Average Error: 37.6 → 26.1
Time: 17.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.1639866401777603 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \le 1.9683258718073588 \cdot 10^{-248}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im - re\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.1639866401777603 \cdot 10^{+111}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\

\mathbf{elif}\;re \le 1.9683258718073588 \cdot 10^{-248}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(im - re\right) \cdot 2.0}\\

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

\end{array}
double f(double re, double im) {
        double r478106 = 0.5;
        double r478107 = 2.0;
        double r478108 = re;
        double r478109 = r478108 * r478108;
        double r478110 = im;
        double r478111 = r478110 * r478110;
        double r478112 = r478109 + r478111;
        double r478113 = sqrt(r478112);
        double r478114 = r478113 - r478108;
        double r478115 = r478107 * r478114;
        double r478116 = sqrt(r478115);
        double r478117 = r478106 * r478116;
        return r478117;
}
↓
double f(double re, double im) {
        double r478118 = re;
        double r478119 = -1.1639866401777603e+111;
        bool r478120 = r478118 <= r478119;
        double r478121 = -2.0;
        double r478122 = r478121 * r478118;
        double r478123 = 2.0;
        double r478124 = r478122 * r478123;
        double r478125 = sqrt(r478124);
        double r478126 = 0.5;
        double r478127 = r478125 * r478126;
        double r478128 = -6.3441075981154e-203;
        bool r478129 = r478118 <= r478128;
        double r478130 = im;
        double r478131 = r478130 * r478130;
        double r478132 = r478118 * r478118;
        double r478133 = r478131 + r478132;
        double r478134 = sqrt(r478133);
        double r478135 = r478134 - r478118;
        double r478136 = r478123 * r478135;
        double r478137 = sqrt(r478136);
        double r478138 = r478126 * r478137;
        double r478139 = 1.9683258718073588e-248;
        bool r478140 = r478118 <= r478139;
        double r478141 = r478130 - r478118;
        double r478142 = r478141 * r478123;
        double r478143 = sqrt(r478142);
        double r478144 = r478126 * r478143;
        double r478145 = r478131 * r478123;
        double r478146 = sqrt(r478145);
        double r478147 = r478134 + r478118;
        double r478148 = sqrt(r478147);
        double r478149 = r478146 / r478148;
        double r478150 = r478126 * r478149;
        double r478151 = r478140 ? r478144 : r478150;
        double r478152 = r478129 ? r478138 : r478151;
        double r478153 = r478120 ? r478127 : r478152;
        return r478153;
}

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 < -1.1639866401777603e+111

    1. Initial program 51.2

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

    2. Taylor expanded around -inf 9.7

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

    if -1.1639866401777603e+111 < re < -6.3441075981154e-203

    1. Initial program 17.7

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

    if -6.3441075981154e-203 < re < 1.9683258718073588e-248

    1. Initial program 27.2

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

    2. Taylor expanded around 0 31.4

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

    if 1.9683258718073588e-248 < re

    1. Initial program 46.9

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

    3. Applied flip--46.9

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

    4. Applied associate-*r/46.9

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

    5. Applied sqrt-div47.0

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

    6. Simplified34.9

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

  4. Final simplification26.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.1639866401777603 \cdot 10^{+111}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -6.3441075981154 \cdot 10^{-203}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \le 1.9683258718073588 \cdot 10^{-248}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im - re\right) \cdot 2.0}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(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)))))