Average Error: 33.4 → 6.7
Time: 26.1s
Precision: binary64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
↓
\[\begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{if}\;y.re \leq -1.054865446815132 \cdot 10^{+76}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}}\\ t_0 \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{t_1 \cdot \left(t_1 \cdot t_1\right)}} \end{array}\\ \end{array} \]
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}
t_0 := \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\
\mathbf{if}\;y.re \leq -1.054865446815132 \cdot 10^{+76}:\\
\;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot t_0\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_1 := \sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}}\\
t_0 \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{t_1 \cdot \left(t_1 \cdot t_1\right)}}
\end{array}\\


\end{array}

(FPCore (x.re x.im y.re y.im)
 :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)))))
↓
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (sin (fma (log (hypot x.re x.im)) y.im (* y.re (atan2 x.im x.re))))))
   (if (<= y.re -1.054865446815132e+76)
     (* (pow (hypot x.im x.re) y.re) t_0)
     (let* ((t_1 (cbrt (cbrt (pow (* y.im (atan2 x.im x.re)) 3.0)))))
       (* t_0 (/ (pow (hypot x.re x.im) y.re) (exp (* t_1 (* t_1 t_1)))))))))
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp((log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)) * sin((log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im))) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re));
}
↓
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.054865446815132e+76) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) * t_0;
	} else {
		double t_1 = cbrt(cbrt(pow((y_46_im * atan2(x_46_im, x_46_re)), 3.0)));
		tmp = t_0 * (pow(hypot(x_46_re, x_46_im), y_46_re) / exp(t_1 * (t_1 * t_1)));
	}
	return tmp;
}

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 2 regimes

  2. if y.re < -1.054865446815132e76

    1. Initial program 37.5

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

    2. Simplified9.7

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

    3. Taylor expanded in y.im around 0 1.2

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{{x.re}^{2} + {x.im}^{2}}\right) \cdot y.re}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

    4. Simplified1.2

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

    if -1.054865446815132e76 < y.re

    1. Initial program 32.3

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

    2. Simplified8.2

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

    3. Applied add-cbrt-cube_binary648.2

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

    4. Simplified8.2

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

    5. Applied add-cube-cbrt_binary648.2

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{\left(\sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}}\right) \cdot \sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}}}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.054865446815132 \cdot 10^{+76}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \cdot \left(\sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \cdot \sqrt[3]{\sqrt[3]{{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021357 
(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)))))