Average Error: 37.5 → 22.3
Time: 20.6s
Precision: 64
\[0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;re \le -6.527134745732682 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \le 5.0588064944949425 \cdot 10^{-224}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im - re\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 1.2645035986882906 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}} \cdot \sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{re + re}} \cdot 0.5\\ \end{array}\]
0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{re \cdot re + im \cdot im} - re\right)}
↓
\begin{array}{l}
\mathbf{if}\;re \le -6.527134745732682 \cdot 10^{+138}:\\
\;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\

\mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\
\;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\

\mathbf{elif}\;re \le 5.0588064944949425 \cdot 10^{-224}:\\
\;\;\;\;0.5 \cdot \sqrt{\left(im - re\right) \cdot 2.0}\\

\mathbf{elif}\;re \le 1.2645035986882906 \cdot 10^{+68}:\\
\;\;\;\;0.5 \cdot \left(\sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}} \cdot \sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}}\right)\\

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

\end{array}
double f(double re, double im) {
        double r706989 = 0.5;
        double r706990 = 2.0;
        double r706991 = re;
        double r706992 = r706991 * r706991;
        double r706993 = im;
        double r706994 = r706993 * r706993;
        double r706995 = r706992 + r706994;
        double r706996 = sqrt(r706995);
        double r706997 = r706996 - r706991;
        double r706998 = r706990 * r706997;
        double r706999 = sqrt(r706998);
        double r707000 = r706989 * r706999;
        return r707000;
}
↓
double f(double re, double im) {
        double r707001 = re;
        double r707002 = -6.527134745732682e+138;
        bool r707003 = r707001 <= r707002;
        double r707004 = -2.0;
        double r707005 = r707004 * r707001;
        double r707006 = 2.0;
        double r707007 = r707005 * r707006;
        double r707008 = sqrt(r707007);
        double r707009 = 0.5;
        double r707010 = r707008 * r707009;
        double r707011 = -1.0853955874561044e-276;
        bool r707012 = r707001 <= r707011;
        double r707013 = im;
        double r707014 = r707013 * r707013;
        double r707015 = r707001 * r707001;
        double r707016 = r707014 + r707015;
        double r707017 = sqrt(r707016);
        double r707018 = r707017 - r707001;
        double r707019 = r707006 * r707018;
        double r707020 = sqrt(r707019);
        double r707021 = r707009 * r707020;
        double r707022 = 5.0588064944949425e-224;
        bool r707023 = r707001 <= r707022;
        double r707024 = r707013 - r707001;
        double r707025 = r707024 * r707006;
        double r707026 = sqrt(r707025);
        double r707027 = r707009 * r707026;
        double r707028 = 1.2645035986882906e+68;
        bool r707029 = r707001 <= r707028;
        double r707030 = r707014 * r707006;
        double r707031 = sqrt(r707030);
        double r707032 = r707017 + r707001;
        double r707033 = sqrt(r707032);
        double r707034 = r707031 / r707033;
        double r707035 = sqrt(r707034);
        double r707036 = r707035 * r707035;
        double r707037 = r707009 * r707036;
        double r707038 = r707001 + r707001;
        double r707039 = sqrt(r707038);
        double r707040 = r707031 / r707039;
        double r707041 = r707040 * r707009;
        double r707042 = r707029 ? r707037 : r707041;
        double r707043 = r707023 ? r707027 : r707042;
        double r707044 = r707012 ? r707021 : r707043;
        double r707045 = r707003 ? r707010 : r707044;
        return r707045;
}

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 5 regimes

  2. if re < -6.527134745732682e+138

    1. Initial program 57.2

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

    2. Taylor expanded around -inf 8.5

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

    if -6.527134745732682e+138 < re < -1.0853955874561044e-276

    1. Initial program 19.1

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

    if -1.0853955874561044e-276 < re < 5.0588064944949425e-224

    1. Initial program 31.5

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

    2. Taylor expanded around 0 32.8

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

    if 5.0588064944949425e-224 < re < 1.2645035986882906e+68

    1. Initial program 38.1

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

    3. Applied flip--38.0

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

    4. Applied associate-*r/38.0

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

    5. Applied sqrt-div38.2

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

    6. Simplified30.6

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im\right) \cdot 2.0 + 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt30.7

      \[\leadsto 0.5 \cdot \color{blue}{\left(\sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0 + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}} \cdot \sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0 + 0}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}}\right)}\]

    if 1.2645035986882906e+68 < re

    1. Initial program 58.1

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

    3. Applied flip--58.1

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

    4. Applied associate-*r/58.1

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

    5. Applied sqrt-div58.1

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

    6. Simplified40.9

      \[\leadsto 0.5 \cdot \frac{\color{blue}{\sqrt{\left(im \cdot im\right) \cdot 2.0 + 0}}}{\sqrt{\sqrt{re \cdot re + im \cdot im} + re}}\]

    7. Taylor expanded around inf 22.9

      \[\leadsto 0.5 \cdot \frac{\sqrt{\left(im \cdot im\right) \cdot 2.0 + 0}}{\sqrt{\color{blue}{re} + re}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -6.527134745732682 \cdot 10^{+138}:\\ \;\;\;\;\sqrt{\left(-2 \cdot re\right) \cdot 2.0} \cdot 0.5\\ \mathbf{elif}\;re \le -1.0853955874561044 \cdot 10^{-276}:\\ \;\;\;\;0.5 \cdot \sqrt{2.0 \cdot \left(\sqrt{im \cdot im + re \cdot re} - re\right)}\\ \mathbf{elif}\;re \le 5.0588064944949425 \cdot 10^{-224}:\\ \;\;\;\;0.5 \cdot \sqrt{\left(im - re\right) \cdot 2.0}\\ \mathbf{elif}\;re \le 1.2645035986882906 \cdot 10^{+68}:\\ \;\;\;\;0.5 \cdot \left(\sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}} \cdot \sqrt{\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{\sqrt{im \cdot im + re \cdot re} + re}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(im \cdot im\right) \cdot 2.0}}{\sqrt{re + re}} \cdot 0.5\\ \end{array}\]

Reproduce

herbie shell --seed 2019151 
(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)))))