Average Error: 33.8 → 8.4
Time: 15.2s
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;x.re \le -3.853834220586221 \cdot 10^{184}:\\ \;\;\;\;\frac{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)}}\\ \mathbf{elif}\;x.re \le 2.2092076921331203 \cdot 10^{89}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log x.re \cdot y.im\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log x.re}}\\ \end{array}\]
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 -3.853834220586221 \cdot 10^{184}:\\
\;\;\;\;\frac{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)}}\\

\mathbf{elif}\;x.re \le 2.2092076921331203 \cdot 10^{89}:\\
\;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log x.re \cdot y.im\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log x.re}}\\

\end{array}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r21959 = x_re;
        double r21960 = r21959 * r21959;
        double r21961 = x_im;
        double r21962 = r21961 * r21961;
        double r21963 = r21960 + r21962;
        double r21964 = sqrt(r21963);
        double r21965 = log(r21964);
        double r21966 = y_re;
        double r21967 = r21965 * r21966;
        double r21968 = atan2(r21961, r21959);
        double r21969 = y_im;
        double r21970 = r21968 * r21969;
        double r21971 = r21967 - r21970;
        double r21972 = exp(r21971);
        double r21973 = r21965 * r21969;
        double r21974 = r21968 * r21966;
        double r21975 = r21973 + r21974;
        double r21976 = sin(r21975);
        double r21977 = r21972 * r21976;
        return r21977;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r21978 = x_re;
        double r21979 = -3.853834220586221e+184;
        bool r21980 = r21978 <= r21979;
        double r21981 = x_im;
        double r21982 = atan2(r21981, r21978);
        double r21983 = y_re;
        double r21984 = r21982 * r21983;
        double r21985 = y_im;
        double r21986 = -1.0;
        double r21987 = r21986 / r21978;
        double r21988 = log(r21987);
        double r21989 = r21985 * r21988;
        double r21990 = r21984 - r21989;
        double r21991 = sin(r21990);
        double r21992 = r21982 * r21985;
        double r21993 = r21983 * r21988;
        double r21994 = -r21993;
        double r21995 = r21992 - r21994;
        double r21996 = exp(r21995);
        double r21997 = r21991 / r21996;
        double r21998 = 2.2092076921331203e+89;
        bool r21999 = r21978 <= r21998;
        double r22000 = hypot(r21978, r21981);
        double r22001 = pow(r22000, r21983);
        double r22002 = log(r22000);
        double r22003 = r22002 * r21985;
        double r22004 = sin(r22003);
        double r22005 = cos(r21984);
        double r22006 = r22004 * r22005;
        double r22007 = cos(r22003);
        double r22008 = sin(r21984);
        double r22009 = r22007 * r22008;
        double r22010 = r22006 + r22009;
        double r22011 = r22001 * r22010;
        double r22012 = exp(r21992);
        double r22013 = r22011 / r22012;
        double r22014 = log(r21978);
        double r22015 = r22014 * r21985;
        double r22016 = fma(r21982, r21983, r22015);
        double r22017 = sin(r22016);
        double r22018 = r21983 * r22014;
        double r22019 = r21992 - r22018;
        double r22020 = exp(r22019);
        double r22021 = r22017 / r22020;
        double r22022 = r21999 ? r22013 : r22021;
        double r22023 = r21980 ? r21997 : r22022;
        return r22023;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  1. Split input into 3 regimes

  2. if x.re < -3.853834220586221e+184

    1. Initial program 64.0

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

    2. Simplified7.2

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

    3. Taylor expanded around -inf 8.4

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

    4. Simplified1.5

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

    if -3.853834220586221e+184 < x.re < 2.2092076921331203e+89

    1. Initial program 24.6

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

    2. Simplified8.1

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

    4. Applied fma-udef8.1

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

    5. Applied sin-sum8.1

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

    if 2.2092076921331203e+89 < x.re

    1. Initial program 52.0

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

    2. Simplified13.7

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

    3. Taylor expanded around inf 15.6

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

    4. Simplified13.1

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

  4. Final simplification8.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \le -3.853834220586221 \cdot 10^{184}:\\ \;\;\;\;\frac{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \left(-y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)}}\\ \mathbf{elif}\;x.re \le 2.2092076921331203 \cdot 10^{89}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \left(\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) + \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log x.re \cdot y.im\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log x.re}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (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)))))