Average Error: 38.8 → 24.5
Time: 4.4s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -2.27109209614237641 \cdot 10^{59}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(-2 \cdot re\right)}\\ \mathbf{elif}\;re \le -1.6126997418755618 \cdot 10^{-300}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\\ \mathbf{elif}\;re \le 1.0984713903285393 \cdot 10^{132}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2}}{\sqrt{re \cdot re + im \cdot im} + re}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \frac{{im}^{2} \cdot \frac{1}{2}}{re}}\\ \end{array}\]
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(-2 \cdot 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 r17064 = 0.5;
        double r17065 = 2.0;
        double r17066 = re;
        double r17067 = r17066 * r17066;
        double r17068 = im;
        double r17069 = r17068 * r17068;
        double r17070 = r17067 + r17069;
        double r17071 = sqrt(r17070);
        double r17072 = r17071 - r17066;
        double r17073 = r17065 * r17072;
        double r17074 = sqrt(r17073);
        double r17075 = r17064 * r17074;
        return r17075;
}
↓
double f(double re, double im) {
        double r17076 = re;
        double r17077 = -2.2710920961423764e+59;
        bool r17078 = r17076 <= r17077;
        double r17079 = 0.5;
        double r17080 = 2.0;
        double r17081 = -2.0;
        double r17082 = r17081 * r17076;
        double r17083 = r17080 * r17082;
        double r17084 = sqrt(r17083);
        double r17085 = r17079 * r17084;
        double r17086 = -1.612699741875562e-300;
        bool r17087 = r17076 <= r17086;
        double r17088 = r17076 * r17076;
        double r17089 = im;
        double r17090 = r17089 * r17089;
        double r17091 = r17088 + r17090;
        double r17092 = sqrt(r17091);
        double r17093 = r17092 - r17076;
        double r17094 = r17080 * r17093;
        double r17095 = sqrt(r17094);
        double r17096 = r17079 * r17095;
        double r17097 = 1.0984713903285393e+132;
        bool r17098 = r17076 <= r17097;
        double r17099 = 2.0;
        double r17100 = pow(r17089, r17099);
        double r17101 = r17092 + r17076;
        double r17102 = r17100 / r17101;
        double r17103 = r17080 * r17102;
        double r17104 = sqrt(r17103);
        double r17105 = r17079 * r17104;
        double r17106 = 0.5;
        double r17107 = r17100 * r17106;
        double r17108 = r17107 / r17076;
        double r17109 = r17080 * r17108;
        double r17110 = sqrt(r17109);
        double r17111 = r17079 * r17110;
        double r17112 = r17098 ? r17105 : r17111;
        double r17113 = r17087 ? r17096 : r17112;
        double r17114 = r17078 ? r17085 : r17113;
        return r17114;
}

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. Using strategy rm

    3. Applied add-cube-cbrt45.2

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

    4. Taylor expanded around -inf 12.8

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

    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

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

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)))))