17141 : r17147;
        double r17149 = r17123 ? r17132 : r17148;
        double r17150 = r17113 ? r17121 : r17149;
        return r17150;
}

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 4 regimes

  2. if re < -2.2710920961423764e+59

    1. Initial program 45.1

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

    2. Taylor expanded around -inf 12.8

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

    if -2.2710920961423764e+59 < re < -1.612699741875562e-300

    1. Initial program 21.9

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

    if -1.612699741875562e-300 < re < 1.0984713903285393e+132

    1. Initial program 39.8

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

    3. Applied flip--39.6

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

    4. Simplified31.0

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

    if 1.0984713903285393e+132 < re

    1. Initial program 62.5

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

    3. Applied add-exp-log62.5

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

    5. Applied add-cube-cbrt62.5

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

    6. Applied exp-prod62.5

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

    7. Taylor expanded around inf 45.5

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

    8. Simplified30.5

      \[\leadsto 0.5 \cdot \sqrt{2 \cdot \color{blue}{\frac{{im}^{2} \cdot \frac{1}{2}}{re}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification24.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 1.0984713903285393 \cdot 10^{132}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} \cdot \frac{1}{2}}{re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 
(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)))))