powComplex, imaginary part

Percentage Accurate: 40.5% → 78.5%

Time: 43.8s

Alternatives: 23

Speedup: 2.0×

• 34.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * sin(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * 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(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

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

Initial Program: 40.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * sin(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * 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(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

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

Alternative 1: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, t_0\right)}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_4 := e^{t_3 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_5 := t_4 \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.im, t_3, t_2\right)}\right)}^{3}\right)\\ t_6 := {\left(\sqrt[3]{{t_1}^{2}}\right)}^{4}\\ \mathbf{if}\;y.re \leq -1.56 \cdot 10^{+87}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-198}:\\ \;\;\;\;t_4 \cdot \sin \left(t_6 \cdot \sqrt[3]{t_1}\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+267}:\\ \;\;\;\;t_5\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{+299}:\\ \;\;\;\;t_4 \cdot \sin t_0\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \sin \left(t_6 \cdot {\left(t_2 + y.im \cdot \log x.im\right)}^{0.1111111111111111}\right)\\ \end{array} \end{array} \]

FPCore

(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 (atan2 x.im x.re) y.re t_0)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (log (hypot x.re x.im)))
        (t_4 (exp (- (* t_3 y.re) (* y.im (atan2 x.im x.re)))))
        (t_5 (* t_4 (sin (pow (cbrt (fma y.im t_3 t_2)) 3.0))))
        (t_6 (pow (cbrt (pow t_1 2.0)) 4.0)))
   (if (<= y.re -1.56e+87)
     t_5
     (if (<= y.re 2.7e-198)
       (* t_4 (sin (* t_6 (cbrt t_1))))
       (if (<= y.re 5.8e+267)
         t_5
         (if (<= y.re 2.35e+299)
           (* t_4 (sin t_0))
           (*
            t_4
            (sin
             (*
              t_6
              (pow (+ t_2 (* y.im (log x.im))) 0.1111111111111111))))))))))

C

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(atan2(x_46_im, x_46_re), y_46_re, t_0));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = log(hypot(x_46_re, x_46_im));
	double t_4 = exp(((t_3 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_5 = t_4 * sin(pow(cbrt(fma(y_46_im, t_3, t_2)), 3.0));
	double t_6 = pow(cbrt(pow(t_1, 2.0)), 4.0);
	double tmp;
	if (y_46_re <= -1.56e+87) {
		tmp = t_5;
	} else if (y_46_re <= 2.7e-198) {
		tmp = t_4 * sin((t_6 * cbrt(t_1)));
	} else if (y_46_re <= 5.8e+267) {
		tmp = t_5;
	} else if (y_46_re <= 2.35e+299) {
		tmp = t_4 * sin(t_0);
	} else {
		tmp = t_4 * sin((t_6 * pow((t_2 + (y_46_im * log(x_46_im))), 0.1111111111111111)));
	}
	return tmp;
}

Julia

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(atan(x_46_im, x_46_re), y_46_re, t_0))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = log(hypot(x_46_re, x_46_im))
	t_4 = exp(Float64(Float64(t_3 * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_5 = Float64(t_4 * sin((cbrt(fma(y_46_im, t_3, t_2)) ^ 3.0)))
	t_6 = cbrt((t_1 ^ 2.0)) ^ 4.0
	tmp = 0.0
	if (y_46_re <= -1.56e+87)
		tmp = t_5;
	elseif (y_46_re <= 2.7e-198)
		tmp = Float64(t_4 * sin(Float64(t_6 * cbrt(t_1))));
	elseif (y_46_re <= 5.8e+267)
		tmp = t_5;
	elseif (y_46_re <= 2.35e+299)
		tmp = Float64(t_4 * sin(t_0));
	else
		tmp = Float64(t_4 * sin(Float64(t_6 * (Float64(t_2 + Float64(y_46_im * log(x_46_im))) ^ 0.1111111111111111))));
	end
	return tmp
end

