powComplex, imaginary part

Percentage Accurate: 42.2% → 78.4%

Time: 46.6s

Precision: binary64

Cost: 110600

• 35.1% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t_0\right)}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := e^{y.re \cdot t_2 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_4 := t_3 \cdot \sin t_0\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+27}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq 10^{+16}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(t_1 \cdot {t_1}^{2}\right)\\ \mathbf{elif}\;y.re \leq 1.62 \cdot 10^{+184}:\\ \;\;\;\;t_3 \cdot \log \left(e^{\sin \left(\mathsf{fma}\left(t_2, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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 (* y.im (log (hypot x.im x.re))))
        (t_1 (cbrt (fma y.re (atan2 x.im x.re) t_0)))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (exp (- (* y.re t_2) (* (atan2 x.im x.re) y.im))))
        (t_4 (* t_3 (sin t_0))))
   (if (<= y.re -3.6e+27)
     t_4
     (if (<= y.re 1e+16)
       (*
        (/ (pow (hypot x.re x.im) y.re) (pow (exp y.im) (atan2 x.im x.re)))
        (sin (* t_1 (pow t_1 2.0))))
       (if (<= y.re 1.62e+184)
         (* t_3 (log (exp (sin (fma t_2 y.im (* y.re (atan2 x.im x.re)))))))
         t_4)))))

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 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_1 = cbrt(fma(y_46_re, atan2(x_46_im, x_46_re), t_0));
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = exp(((y_46_re * t_2) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_4 = t_3 * sin(t_0);
	double tmp;
	if (y_46_re <= -3.6e+27) {
		tmp = t_4;
	} else if (y_46_re <= 1e+16) {
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / pow(exp(y_46_im), atan2(x_46_im, x_46_re))) * sin((t_1 * pow(t_1, 2.0)));
	} else if (y_46_re <= 1.62e+184) {
		tmp = t_3 * log(exp(sin(fma(t_2, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))))));
	} else {
		tmp = t_4;
	}
	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 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_1 = cbrt(fma(y_46_re, atan(x_46_im, x_46_re), t_0))
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = exp(Float64(Float64(y_46_re * t_2) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_4 = Float64(t_3 * sin(t_0))
	tmp = 0.0
	if (y_46_re <= -3.6e+27)
		tmp = t_4;
	elseif (y_46_re <= 1e+16)
		tmp = Float64(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / (exp(y_46_im) ^ atan(x_46_im, x_46_re))) * sin(Float64(t_1 * (t_1 ^ 2.0))));
	elseif (y_46_re <= 1.62e+184)
		tmp = Float64(t_3 * log(exp(sin(fma(t_2, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))))));
	else
		tmp = t_4;
	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[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(y$46$re * t$95$2), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.6e+27], t$95$4, If[LessEqual[y$46$re, 1e+16], N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.62e+184], N[(t$95$3 * N[Log[N[Exp[N[Sin[N[(t$95$2 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.59999999999999983e27 or 1.61999999999999999e184 < y.re

   1. Initial program 53.2%

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

   2. Simplified 74.7%

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

      [Start] 53.2	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. Taylor expanded in y.re around 0 58.2%

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

   4. Simplified 84.8%

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

      [Start] 58.2	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 58.2	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]
      unpow2 [=>] 58.2	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]
      hypot-def [=>] 84.8	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   if -3.59999999999999983e27 < y.re < 1e16

   1. Initial program 49.5%

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

   2. Simplified 86.2%

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

      [Start] 49.5	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      exp-diff [=>] 49.5	\[ \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      +-rgt-identity [<=] 49.5	\[ \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      +-rgt-identity [=>] 49.5	\[ \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      exp-to-pow [=>] 49.5	\[ \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      hypot-def [=>] 49.5	\[ \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      *-commutative [=>] 49.5	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      exp-prod [=>] 49.5	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
      +-commutative [=>] 49.5	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]
      *-commutative [=>] 49.5	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Applied egg-rr 88.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}\right)} \]
      Step-by-step derivation

      [Start] 86.2	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \]
      add-cube-cbrt [=>] 88.1	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}\right)} \]
      pow2 [=>] 88.1	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}\right)}^{2}} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}\right) \]
      hypot-udef [=>] 50.3	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}\right) \]
      +-commutative [=>] 50.3	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}\right) \]
      hypot-def [=>] 88.1	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}\right) \]
      hypot-udef [=>] 50.3	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)}\right) \]
      +-commutative [=>] 50.3	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)}\right) \]
      hypot-def [=>] 88.1	\[ \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}\right)}^{2} \cdot \sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot y.im\right)}\right) \]

   if 1e16 < y.re < 1.61999999999999999e184

   1. Initial program 47.5%

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

   2. Simplified 85.0%

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

      [Start] 47.5	\[ e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. Applied egg-rr 85.0%

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

      [Start] 85.0	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
      add-log-exp [=>] 85.0	\[ e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\log \left(e^{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+27}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 10^{+16}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \cdot {\left(\sqrt[3]{\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}\right)\\ \mathbf{elif}\;y.re \leq 1.62 \cdot 10^{+184}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \log \left(e^{\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	81.0%
Cost	58820

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{y.re \cdot t_0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq 1.8 \cdot 10^{+184}:\\ \;\;\;\;t_1 \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 2
Accuracy	80.2%
Cost	52617

