powComplex, imaginary part

Percentage Accurate: 39.5% → 79.1%

Time: 31.5s

Precision: binary64

Cost: 71552

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

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

FPCore

(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.re x.im))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (sin (fma t_0 y.im (pow (cbrt (* y.re (atan2 x.im x.re))) 3.0))))))

C

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_re, x_46_im));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(fma(t_0, y_46_im, pow(cbrt((y_46_re * atan2(x_46_im, x_46_re))), 3.0)));
}

Julia

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_re, x_46_im))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(fma(t_0, y_46_im, (cbrt(Float64(y_46_re * atan(x_46_im, x_46_re))) ^ 3.0))))
end

Wolfram

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$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 * y$46$im + N[Power[N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 32.5%

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

2. Simplified 78.1%

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

   [Start] 32.5%	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

3. Applied egg-rr 82.2%

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

   [Start] 78.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   *-commutative [<=] 78.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)\right) \]
   add-cube-cbrt [=>] 82.6%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re} \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \cdot \sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}}\right)\right) \]
   pow3 [=>] 82.2%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{{\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)}^{3}}\right)\right) \]

4. Final simplification 82.2%

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

Alternatives

Alternative 1
Accuracy	79.1%
Cost	71552

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

Alternative 2
Accuracy	78.9%
Cost	118212

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_4 := e^{t_2 \cdot y.re - t_3}\\ t_5 := \sin \left(\mathsf{fma}\left(t_2, y.im, t_0\right)\right)\\ \mathbf{if}\;e^{y.re \cdot t_1 - t_3} \cdot \sin \left(y.im \cdot t_1 + t_0\right) \leq -\infty:\\ \;\;\;\;t_4 \cdot \left|t_5\right|\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot t_5\\ \end{array} \]

Alternative 3
Accuracy	79.2%
Cost	58688

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

Alternative 4
Accuracy	75.1%
Cost	52292

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq -5.6 \cdot 10^{-17}:\\ \;\;\;\;e^{t_2 \cdot y.re - t_0} \cdot {\left(\sqrt[3]{\sin t_1}\right)}^{3}\\ \mathbf{elif}\;y.re \leq 3.15 \cdot 10^{-24}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t_2 \cdot y.im\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \sin \left(\left|t_1\right|\right)\\ \end{array} \]

Alternative 5
Accuracy	72.9%
Cost	46544

\[\begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_0 \cdot \sin \left(\left|t_1\right|\right)\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{+18}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+69}:\\ \;\;\;\;t_0 \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{+191}:\\ \;\;\;\;\sin t_1 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Accuracy	73.8%
Cost	46148

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -0.8:\\ \;\;\;\;t_1 \cdot \left|t_0\right|\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+65} \lor \neg \left(y.re \leq 1.25 \cdot 10^{+196}\right):\\ \;\;\;\;t_1 \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 7
Accuracy	73.2%
Cost	45896

\[\begin{array}{l} t_0 := \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{-16}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+27}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+67} \lor \neg \left(y.re \leq 4.9 \cdot 10^{+191}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 8
Accuracy	72.1%
Cost	45768

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+92}:\\ \;\;\;\;t_0 \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-13}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 9
Accuracy	62.4%
Cost	40344

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log x.re - t_0}\\ t_2 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ t_3 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_4 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_5 := e^{y.re \cdot \log \left(-x.re\right) - t_0}\\ t_6 := t_3 \cdot t_5\\ \mathbf{if}\;x.re \leq -2.8 \cdot 10^{+47}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;x.re \leq -290000000000:\\ \;\;\;\;t_4 \cdot t_5\\ \mathbf{elif}\;x.re \leq -1.9 \cdot 10^{-12}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;x.re \leq 2.05 \cdot 10^{-98}:\\ \;\;\;\;t_2 \cdot t_3\\ \mathbf{elif}\;x.re \leq 6 \cdot 10^{-56}:\\ \;\;\;\;\left|t_4\right| \cdot t_1\\ \mathbf{elif}\;x.re \leq 4.5:\\ \;\;\;\;t_2 \cdot \sin \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;x.re \leq 7 \cdot 10^{+136}:\\ \;\;\;\;t_4 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot t_1\\ \end{array} \]

Alternative 10
Accuracy	59.6%
Cost	40212

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := t_1 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ t_3 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_4 := e^{y.re \cdot \log x.im - t_0}\\ \mathbf{if}\;x.im \leq -2.5 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq -2.02 \cdot 10^{-224}:\\ \;\;\;\;t_3 \cdot e^{y.re \cdot \log \left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq 5.4 \cdot 10^{-160}:\\ \;\;\;\;t_1 \cdot t_4\\ \mathbf{elif}\;x.im \leq 2.2 \cdot 10^{-6}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot t_4\\ \end{array} \]

Alternative 11
Accuracy	60.0%
Cost	39892

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{y.re \cdot \log x.re - t_0}\\ t_2 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_3 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_4 := e^{y.re \cdot \log \left(-x.re\right) - t_0}\\ t_5 := t_2 \cdot t_4\\ \mathbf{if}\;x.re \leq -9.5 \cdot 10^{+46}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq -3.7 \cdot 10^{-81}:\\ \;\;\;\;t_3 \cdot t_4\\ \mathbf{elif}\;x.re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq 2.05 \cdot 10^{-125}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log \left(x.re + 0.5 \cdot \frac{x.im \cdot x.im}{x.re}\right) - t_0}\\ \mathbf{elif}\;x.re \leq 2.65 \cdot 10^{-56}:\\ \;\;\;\;\left|t_3\right| \cdot t_1\\ \mathbf{elif}\;x.re \leq 2.65:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{+136}:\\ \;\;\;\;t_3 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_1\\ \end{array} \]

Alternative 12
Accuracy	59.3%
Cost	33812

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := t_1 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ t_3 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_4 := e^{y.re \cdot \log x.im - t_0}\\ \mathbf{if}\;x.im \leq -3.4 \cdot 10^{-49}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq -1.5 \cdot 10^{-226}:\\ \;\;\;\;t_3 \cdot e^{y.re \cdot \log \left(\frac{\left(x.re \cdot x.re\right) \cdot -0.5}{x.im} - x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.im \leq 6.6 \cdot 10^{-110}:\\ \;\;\;\;t_1 \cdot t_4\\ \mathbf{elif}\;x.im \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \left(y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot t_4\\ \end{array} \]

Alternative 13
Accuracy	58.8%
Cost	33548

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := e^{y.re \cdot \log x.im - t_0}\\ \mathbf{if}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;x.im \leq 1.12 \cdot 10^{-14}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \left(y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t_2\\ \end{array} \]

Alternative 14
Accuracy	54.4%
Cost	33228

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+18}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot 0\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-268}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-307}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.im - t_0}\\ \mathbf{elif}\;y.re \leq 10^{+28}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin t_1 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 15
Accuracy	57.6%
Cost	33096

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq -1.42 \cdot 10^{-145}:\\ \;\;\;\;\sin t_1 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.im - t_0}\\ \end{array} \]

Alternative 16
Accuracy	59.7%
Cost	33028

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log x.im - t_0}\\ \end{array} \]

Alternative 17
Accuracy	54.4%
Cost	26756

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{+18}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+29}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 18
Accuracy	51.7%
Cost	20105

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+18} \lor \neg \left(y.re \leq 7.4 \cdot 10^{+27}\right):\\ \;\;\;\;\sin t_0 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 19
Accuracy	30.9%
Cost	19648

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

Reproduce

herbie shell --seed 2023229 
(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)))))