Wolfram

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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + t$95$0), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sin[N[Power[N[Power[N[(y$46$im * t$95$3 + t$95$2), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], 1/3], $MachinePrecision], 4.0], $MachinePrecision]}, If[LessEqual[y$46$re, -1.56e+87], t$95$5, If[LessEqual[y$46$re, 2.7e-198], N[(t$95$4 * N[Sin[N[(t$95$6 * N[Power[t$95$1, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e+267], t$95$5, If[LessEqual[y$46$re, 2.35e+299], N[(t$95$4 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[Sin[N[(t$95$6 * N[Power[N[(t$95$2 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.1111111111111111], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.5600000000000001e87 or 2.7000000000000002e-198 < y.re < 5.79999999999999966e267

   1. Initial program 33.1%

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

   3. Simplified 76.6%

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

      1. add-cube-cbrt 80.1%

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

      2. pow3 85.4%

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

      3. fma-udef 85.4%

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

      4. *-commutative 85.4%

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

      5. *-commutative 85.4%

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

      6. fma-def 85.4%

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

   5. Applied egg-rr 85.4%

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

   if -1.5600000000000001e87 < y.re < 2.7000000000000002e-198

   1. Initial program 40.7%

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

   3. Simplified 73.4%

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

      1. add-cube-cbrt 74.8%

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

      2. pow3 73.0%

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

      3. fma-udef 72.9%

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

      4. *-commutative 72.9%

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

      5. *-commutative 72.9%

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

      6. fma-def 73.0%

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

   5. Applied egg-rr 73.0%

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

      1. unpow3 74.8%

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

      2. add-cube-cbrt 75.2%

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

      3. associate-*r* 74.3%

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

   7. Applied egg-rr 77.1%

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

   9. Simplified 84.1%

      \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left({\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)} \]

   if 5.79999999999999966e267 < y.re < 2.35e299

   1. Initial program 44.4%

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

   3. Simplified 66.7%

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

   4. Taylor expanded in y.im around inf 44.4%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

      3. hypot-def 88.9%

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

   6. Simplified 88.9%

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

   if 2.35e299 < y.re

   1. Initial program 33.3%

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

   3. Simplified 33.3%

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

      1. add-cube-cbrt 33.3%

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

      2. pow3 66.7%

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

      3. fma-udef 66.7%

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

      4. *-commutative 66.7%

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

      5. *-commutative 66.7%

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

      6. fma-def 66.7%

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

   5. Applied egg-rr 66.7%

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

      1. unpow3 33.3%

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

      2. add-cube-cbrt 33.3%

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

      3. associate-*r* 33.3%

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

   7. Applied egg-rr 33.3%

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

   9. Simplified 66.7%

      \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left({\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)} \]

   10. Taylor expanded in x.re around 0 100.0%

       \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)}^{0.1111111111111111}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.56 \cdot 10^{+87}:\\ \;\;\;\;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(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-198}:\\ \;\;\;\;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]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+267}:\\ \;\;\;\;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(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{+299}:\\ \;\;\;\;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{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({\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot {\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)}^{0.1111111111111111}\right)\\ \end{array} \]

Alternative 2: 78.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot t_0\right)}}\\ t_2 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_3 := t_2 \cdot \sin \left(t_1 \cdot {\left({t_1}^{2}\right)}^{4}\right)\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{-85}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-56}:\\ \;\;\;\;t_2 \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.im, t_0, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{elif}\;y.im \leq 5.9 \cdot 10^{+40}:\\ \;\;\;\;t_2 \cdot \sin \left(\left|\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im)))
        (t_1 (cbrt (cbrt (fma (atan2 x.im x.re) y.re (* y.im t_0)))))
        (t_2 (exp (- (* t_0 y.re) (* y.im (atan2 x.im x.re)))))
        (t_3 (* t_2 (sin (* t_1 (pow (pow t_1 2.0) 4.0))))))
   (if (<= y.im -1.15e-85)
     t_3
     (if (<= y.im 5.2e-56)
       (* t_2 (sin (pow (cbrt (fma y.im t_0 (* y.re (atan2 x.im x.re)))) 3.0)))
       (if (<= y.im 5.9e+40)
         (*
          t_2
          (sin
           (fabs
            (fma (atan2 x.im x.re) y.re (* y.im (log (hypot x.im x.re)))))))
         t_3)))))

