Average Error: 33.2 → 20.7
Time: 33.1s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -2.8995865521776026 \cdot 10^{-105}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le 3.7259888111147467 \cdot 10^{-302}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \le 3.9300468248158223 \cdot 10^{-82}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right) \cdot \left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\right)}\\ \mathbf{elif}\;x.re \le 9416.095709505209:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\sqrt[3]{\log x.re \cdot y.im} \cdot \sqrt[3]{\log x.re \cdot y.im}\right) \cdot \sqrt[3]{\log x.re \cdot y.im}\right) \cdot e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]
e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)
↓
\begin{array}{l}
\mathbf{if}\;x.re \le -2.8995865521776026 \cdot 10^{-105}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\

\mathbf{elif}\;x.re \le 3.7259888111147467 \cdot 10^{-302}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\

\mathbf{elif}\;x.re \le 3.9300468248158223 \cdot 10^{-82}:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right) \cdot \left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\right)}\\

\mathbf{elif}\;x.re \le 9416.095709505209:\\
\;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\

\mathbf{else}:\\
\;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\sqrt[3]{\log x.re \cdot y.im} \cdot \sqrt[3]{\log x.re \cdot y.im}\right) \cdot \sqrt[3]{\log x.re \cdot y.im}\right) \cdot e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r587979 = x_re;
        double r587980 = r587979 * r587979;
        double r587981 = x_im;
        double r587982 = r587981 * r587981;
        double r587983 = r587980 + r587982;
        double r587984 = sqrt(r587983);
        double r587985 = log(r587984);
        double r587986 = y_re;
        double r587987 = r587985 * r587986;
        double r587988 = atan2(r587981, r587979);
        double r587989 = y_im;
        double r587990 = r587988 * r587989;
        double r587991 = r587987 - r587990;
        double r587992 = exp(r587991);
        double r587993 = r587985 * r587989;
        double r587994 = r587988 * r587986;
        double r587995 = r587993 + r587994;
        double r587996 = sin(r587995);
        double r587997 = r587992 * r587996;
        return r587997;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r587998 = x_re;
        double r587999 = -2.8995865521776026e-105;
        bool r588000 = r587998 <= r587999;
        double r588001 = y_re;
        double r588002 = r587998 * r587998;
        double r588003 = x_im;
        double r588004 = r588003 * r588003;
        double r588005 = r588002 + r588004;
        double r588006 = sqrt(r588005);
        double r588007 = log(r588006);
        double r588008 = r588001 * r588007;
        double r588009 = atan2(r588003, r587998);
        double r588010 = y_im;
        double r588011 = r588009 * r588010;
        double r588012 = r588008 - r588011;
        double r588013 = exp(r588012);
        double r588014 = r588009 * r588001;
        double r588015 = -r587998;
        double r588016 = log(r588015);
        double r588017 = r588010 * r588016;
        double r588018 = r588014 + r588017;
        double r588019 = sin(r588018);
        double r588020 = r588013 * r588019;
        double r588021 = 3.7259888111147467e-302;
        bool r588022 = r587998 <= r588021;
        double r588023 = r588010 * r588007;
        double r588024 = r588023 + r588014;
        double r588025 = sin(r588024);
        double r588026 = r588013 * r588025;
        double r588027 = 3.9300468248158223e-82;
        bool r588028 = r587998 <= r588027;
        double r588029 = log(r587998);
        double r588030 = r588029 * r588010;
        double r588031 = r588014 + r588030;
        double r588032 = sin(r588031);
        double r588033 = r588032 * r588032;
        double r588034 = r588032 * r588033;
        double r588035 = cbrt(r588034);
        double r588036 = r588013 * r588035;
        double r588037 = 9416.095709505209;
        bool r588038 = r587998 <= r588037;
        double r588039 = cbrt(r588030);
        double r588040 = r588039 * r588039;
        double r588041 = r588040 * r588039;
        double r588042 = r588014 + r588041;
        double r588043 = sin(r588042);
        double r588044 = r588029 * r588001;
        double r588045 = r588044 - r588011;
        double r588046 = exp(r588045);
        double r588047 = r588043 * r588046;
        double r588048 = r588038 ? r588026 : r588047;
        double r588049 = r588028 ? r588036 : r588048;
        double r588050 = r588022 ? r588026 : r588049;
        double r588051 = r588000 ? r588020 : r588050;
        return r588051;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if x.re < -2.8995865521776026e-105

    1. Initial program 33.7

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around -inf 20.0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    3. Simplified20.0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    if -2.8995865521776026e-105 < x.re < 3.7259888111147467e-302 or 3.9300468248158223e-82 < x.re < 9416.095709505209

    1. Initial program 25.3

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    if 3.7259888111147467e-302 < x.re < 3.9300468248158223e-82

    1. Initial program 26.5

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around inf 22.2

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\]

    3. Simplified22.2

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)}\]
    4. Using strategy rm

    5. Applied add-cbrt-cube27.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)}}\]

    if 9416.095709505209 < x.re

    1. Initial program 44.4

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]

    2. Taylor expanded around inf 28.3

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)}\]

    3. Simplified28.3

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)}\]

    4. Taylor expanded around inf 13.0

      \[\leadsto e^{\log \color{blue}{x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log x.re \cdot y.im\right)\]
    5. Using strategy rm

    6. Applied add-cube-cbrt13.3

      \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(\sqrt[3]{\log x.re \cdot y.im} \cdot \sqrt[3]{\log x.re \cdot y.im}\right) \cdot \sqrt[3]{\log x.re \cdot y.im}}\right)\]
  3. Recombined 4 regimes into one program.

  4. Final simplification20.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -2.8995865521776026 \cdot 10^{-105}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \log \left(-x.re\right)\right)\\ \mathbf{elif}\;x.re \le 3.7259888111147467 \cdot 10^{-302}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \le 3.9300468248158223 \cdot 10^{-82}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt[3]{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right) \cdot \left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log x.re \cdot y.im\right)\right)}\\ \mathbf{elif}\;x.re \le 9416.095709505209:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \left(\sqrt[3]{\log x.re \cdot y.im} \cdot \sqrt[3]{\log x.re \cdot y.im}\right) \cdot \sqrt[3]{\log x.re \cdot y.im}\right) \cdot e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))