powComplex, imaginary part

Percentage Accurate: 41.1% → 79.7%

Time: 44.9s

Precision: binary64

Cost: 71816

• 36.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-120}:\\ \;\;\;\;t_1 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+65}:\\ \;\;\;\;t_1 \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(t_0, y.im, t_2\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin t_2\\ \end{array} \]

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)))
        (t_1 (exp (- (* t_0 y.re) (* y.im (atan2 x.im x.re)))))
        (t_2 (* y.re (atan2 x.im x.re))))
   (if (<= y.im -8.2e-120)
     (* t_1 (sin (* y.im (log (hypot x.im x.re)))))
     (if (<= y.im 5.2e+65)
       (* t_1 (sin (pow (cbrt (fma t_0 y.im t_2)) 3.0)))
       (* t_1 (sin t_2))))))

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));
	double t_1 = exp(((t_0 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= -8.2e-120) {
		tmp = t_1 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else if (y_46_im <= 5.2e+65) {
		tmp = t_1 * sin(pow(cbrt(fma(t_0, y_46_im, t_2)), 3.0));
	} else {
		tmp = t_1 * sin(t_2);
	}
	return tmp;
}

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))
	t_1 = exp(Float64(Float64(t_0 * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= -8.2e-120)
		tmp = Float64(t_1 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	elseif (y_46_im <= 5.2e+65)
		tmp = Float64(t_1 * sin((cbrt(fma(t_0, y_46_im, t_2)) ^ 3.0)));
	else
		tmp = Float64(t_1 * sin(t_2));
	end
	return tmp
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]}, Block[{t$95$1 = N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.2e-120], N[(t$95$1 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.2e+65], N[(t$95$1 * N[Sin[N[Power[N[Power[N[(t$95$0 * y$46$im + t$95$2), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.20000000000000068e-120

   1. Initial program 37.6%

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

   2. Simplified 77.2%

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

      [Start] 37.6%	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. Taylor expanded in y.im around inf 40.1%

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

   4. Simplified 82.5%

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

      [Start] 40.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 40.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 40.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]
      hypot-def [=>] 82.5%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   if -8.20000000000000068e-120 < y.im < 5.20000000000000005e65

   1. Initial program 44.3%

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

   2. Simplified 91.1%

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

      [Start] 44.3%	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. Applied egg-rr 93.9%

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

      [Start] 91.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      fma-udef [=>] 91.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
      hypot-udef [=>] 44.3%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
      *-commutative [<=] 44.3%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]
      add-cube-cbrt [=>] 46.1%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)} \]
      pow3 [=>] 46.9%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)}^{3}\right)} \]
      hypot-udef [<=] 93.9%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)}^{3}\right) \]
      *-commutative [=>] 93.9%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{3}\right) \]
      fma-udef [<=] 93.9%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{\color{blue}{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right)}^{3}\right) \]
      *-commutative [<=] 93.9%	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)}\right)}^{3}\right) \]

   if 5.20000000000000005e65 < y.im

   1. Initial program 30.8%

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

   2. Simplified 61.6%

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

      [Start] 30.8%	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. Taylor expanded in y.im around 0 80.9%

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

4. Final simplification 87.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-120}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+65}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	79.7%
Cost	71816

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-120}:\\ \;\;\;\;t_1 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{+65}:\\ \;\;\;\;t_1 \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(t_0, y.im, t_2\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin t_2\\ \end{array} \]

Alternative 2
Accuracy	79.9%
Cost	78212

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq 5.2 \cdot 10^{+65}:\\ \;\;\;\;t_1 \cdot \sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt[3]{y.re}\right)}^{2}, \sqrt[3]{y.re}, y.im \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 3
Accuracy	78.3%
Cost	58632

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-121}:\\ \;\;\;\;t_1 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+47}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{\mathsf{fma}\left(t_0, y.im, t_2\right)}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin t_2\\ \end{array} \]

Alternative 4
Accuracy	78.7%
Cost	52616

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{t_0 \cdot y.re - t_1}\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{-114}:\\ \;\;\;\;t_2 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+47}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{t_1 + 1} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, t_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sin t_3\\ \end{array} \]

Alternative 5
Accuracy	77.8%
Cost	45828

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := e^{t_1 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -310:\\ \;\;\;\;t_2 \cdot \sin \left(\left|t_0\right|\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+47}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(t_1, y.im, t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sin t_0\\ \end{array} \]

Alternative 6
Accuracy	78.6%
Cost	45828

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -1.42 \cdot 10^{-115}:\\ \;\;\;\;t_1 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+47}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, t_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin t_2\\ \end{array} \]

Alternative 7
Accuracy	77.9%
Cost	45769

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -390 \lor \neg \left(y.im \leq 7 \cdot 10^{+47}\right):\\ \;\;\;\;e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, t_1\right)\right)\\ \end{array} \]

Alternative 8
Accuracy	71.4%
Cost	39888

\[\begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_3 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - t_0} \cdot \sin t_1\\ t_4 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \frac{t_2}{t_0 + 1}\\ \mathbf{if}\;y.im \leq -470:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-181}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-212}:\\ \;\;\;\;t_2 \cdot \sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+47}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Accuracy	71.4%
Cost	39824

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_2 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin t_0\\ t_3 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_1\\ \mathbf{if}\;y.im \leq -450:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-215}:\\ \;\;\;\;t_1 \cdot \sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+47}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 10
Accuracy	58.0%
Cost	33420

\[\begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t_1 \cdot e^{\log \left(\frac{-1}{x.re}\right) \cdot \left(-y.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_3 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_0\\ \mathbf{if}\;x.re \leq -1.3 \cdot 10^{-135}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq -7.6 \cdot 10^{-238}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;t_0 \cdot \sin \left(t_1 + y.im \cdot \log x.re\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 11
Accuracy	57.3%
Cost	33360

\[\begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_3 := t_2 \cdot \frac{t_0}{t_1 + 1}\\ t_4 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_0\\ \mathbf{if}\;y.im \leq -2.85 \cdot 10^{+228}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{-181}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-233}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{+165}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log x.im - t_1}\\ \end{array} \]

Alternative 12
Accuracy	56.1%
Cost	33228

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_2 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_1\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-181}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-233}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+165}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]

Alternative 13
Accuracy	56.4%
Cost	33156

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;x.im \leq -6.2 \cdot 10^{-53}:\\ \;\;\;\;\sin t_0 \cdot e^{\log \left(\frac{-1}{x.im}\right) \cdot \left(-y.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.im \leq 5.6 \cdot 10^{-211}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(t_0 + y.im \cdot \log x.im\right)\\ \end{array} \]

Alternative 14
Accuracy	54.1%
Cost	32777

\[\begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -5.5 \cdot 10^{-181} \lor \neg \left(y.im \leq 4.2 \cdot 10^{-233}\right):\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t_0\\ \end{array} \]

Alternative 15
Accuracy	45.4%
Cost	32644

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq 4.2 \cdot 10^{+16}:\\ \;\;\;\;\sin t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(\left|t_0\right|\right)\\ \end{array} \]

Alternative 16
Accuracy	45.6%
Cost	26112

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

Reproduce

herbie shell --seed 2023263 
(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)))))