C

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 = cbrt(cbrt(fma(atan2(x_46_im, x_46_re), y_46_re, (y_46_im * t_0))));
	double t_2 = exp(((t_0 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_3 = t_2 * sin((t_1 * pow(pow(t_1, 2.0), 4.0)));
	double tmp;
	if (y_46_im <= -1.15e-85) {
		tmp = t_3;
	} else if (y_46_im <= 5.2e-56) {
		tmp = t_2 * sin(pow(cbrt(fma(y_46_im, t_0, (y_46_re * atan2(x_46_im, x_46_re)))), 3.0));
	} else if (y_46_im <= 5.9e+40) {
		tmp = t_2 * sin(fabs(fma(atan2(x_46_im, x_46_re), y_46_re, (y_46_im * log(hypot(x_46_im, x_46_re))))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

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 = cbrt(cbrt(fma(atan(x_46_im, x_46_re), y_46_re, Float64(y_46_im * t_0))))
	t_2 = exp(Float64(Float64(t_0 * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_3 = Float64(t_2 * sin(Float64(t_1 * ((t_1 ^ 2.0) ^ 4.0))))
	tmp = 0.0
	if (y_46_im <= -1.15e-85)
		tmp = t_3;
	elseif (y_46_im <= 5.2e-56)
		tmp = Float64(t_2 * sin((cbrt(fma(y_46_im, t_0, Float64(y_46_re * atan(x_46_im, x_46_re)))) ^ 3.0)));
	elseif (y_46_im <= 5.9e+40)
		tmp = Float64(t_2 * sin(abs(fma(atan(x_46_im, x_46_re), y_46_re, Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

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[Power[N[Power[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = 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$3 = N[(t$95$2 * N[Sin[N[(t$95$1 * N[Power[N[Power[t$95$1, 2.0], $MachinePrecision], 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.15e-85], t$95$3, If[LessEqual[y$46$im, 5.2e-56], N[(t$95$2 * N[Sin[N[Power[N[Power[N[(y$46$im * t$95$0 + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.9e+40], N[(t$95$2 * N[Sin[N[Abs[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.15e-85 or 5.8999999999999999e40 < y.im

   1. Initial program 33.3%

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

   3. Simplified 65.2%

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

      1. add-cube-cbrt 69.1%

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

      2. pow3 69.8%

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

      3. fma-udef 69.8%

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

      4. *-commutative 69.8%

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

      5. *-commutative 69.8%

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

      6. fma-def 69.8%

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

   5. Applied egg-rr 69.8%

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

      1. unpow3 69.1%

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

      2. add-cube-cbrt 69.0%

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

      3. associate-*r* 66.8%

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

   7. Applied egg-rr 70.6%

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

   9. Simplified 75.1%

      \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left({\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 75.1%

          \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)\right)\right)} \]

       2. expm1-udef 70.9%

          \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left({\left(\sqrt[3]{{\left(\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}\right)}^{2}}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)\right)} - 1\right)} \]

   11. Applied egg-rr 74.5%

       \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left({\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)}^{2}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)\right)} - 1\right)} \]
   12. Step-by-step derivation

       1. expm1-def 78.7%

          \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left({\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)}^{2}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)\right)\right)} \]

       2. expm1-log1p 78.7%

          \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left({\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)}^{2}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}\right)} \]

   13. Simplified 78.7%

       \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left({\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}}\right)}^{2}\right)}^{4} \cdot \sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}}\right)} \]

