Average Error: 33.7 → 3.7
Time: 12.3s
Precision: binary64
\[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
↓
\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ e^{y.re \cdot t_0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, t_0, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \end{array} \]
(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 (log (hypot x.im x.re))))
   (*
    (exp (- (* y.re t_0) (* (atan2 x.im x.re) y.im)))
    (sin (fma y.im t_0 (* y.re (atan2 x.im x.re)))))))
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 = log(hypot(x_46_im, x_46_re));
	return exp(((y_46_re * t_0) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(fma(y_46_im, t_0, (y_46_re * atan2(x_46_im, x_46_re))));
}

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end
↓
function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	return Float64(exp(Float64(Float64(y_46_re * t_0) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(fma(y_46_im, t_0, Float64(y_46_re * atan(x_46_im, x_46_re)))))
end

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]
↓
code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$0 + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

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 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\
e^{y.re \cdot t_0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, t_0, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)
\end{array}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

  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. Simplified9.7

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

  3. Taylor expanded in x.im around 0 9.0

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

  4. Applied egg-rr3.7

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

  5. Applied egg-rr3.9

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

  6. Taylor expanded in y.im around inf 31.1

    \[\leadsto e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, {\left({\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{0.3333333333333333}\right)}^{3}\right)\right)} \]

  7. Simplified3.7

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

  8. Final simplification3.7

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

Reproduce

herbie shell --seed 2022150 
(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)))))