Average Error: 33.5 → 5.8
Time: 15.2s
Precision: binary64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
↓
\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := e^{t_1 \cdot y.re - t_2} \cdot \cos \left(t_1 \cdot y.im + t_0\right)\\ \mathbf{if}\;t_3 \leq 0.8068509587936992:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)}\\ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{t_2}} \cdot \cos \left(t_4 \cdot \left(t_4 \cdot t_4\right)\right) \end{array}\\ \end{array} \]
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 \cos \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 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\
t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\
t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\
t_3 := e^{t_1 \cdot y.re - t_2} \cdot \cos \left(t_1 \cdot y.im + t_0\right)\\
\mathbf{if}\;t_3 \leq 0.8068509587936992:\\
\;\;\;\;t_3\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_4 := \sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)}\\
\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{t_2}} \cdot \cos \left(t_4 \cdot \left(t_4 \cdot t_4\right)\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)))
  (cos
   (+
    (* (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 (* y.re (atan2 x.im x.re)))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (* (exp (- (* t_1 y.re) t_2)) (cos (+ (* t_1 y.im) t_0)))))
   (if (<= t_3 0.8068509587936992)
     t_3
     (let* ((t_4 (cbrt (fma (log (hypot x.re x.im)) y.im t_0))))
       (*
        (/ (pow (hypot x.re x.im) y.re) (exp t_2))
        (cos (* t_4 (* t_4 t_4))))))))
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)) * cos((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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(sqrt((x_46_re * x_46_re) + (x_46_im * x_46_im)));
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = exp((t_1 * y_46_re) - t_2) * cos((t_1 * y_46_im) + t_0);
	double tmp;
	if (t_3 <= 0.8068509587936992) {
		tmp = t_3;
	} else {
		double t_4 = cbrt(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0));
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / exp(t_2)) * cos(t_4 * (t_4 * t_4));
	}
	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 (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 0.80685095879369917

    1. Initial program 3.9

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

    if 0.80685095879369917 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

    1. Initial program 46.0

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

    2. Simplified6.6

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

    3. Applied add-cube-cbrt_binary646.6

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

  4. Final simplification5.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 0.8068509587936992:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \cos \left(\sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022082 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))