   if -1.15e-85 < y.im < 5.19999999999999994e-56

   1. Initial program 41.8%

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

   3. Simplified 85.7%

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

      1. add-cube-cbrt 84.8%

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

      2. pow3 88.5%

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

      3. fma-udef 88.5%

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

      4. *-commutative 88.5%

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

      5. *-commutative 88.5%

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

      6. fma-def 88.5%

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

   5. Applied egg-rr 88.5%

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

   if 5.19999999999999994e-56 < y.im < 5.8999999999999999e40

   1. Initial program 31.3%

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

   3. Simplified 75.0%

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

      1. add-sqr-sqrt 68.6%

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

      2. sqrt-unprod 68.8%

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

      3. pow2 68.8%

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

      4. fma-udef 68.8%

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

      5. *-commutative 68.8%

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

      6. *-commutative 68.8%

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

      7. fma-def 68.8%

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

   5. Applied egg-rr 68.8%

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

      1. unpow2 68.8%

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

      2. rem-sqrt-square 100.0%

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

      3. fma-def 100.0%

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

      4. +-commutative 100.0%

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

      5. fma-def 99.9%

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

      6. hypot-def 31.2%

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

      7. unpow2 31.2%

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

      8. unpow2 31.2%

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

      9. +-commutative 31.2%

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

      10. unpow2 31.2%

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

      11. unpow2 31.2%

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

      12. hypot-def 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{-85}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}}\right)}^{2}\right)}^{4}\right)\\ \mathbf{elif}\;y.im \leq 5.2 \cdot 10^{-56}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{elif}\;y.im \leq 5.9 \cdot 10^{+40}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\left|\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}} \cdot {\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}}\right)}^{2}\right)}^{4}\right)\\ \end{array} \]

Alternative 3: 79.3% accurate, 0.7× speedup

Math

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

FPCore

(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) (* y.im (atan2 x.im x.re))))
    (sin (pow (cbrt (fma y.im t_0 (* 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) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	return exp(((t_0 * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(pow(cbrt(fma(y_46_im, t_0, (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)
	t_0 = log(hypot(x_46_re, x_46_im))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin((cbrt(fma(y_46_im, t_0, 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_] := 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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[Power[N[Power[N[(y$46$im * t$95$0 + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.8%

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

3. Simplified 74.4%

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

   1. add-cube-cbrt 75.6%

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

   2. pow3 77.9%

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

   3. fma-udef 77.9%

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

   4. *-commutative 77.9%

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

   5. *-commutative 77.9%

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

   6. fma-def 77.9%

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

5. Applied egg-rr 77.9%

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

6. Final simplification 77.9%

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

Alternative 4: 78.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{t_0 \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.re \leq -5.2 \cdot 10^{+182}:\\ \;\;\;\;t_1 \cdot \sin \left(\left|\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(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))))))
   (if (<= x.re -5.2e+182)
     (*
      t_1
      (sin
       (fabs (fma (atan2 x.im x.re) y.re (* y.im (log (hypot x.im x.re)))))))
     (* t_1 (sin (fma t_0 y.im (* y.re (atan2 x.im x.re))))))))

C

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 tmp;
	if (x_46_re <= -5.2e+182) {
		tmp = t_1 * sin(fabs(fma(atan2(x_46_im, x_46_re), y_46_re, (y_46_im * log(hypot(x_46_im, x_46_re))))));
	} else {
		tmp = t_1 * sin(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

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))))
	tmp = 0.0
	if (x_46_re <= -5.2e+182)
		tmp = Float64(t_1 * sin(abs(fma(atan(x_46_im, x_46_re), y_46_re, Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))));
	else
		tmp = Float64(t_1 * sin(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

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]}, If[LessEqual[x$46$re, -5.2e+182], N[(t$95$1 * N[Sin[N[Abs[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -5.2e182

   1. Initial program 0.0%

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

   3. Simplified 56.5%

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

      1. add-sqr-sqrt 39.1%

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

      2. sqrt-unprod 35.3%

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

      3. pow2 35.3%

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

      4. fma-udef 35.3%

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

      5. *-commutative 35.3%

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

      6. *-commutative 35.3%

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

      7. fma-def 35.3%

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

   5. Applied egg-rr 35.3%

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

      1. unpow2 35.3%

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

      2. rem-sqrt-square 74.2%

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

      3. fma-def 74.2%

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

      4. +-commutative 74.2%

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

      5. fma-def 74.2%

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

      6. hypot-def 0.0%

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

      7. unpow2 0.0%

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

      8. unpow2 0.0%

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

      9. +-commutative 0.0%

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

      10. unpow2 0.0%

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

      11. unpow2 0.0%

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

      12. hypot-def 74.2%

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

   7. Simplified 74.2%

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

   if -5.2e182 < x.re

   1. Initial program 40.4%

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

   3. Simplified 76.1%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5.2 \cdot 10^{+182}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\left|\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternative 5: 79.8% accurate, 0.9× speedup

Math

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

FPCore

(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) (* y.im (atan2 x.im x.re))))
    (sin (fma t_0 y.im (* y.re (atan2 x.im x.re)))))))

