Average Error: 39.0 → 27.3
Time: 9.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -3.94465399359600175861210479053597958901 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le \frac{-5938511416015121}{1.850745787979017418800567970827224916526 \cdot 10^{224}}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt[3]{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -3.94465399359600175861210479053597958901 \cdot 10^{108}:\\
\;\;\;\;\frac{1}{2} \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\

\mathbf{elif}\;re \le \frac{-5938511416015121}{1.850745787979017418800567970827224916526 \cdot 10^{224}}:\\
\;\;\;\;\frac{1}{2} \cdot \sqrt[3]{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{3}}\\

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

\end{array}
double f(double re, double im) {
        double r29154 = 0.5;
        double r29155 = 2.0;
        double r29156 = re;
        double r29157 = r29156 * r29156;
        double r29158 = im;
        double r29159 = r29158 * r29158;
        double r29160 = r29157 + r29159;
        double r29161 = sqrt(r29160);
        double r29162 = r29161 - r29156;
        double r29163 = r29155 * r29162;
        double r29164 = sqrt(r29163);
        double r29165 = r29154 * r29164;
        return r29165;
}
↓
double f(double re, double im) {
        double r29166 = re;
        double r29167 = -3.944653993596002e+108;
        bool r29168 = r29166 <= r29167;
        double r29169 = 1.0;
        double r29170 = 2.0;
        double r29171 = r29169 / r29170;
        double r29172 = -1.0;
        double r29173 = r29172 * r29166;
        double r29174 = r29173 - r29166;
        double r29175 = r29170 * r29174;
        double r29176 = sqrt(r29175);
        double r29177 = r29171 * r29176;
        double r29178 = -5938511416015121.0;
        double r29179 = 1.8507457879790174e+224;
        double r29180 = r29178 / r29179;
        bool r29181 = r29166 <= r29180;
        double r29182 = r29166 * r29166;
        double r29183 = im;
        double r29184 = r29183 * r29183;
        double r29185 = r29182 + r29184;
        double r29186 = sqrt(r29185);
        double r29187 = r29186 - r29166;
        double r29188 = r29170 * r29187;
        double r29189 = sqrt(r29188);
        double r29190 = 3.0;
        double r29191 = pow(r29189, r29190);
        double r29192 = cbrt(r29191);
        double r29193 = r29171 * r29192;
        double r29194 = 2.0;
        double r29195 = pow(r29183, r29194);
        double r29196 = r29186 + r29166;
        double r29197 = r29195 / r29196;
        double r29198 = r29170 * r29197;
        double r29199 = sqrt(r29198);
        double r29200 = r29171 * r29199;
        double r29201 = r29181 ? r29193 : r29200;
        double r29202 = r29168 ? r29177 : r29201;
        return r29202;
}

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 3 regimes

  2. if re < -3.944653993596002e+108

    1. Initial program 53.5

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

    2. Taylor expanded around -inf 10.6

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

    if -3.944653993596002e+108 < re < -3.208712646862146e-209

    1. Initial program 18.4

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

    3. Applied add-cbrt-cube18.8

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

    4. Simplified18.8

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

    if -3.208712646862146e-209 < re

    1. Initial program 43.9

      \[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
    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. Simplified35.6

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

  4. Final simplification27.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -3.94465399359600175861210479053597958901 \cdot 10^{108}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{2 \cdot \left(-1 \cdot re - re\right)}\\ \mathbf{elif}\;re \le \frac{-5938511416015121}{1.850745787979017418800567970827224916526 \cdot 10^{224}}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt[3]{{\left(\sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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)))))