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.im \leq -0.052 \lor \neg \left(y.im \leq 2 \cdot 10^{-122}\right):\\ \;\;\;\;e^{y.re \cdot t_0 - t_1} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\frac{t_1 + 1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]

Alternative 3
Accuracy	80.2%
Cost	45961

\[\begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.im \leq -0.052 \lor \neg \left(y.im \leq 10^{-122}\right):\\ \;\;\;\;e^{y.re \cdot t_0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]

Alternative 4
Accuracy	78.9%
Cost	45896

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := e^{y.re \cdot t_1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.im \leq -16000:\\ \;\;\;\;t_2 \cdot t_0\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(t_1, y.im, t_0\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sin t_0\\ \end{array} \]

Alternative 5
Accuracy	72.1%
Cost	39824

\[\begin{array}{l} t_0 := \frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ t_1 := e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \sin t_2\\ \mathbf{if}\;y.im \leq -160000:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-238}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-191}:\\ \;\;\;\;t_3 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_3\\ \end{array} \]

Alternative 6
Accuracy	57.3%
Cost	33432

\[\begin{array}{l} t_0 := \frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_1\right)\right)\\ t_3 := \sin t_1\\ \mathbf{if}\;y.re \leq -82000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq -1.95 \cdot 10^{-184}:\\ \;\;\;\;\sqrt[3]{{t_1}^{3}}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+30}:\\ \;\;\;\;t_3 \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+184}:\\ \;\;\;\;t_3 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	72.4%
Cost	33424

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ t_2 := e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t_0\\ \mathbf{if}\;y.im \leq -10000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-240}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-196}:\\ \;\;\;\;\sin t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 560000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 8
Accuracy	56.3%
Cost	32976

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t_0\right)\right)\\ t_2 := \sin t_0\\ t_3 := t_2 \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.1 \cdot 10^{-17}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq -3.7 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-186}:\\ \;\;\;\;\sqrt[3]{{t_0}^{3}}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-31}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+31}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 9
Accuracy	47.2%
Cost	27100

\[\begin{array}{l} t_0 := y.im \cdot \log x.re\\ t_1 := \sin t_0\\ t_2 := \frac{t_1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := t_3 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_5 := {x.re}^{y.re} \cdot \sin \left(t_3 + t_0\right)\\ \mathbf{if}\;x.re \leq 1.26 \cdot 10^{-303}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.re \leq 9.6 \cdot 10^{-237}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq 1.08 \cdot 10^{-147}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq 4 \cdot 10^{-109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq 7.8 \cdot 10^{-16}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.re \leq 1.92 \cdot 10^{+56}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{+100}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]

Alternative 10
Accuracy	46.6%
Cost	26968

\[\begin{array}{l} t_0 := \sin \left(y.im \cdot \log x.re\right)\\ t_1 := \frac{t_0}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_4 := \sin t_2\\ \mathbf{if}\;x.re \leq 2 \cdot 10^{-306}:\\ \;\;\;\;t_2 \cdot t_3\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{-238}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 3.4 \cdot 10^{-158}:\\ \;\;\;\;t_4 \cdot {x.re}^{y.re}\\ \mathbf{elif}\;x.re \leq 2.1 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;t_4 \cdot t_3\\ \mathbf{elif}\;x.re \leq 2.5 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{\frac{1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]

Alternative 11
Accuracy	46.6%
Cost	26508

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ t_2 := t_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.9 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-108}:\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(t_0\right)\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+32}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 12
Accuracy	47.3%
Cost	26508

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -9.2 \cdot 10^{-141}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+31}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 13
Accuracy	45.0%
Cost	20040

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-149}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{t_0}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin t_0}{{\left(\frac{1}{x.re}\right)}^{y.re}}\\ \end{array} \]

Alternative 14
Accuracy	38.3%
Cost	19984

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \mathsf{expm1}\left(t_0\right)\\ t_2 := \mathsf{log1p}\left(t_1\right)\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+67}:\\ \;\;\;\;\sin t_0 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-107}:\\ \;\;\;\;\log \left(1 + t_1\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-17}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 15
Accuracy	39.9%
Cost	19913

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -0.122 \lor \neg \left(y.re \leq 1.05 \cdot 10^{-20}\right):\\ \;\;\;\;\sin t_0 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{t_0}^{3}}\\ \end{array} \]

Alternative 16
Accuracy	45.0%
Cost	19912

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{-148}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-20}:\\ \;\;\;\;\sqrt[3]{{t_0}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 17
Accuracy	36.0%
Cost	19785

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -5.7 \cdot 10^{+110} \lor \neg \left(y.re \leq 8 \cdot 10^{-18}\right):\\ \;\;\;\;t_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{t_0}^{3}}\\ \end{array} \]

Alternative 18
Accuracy	32.9%
Cost	19721

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -5.2 \cdot 10^{+144} \lor \neg \left(y.re \leq 1.02 \cdot 10^{-17}\right):\\ \;\;\;\;t_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \end{array} \]

Alternative 19
Accuracy	36.7%
Cost	13513

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -90000000000 \lor \neg \left(y.re \leq 1.02 \cdot 10^{-17}\right):\\ \;\;\;\;t_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 20
Accuracy	14.2%
Cost	13124

\[\begin{array}{l} \mathbf{if}\;x.re \leq 2.85 \cdot 10^{+253}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 21
Accuracy	13.7%
Cost	6656

\[y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} \]

Reproduce

herbie shell --seed 2023160 
(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)))))