C

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) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
}

Julia

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(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))))
end

Wolfram

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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 36.8%

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

3. Simplified 74.4%

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

4. Final simplification 74.4%

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

Alternative 6: 71.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{x.im}\right)\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_4 := e^{t_3 \cdot y.re - t_1}\\ \mathbf{if}\;x.im \leq -7.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sin \left(t_2 - y.im \cdot t_0\right)}{e^{t_1 + y.re \cdot t_0}}\\ \mathbf{elif}\;x.im \leq 2.15 \cdot 10^{-302}:\\ \;\;\;\;t_4 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{-83}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_3, y.im, t_2\right)\right) \cdot e^{y.re \cdot \log x.im - t_1}\\ \mathbf{elif}\;x.im \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_1} \cdot \sin t_2\\ \mathbf{else}:\\ \;\;\;\;t_4 \cdot \sin \left(t_2 + y.im \cdot \log x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (/ -1.0 x.im)))
        (t_1 (* y.im (atan2 x.im x.re)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (log (hypot x.re x.im)))
        (t_4 (exp (- (* t_3 y.re) t_1))))
   (if (<= x.im -7.7e-32)
     (/ (sin (- t_2 (* y.im t_0))) (exp (+ t_1 (* y.re t_0))))
     (if (<= x.im 2.15e-302)
       (* t_4 (sin (* y.im (log (hypot x.im x.re)))))
       (if (<= x.im 6e-83)
         (* (sin (fma t_3 y.im t_2)) (exp (- (* y.re (log x.im)) t_1)))
         (if (<= x.im 1.7e-9)
           (*
            (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_1))
            (sin t_2))
           (* t_4 (sin (+ t_2 (* y.im (log x.im)))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log((-1.0 / x_46_im));
	double t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = log(hypot(x_46_re, x_46_im));
	double t_4 = exp(((t_3 * y_46_re) - t_1));
	double tmp;
	if (x_46_im <= -7.7e-32) {
		tmp = sin((t_2 - (y_46_im * t_0))) / exp((t_1 + (y_46_re * t_0)));
	} else if (x_46_im <= 2.15e-302) {
		tmp = t_4 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else if (x_46_im <= 6e-83) {
		tmp = sin(fma(t_3, y_46_im, t_2)) * exp(((y_46_re * log(x_46_im)) - t_1));
	} else if (x_46_im <= 1.7e-9) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1)) * sin(t_2);
	} else {
		tmp = t_4 * sin((t_2 + (y_46_im * log(x_46_im))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(Float64(-1.0 / x_46_im))
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = log(hypot(x_46_re, x_46_im))
	t_4 = exp(Float64(Float64(t_3 * y_46_re) - t_1))
	tmp = 0.0
	if (x_46_im <= -7.7e-32)
		tmp = Float64(sin(Float64(t_2 - Float64(y_46_im * t_0))) / exp(Float64(t_1 + Float64(y_46_re * t_0))));
	elseif (x_46_im <= 2.15e-302)
		tmp = Float64(t_4 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	elseif (x_46_im <= 6e-83)
		tmp = Float64(sin(fma(t_3, y_46_im, t_2)) * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_1)));
	elseif (x_46_im <= 1.7e-9)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_1)) * sin(t_2));
	else
		tmp = Float64(t_4 * sin(Float64(t_2 + Float64(y_46_im * log(x_46_im)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -7.7e-32], N[(N[Sin[N[(t$95$2 - N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$1 + N[(y$46$re * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.15e-302], N[(t$95$4 * 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[x$46$im, 6e-83], N[(N[Sin[N[(t$95$3 * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.7e-9], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision], N[(t$95$4 * N[Sin[N[(t$95$2 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.im < -7.6999999999999997e-32

   1. Initial program 31.9%

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

      1. exp-diff 27.5%

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

      2. +-rgt-identity 27.5%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 27.5%

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

      4. exp-to-pow 27.5%

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

      5. hypot-def 27.5%

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

      6. *-commutative 27.5%

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

      7. exp-prod 26.1%

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

      8. fma-def 26.1%

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

      9. hypot-def 57.9%

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

      10. *-commutative 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in x.im around -inf 60.2%

