Average Error: 31.4 → 0.2
Time: 3.9m
Precision: 64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\]
↓
\[\frac{\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) + \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}}\]
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)
↓
\frac{\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right) + \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}}
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1154151 = x_re;
        double r1154152 = r1154151 * r1154151;
        double r1154153 = x_im;
        double r1154154 = r1154153 * r1154153;
        double r1154155 = r1154152 + r1154154;
        double r1154156 = sqrt(r1154155);
        double r1154157 = log(r1154156);
        double r1154158 = y_re;
        double r1154159 = r1154157 * r1154158;
        double r1154160 = atan2(r1154153, r1154151);
        double r1154161 = y_im;
        double r1154162 = r1154160 * r1154161;
        double r1154163 = r1154159 - r1154162;
        double r1154164 = exp(r1154163);
        double r1154165 = r1154157 * r1154161;
        double r1154166 = r1154160 * r1154158;
        double r1154167 = r1154165 + r1154166;
        double r1154168 = sin(r1154167);
        double r1154169 = r1154164 * r1154168;
        return r1154169;
}
↓
double f(double x_re, double x_im, double y_re, double y_im) {
        double r1154170 = x_im;
        double r1154171 = x_re;
        double r1154172 = atan2(r1154170, r1154171);
        double r1154173 = y_re;
        double r1154174 = r1154172 * r1154173;
        double r1154175 = cos(r1154174);
        double r1154176 = hypot(r1154171, r1154170);
        double r1154177 = log(r1154176);
        double r1154178 = y_im;
        double r1154179 = r1154177 * r1154178;
        double r1154180 = sin(r1154179);
        double r1154181 = r1154175 * r1154180;
        double r1154182 = sin(r1154174);
        double r1154183 = cos(r1154179);
        double r1154184 = r1154182 * r1154183;
        double r1154185 = r1154181 + r1154184;
        double r1154186 = r1154172 * r1154178;
        double r1154187 = r1154177 * r1154173;
        double r1154188 = r1154186 - r1154187;
        double r1154189 = exp(r1154188);
        double r1154190 = r1154185 / r1154189;
        return r1154190;
}

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. Initial program 31.4

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

  2. Simplified0.2

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

  4. Applied fma-udef0.2

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

  5. Applied sin-sum0.2

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

  6. Final simplification0.2

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

Reproduce

herbie shell --seed 2019132 +o rules:numerics
(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)))))