Average Error: 37.6 → 25.9
Time: 19.7s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.7109531485520302 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -5.7290891404837934 \cdot 10^{-198}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right| - re\right)}\\ \mathbf{elif}\;re \le -4.685030330992167 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.7109531485520302 \cdot 10^{+90}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -5.7290891404837934 \cdot 10^{-198}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right| - re\right)}\\

\mathbf{elif}\;re \le -4.685030330992167 \cdot 10^{-268}:\\
\;\;\;\;\sqrt{2.0 \cdot \left(im - re\right)} \cdot 0.5\\

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

\end{array}
double f(double re, double im) {
        double r950108 = 0.5;
        double r950109 = 2.0;
        double r950110 = re;
        double r950111 = r950110 * r950110;
        double r950112 = im;
        double r950113 = r950112 * r950112;
        double r950114 = r950111 + r950113;
        double r950115 = sqrt(r950114);
        double r950116 = r950115 - r950110;
        double r950117 = r950109 * r950116;
        double r950118 = sqrt(r950117);
        double r950119 = r950108 * r950118;
        return r950119;
}
↓
double f(double re, double im) {
        double r950120 = re;
        double r950121 = -1.7109531485520302e+90;
        bool r950122 = r950120 <= r950121;
        double r950123 = -2.0;
        double r950124 = r950123 * r950120;
        double r950125 = 2.0;
        double r950126 = r950124 * r950125;
        double r950127 = sqrt(r950126);
        double r950128 = 0.5;
        double r950129 = r950127 * r950128;
        double r950130 = -5.7290891404837934e-198;
        bool r950131 = r950120 <= r950130;
        double r950132 = im;
        double r950133 = r950132 * r950132;
        double r950134 = r950120 * r950120;
        double r950135 = r950133 + r950134;
        double r950136 = cbrt(r950135);
        double r950137 = sqrt(r950136);
        double r950138 = fabs(r950136);
        double r950139 = r950137 * r950138;
        double r950140 = r950139 - r950120;
        double r950141 = r950125 * r950140;
        double r950142 = sqrt(r950141);
        double r950143 = r950128 * r950142;
        double r950144 = -4.685030330992167e-268;
        bool r950145 = r950120 <= r950144;
        double r950146 = r950132 - r950120;
        double r950147 = r950125 * r950146;
        double r950148 = sqrt(r950147);
        double r950149 = r950148 * r950128;
        double r950150 = r950125 * r950133;
        double r950151 = sqrt(r950150);
        double r950152 = sqrt(r950135);
        double r950153 = r950120 + r950152;
        double r950154 = sqrt(r950153);
        double r950155 = r950151 / r950154;
        double r950156 = r950128 * r950155;
        double r950157 = r950145 ? r950149 : r950156;
        double r950158 = r950131 ? r950143 : r950157;
        double r950159 = r950122 ? r950129 : r950158;
        return r950159;
}

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 < -1.7109531485520302e+90

    1. Initial program 47.7

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

    2. Taylor expanded around -inf 10.1

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \color{blue}{\left(-2 \cdot re\right)}}\]

    if -1.7109531485520302e+90 < re < -5.7290891404837934e-198

    1. Initial program 17.1

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

    3. Applied add-cube-cbrt17.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\color{blue}{\left(\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}\right) \cdot \sqrt[3]{re \cdot re + im \cdot im}}} - re\right)}\]

    4. Applied sqrt-prod17.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\sqrt{\sqrt[3]{re \cdot re + im \cdot im} \cdot \sqrt[3]{re \cdot re + im \cdot im}} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}}} - re\right)}\]

    5. Simplified17.4

      \[\leadsto 0.5 \cdot \sqrt{2.0 \cdot \left(\color{blue}{\left|\sqrt[3]{re \cdot re + im \cdot im}\right|} \cdot \sqrt{\sqrt[3]{re \cdot re + im \cdot im}} - re\right)}\]

    if -5.7290891404837934e-198 < re < -4.685030330992167e-268

    1. Initial program 27.4

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

    2. Taylor expanded around 0 33.9

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

    if -4.685030330992167e-268 < re

    1. Initial program 44.3

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

    3. Applied flip--44.3

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

    4. Applied associate-*r/44.3

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

    5. Applied sqrt-div44.4

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

    6. Simplified34.2

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2.0 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification25.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.7109531485520302 \cdot 10^{+90}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -5.7290891404837934 \cdot 10^{-198}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{\sqrt[3]{im \cdot im + re \cdot re}} \cdot \left|\sqrt[3]{im \cdot im + re \cdot re}\right| - re\right)}\\ \mathbf{elif}\;re \le -4.685030330992167 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{2.0 \cdot \left(im - re\right)} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2.0 \cdot \left(im \cdot im\right)}}{\sqrt{re + \sqrt{im \cdot im + re \cdot re}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (re im)
  :name "math.sqrt on complex, imaginary part, im greater than 0 branch"
  (* 0.5 (sqrt (* 2.0 (- (sqrt (+ (* re re) (* im im))) re)))))