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

      1. associate-/l* 60.2%

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

      2. +-commutative 60.2%

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

      3. mul-1-neg 60.2%

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

      4. unsub-neg 60.2%

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

      5. *-commutative 60.2%

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

      6. *-commutative 60.2%

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

      7. *-commutative 60.2%

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

      8. div-exp 68.9%

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

      9. *-commutative 68.9%

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

      10. mul-1-neg 68.9%

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

      11. *-commutative 68.9%

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

   6. Simplified 68.9%

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

   if -7.6999999999999997e-32 < x.im < 2.1500000000000001e-302

   1. Initial program 38.7%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in y.im around inf 40.7%

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

      1. unpow2 40.7%

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

      2. unpow2 40.7%

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

      3. hypot-def 67.2%

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

   6. Simplified 67.2%

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

   if 2.1500000000000001e-302 < x.im < 6.00000000000000021e-83

   1. Initial program 43.4%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in x.re around 0 74.8%

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

   if 6.00000000000000021e-83 < x.im < 1.6999999999999999e-9

   1. Initial program 57.0%

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

   2. Taylor expanded in y.im around 0 86.2%

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

   if 1.6999999999999999e-9 < x.im

   1. Initial program 32.3%

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

   3. Simplified 80.3%

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

   4. Taylor expanded in x.re around 0 80.6%

      \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(\log x.im \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7.7 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \log \left(\frac{-1}{x.im}\right)}}\\ \mathbf{elif}\;x.im \leq 2.15 \cdot 10^{-302}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{-83}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.im \leq 1.7 \cdot 10^{-9}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \end{array} \]

Alternative 7: 73.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{-1}{x.im}\right)\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - t_1}\\ \mathbf{if}\;x.im \leq -5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sin \left(t_2 - y.im \cdot t_0\right)}{e^{t_1 + y.re \cdot t_0}}\\ \mathbf{elif}\;x.im \leq 3.1 \cdot 10^{-140}:\\ \;\;\;\;t_3 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sin \left(t_2 + y.im \cdot \log x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (/ -1.0 x.im)))
        (t_1 (* y.im (atan2 x.im x.re)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (exp (- (* (log (hypot x.re x.im)) y.re) t_1))))
   (if (<= x.im -5e-32)
     (/ (sin (- t_2 (* y.im t_0))) (exp (+ t_1 (* y.re t_0))))
     (if (<= x.im 3.1e-140)
       (* t_3 (sin (* y.im (log (hypot x.im x.re)))))
       (* t_3 (sin (+ t_2 (* y.im (log x.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log((-1.0 / x_46_im));
	double t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - t_1));
	double tmp;
	if (x_46_im <= -5e-32) {
		tmp = sin((t_2 - (y_46_im * t_0))) / exp((t_1 + (y_46_re * t_0)));
	} else if (x_46_im <= 3.1e-140) {
		tmp = t_3 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = t_3 * sin((t_2 + (y_46_im * log(x_46_im))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log((-1.0 / x_46_im));
	double t_1 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.exp(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_re) - t_1));
	double tmp;
	if (x_46_im <= -5e-32) {
		tmp = Math.sin((t_2 - (y_46_im * t_0))) / Math.exp((t_1 + (y_46_re * t_0)));
	} else if (x_46_im <= 3.1e-140) {
		tmp = t_3 * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = t_3 * Math.sin((t_2 + (y_46_im * Math.log(x_46_im))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log((-1.0 / x_46_im))
	t_1 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = math.exp(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_re) - t_1))
	tmp = 0
	if x_46_im <= -5e-32:
		tmp = math.sin((t_2 - (y_46_im * t_0))) / math.exp((t_1 + (y_46_re * t_0)))
	elif x_46_im <= 3.1e-140:
		tmp = t_3 * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	else:
		tmp = t_3 * math.sin((t_2 + (y_46_im * math.log(x_46_im))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(Float64(-1.0 / x_46_im))
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - t_1))
	tmp = 0.0
	if (x_46_im <= -5e-32)
		tmp = Float64(sin(Float64(t_2 - Float64(y_46_im * t_0))) / exp(Float64(t_1 + Float64(y_46_re * t_0))));
	elseif (x_46_im <= 3.1e-140)
		tmp = Float64(t_3 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = Float64(t_3 * sin(Float64(t_2 + Float64(y_46_im * log(x_46_im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log((-1.0 / x_46_im));
	t_1 = y_46_im * atan2(x_46_im, x_46_re);
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - t_1));
	tmp = 0.0;
	if (x_46_im <= -5e-32)
		tmp = sin((t_2 - (y_46_im * t_0))) / exp((t_1 + (y_46_re * t_0)));
	elseif (x_46_im <= 3.1e-140)
		tmp = t_3 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = t_3 * sin((t_2 + (y_46_im * log(x_46_im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -5e-32], N[(N[Sin[N[(t$95$2 - N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$1 + N[(y$46$re * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 3.1e-140], N[(t$95$3 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[Sin[N[(t$95$2 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -5e-32

