Average Error: 38.7 → 26.3
Time: 15.1s
Precision: 64
\[0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 2.170454861417431561301436996642453156775 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]
0.5 \cdot \sqrt{2 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\

\mathbf{elif}\;re \le 2.170454861417431561301436996642453156775 \cdot 10^{-276}:\\
\;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\

\mathbf{else}:\\
\;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\

\end{array}
double f(double re, double im) {
        double r921948 = 0.5;
        double r921949 = 2.0;
        double r921950 = re;
        double r921951 = r921950 * r921950;
        double r921952 = im;
        double r921953 = r921952 * r921952;
        double r921954 = r921951 + r921953;
        double r921955 = sqrt(r921954);
        double r921956 = r921955 - r921950;
        double r921957 = r921949 * r921956;
        double r921958 = sqrt(r921957);
        double r921959 = r921948 * r921958;
        return r921959;
}
↓
double f(double re, double im) {
        double r921960 = re;
        double r921961 = -1.9401195934627838e+70;
        bool r921962 = r921960 <= r921961;
        double r921963 = -2.0;
        double r921964 = r921963 * r921960;
        double r921965 = 2.0;
        double r921966 = r921964 * r921965;
        double r921967 = sqrt(r921966);
        double r921968 = 0.5;
        double r921969 = r921967 * r921968;
        double r921970 = 2.1704548614174316e-276;
        bool r921971 = r921960 <= r921970;
        double r921972 = im;
        double r921973 = r921972 * r921972;
        double r921974 = r921960 * r921960;
        double r921975 = r921973 + r921974;
        double r921976 = sqrt(r921975);
        double r921977 = r921976 - r921960;
        double r921978 = r921965 * r921977;
        double r921979 = sqrt(r921978);
        double r921980 = r921968 * r921979;
        double r921981 = r921965 * r921973;
        double r921982 = sqrt(r921981);
        double r921983 = r921976 + r921960;
        double r921984 = sqrt(r921983);
        double r921985 = r921982 / r921984;
        double r921986 = r921968 * r921985;
        double r921987 = r921971 ? r921980 : r921986;
        double r921988 = r921962 ? r921969 : r921987;
        return r921988;
}

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 < -1.9401195934627838e+70

    1. Initial program 46.3

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

    2. Taylor expanded around -inf 12.0

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

    if -1.9401195934627838e+70 < re < 2.1704548614174316e-276

    1. Initial program 22.3

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

    if 2.1704548614174316e-276 < re

    1. Initial program 47.2

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

    3. Applied add-exp-log49.4

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

    5. Applied flip--49.2

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

    6. Applied associate-*r/49.2

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

    7. Applied sqrt-div49.2

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

    8. Simplified36.5

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}}{\sqrt{e^{\log \left(\sqrt{re \cdot re + im \cdot im}\right)} + re}}\]

    9. Simplified35.3

      \[\leadsto 0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im + 0\right)}}{\color{blue}{\sqrt{re + \sqrt{re \cdot re + im \cdot im}}}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification26.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -1.940119593462783780503740532731557409155 \cdot 10^{70}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2} \cdot 0.5\\ \mathbf{elif}\;re \le 2.170454861417431561301436996642453156775 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \sqrt{2 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sqrt{2 \cdot \left(im \cdot im\right)}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(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)))))