   1. Initial program 31.9%

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

      1. exp-diff 27.5%

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

      2. +-rgt-identity 27.5%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 27.5%

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

      4. exp-to-pow 27.5%

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

      5. hypot-def 27.5%

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

      6. *-commutative 27.5%

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

      7. exp-prod 26.1%

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

      8. fma-def 26.1%

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

      9. hypot-def 57.9%

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

      10. *-commutative 57.9%

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

   3. Simplified 57.9%

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

   4. Taylor expanded in x.im around -inf 60.2%

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

      1. associate-/l* 60.2%

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

      2. +-commutative 60.2%

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

      3. mul-1-neg 60.2%

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

      4. unsub-neg 60.2%

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

      5. *-commutative 60.2%

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

      6. *-commutative 60.2%

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

      7. *-commutative 60.2%

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

      8. div-exp 68.9%

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

      9. *-commutative 68.9%

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

      10. mul-1-neg 68.9%

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

      11. *-commutative 68.9%

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

   6. Simplified 68.9%

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

   if -5e-32 < x.im < 3.0999999999999999e-140

   1. Initial program 35.8%

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

   3. Simplified 69.1%

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

   4. Taylor expanded in y.im around inf 37.3%

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

      1. unpow2 37.3%

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

      2. unpow2 37.3%

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

      3. hypot-def 69.5%

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

   6. Simplified 69.5%

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

   if 3.0999999999999999e-140 < x.im

   1. Initial program 40.6%

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

   3. Simplified 81.3%

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

   4. Taylor expanded in x.re around 0 76.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{-32}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \log \left(\frac{-1}{x.im}\right)}}\\ \mathbf{elif}\;x.im \leq 3.1 \cdot 10^{-140}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \end{array} \]

Alternative 8: 70.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_2 := e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - t_0} \cdot t_1\\ t_3 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-129}:\\ \;\;\;\;t_3 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+73}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 10^{+217}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|t_1\right|}{e^{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (sin (* y.im (log (hypot x.im x.re)))))
        (t_2 (* (exp (- (* (log (hypot x.re x.im)) y.re) t_0)) t_1))
        (t_3 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.im -7e-190)
     t_2
     (if (<= y.im 1.15e-129)
       (* t_3 (pow (hypot x.im x.re) y.re))
       (if (<= y.im 1.6e+73)
         t_2
         (if (<= y.im 1e+217)
           (*
            (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
            t_3)
           (/ (fabs t_1) (exp t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_2 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * t_1;
	double t_3 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -7e-190) {
		tmp = t_2;
	} else if (y_46_im <= 1.15e-129) {
		tmp = t_3 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.6e+73) {
		tmp = t_2;
	} else if (y_46_im <= 1e+217) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * t_3;
	} else {
		tmp = fabs(t_1) / exp(t_0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double t_2 = Math.exp(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * t_1;
	double t_3 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -7e-190) {
		tmp = t_2;
	} else if (y_46_im <= 1.15e-129) {
		tmp = t_3 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.6e+73) {
		tmp = t_2;
	} else if (y_46_im <= 1e+217) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * t_3;
	} else {
		tmp = Math.abs(t_1) / Math.exp(t_0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	t_2 = math.exp(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * t_1
	t_3 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -7e-190:
		tmp = t_2
	elif y_46_im <= 1.15e-129:
		tmp = t_3 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_im <= 1.6e+73:
		tmp = t_2
	elif y_46_im <= 1e+217:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * t_3
	else:
		tmp = math.fabs(t_1) / math.exp(t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	t_2 = Float64(exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * t_1)
	t_3 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -7e-190)
		tmp = t_2;
	elseif (y_46_im <= 1.15e-129)
		tmp = Float64(t_3 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 1.6e+73)
		tmp = t_2;
	elseif (y_46_im <= 1e+217)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * t_3);
	else
		tmp = Float64(abs(t_1) / exp(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	t_2 = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * t_1;
	t_3 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -7e-190)
		tmp = t_2;
	elseif (y_46_im <= 1.15e-129)
		tmp = t_3 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_im <= 1.6e+73)
		tmp = t_2;
	elseif (y_46_im <= 1e+217)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * t_3;
	else
		tmp = abs(t_1) / exp(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{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]}, Block[{t$95$2 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -7e-190], t$95$2, If[LessEqual[y$46$im, 1.15e-129], N[(t$95$3 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+73], t$95$2, If[LessEqual[y$46$im, 1e+217], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[Abs[t$95$1], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.9999999999999999e-190 or 1.15e-129 < y.im < 1.59999999999999991e73

   1. Initial program 33.3%

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

   3. Simplified 69.8%

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

   4. Taylor expanded in y.im around inf 31.1%

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

      1. unpow2 31.1%

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

      2. unpow2 31.1%

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

      3. hypot-def 69.2%

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

   6. Simplified 69.2%

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

   if -6.9999999999999999e-190 < y.im < 1.15e-129

   1. Initial program 42.4%

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

      1. exp-diff 42.4%

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

      2. +-rgt-identity 42.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.4%

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

      4. exp-to-pow 42.5%

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

      5. hypot-def 42.5%

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

      6. *-commutative 42.5%

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

      7. exp-prod 42.5%

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

      8. fma-def 42.6%

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

      9. hypot-def 90.5%

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

      10. *-commutative 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y.im around 0 64.3%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 64.3%

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

      2. unpow2 64.3%

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

      3. unpow2 64.3%

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

      4. hypot-def 74.4%

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

   6. Simplified 74.4%

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

   if 1.59999999999999991e73 < y.im < 9.9999999999999996e216

   1. Initial program 34.3%

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

   2. Taylor expanded in y.im around 0 71.8%

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

   if 9.9999999999999996e216 < y.im

   1. Initial program 42.1%

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

      1. exp-diff 31.6%

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

      2. +-rgt-identity 31.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 31.6%

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

      4. exp-to-pow 31.6%

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

      5. hypot-def 31.6%

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

      6. *-commutative 31.6%

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

      7. exp-prod 31.6%

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

      8. fma-def 31.6%

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

      9. hypot-def 58.0%

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

      10. *-commutative 58.0%

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

   3. Simplified 58.0%

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

   4. Taylor expanded in y.re around 0 36.9%

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

      1. add-sqr-sqrt 26.3%

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

      2. sqrt-unprod 42.2%

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

      3. pow2 42.2%

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

      4. add-log-exp 0.0%

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

      5. *-commutative 0.0%

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

      6. exp-prod 0.0%

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

      7. add-exp-log 0.0%

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

      8. +-commutative 0.0%

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

      9. unpow2 0.0%

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

      10. unpow2 0.0%

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

      11. hypot-udef 0.0%

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

   6. Applied egg-rr 0.0%

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

      1. unpow2 0.0%

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

      2. rem-sqrt-square 0.0%

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

      3. log-pow 80.0%

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

      4. hypot-def 42.2%

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

      5. unpow2 42.2%

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

      6. unpow2 42.2%

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

      7. +-commutative 42.2%

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

      8. unpow2 42.2%

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

      9. unpow2 42.2%

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

      10. hypot-def 80.0%

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

   8. Simplified 80.0%

      \[\leadsto \frac{\color{blue}{\left|\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right|}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{-190}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-129}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+73}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 10^{+217}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right|}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 9: 59.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{t_0}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq -2.1 \cdot 10^{+173}:\\ \;\;\;\;\frac{\sin \left(\left|y.im \cdot \log \left(\frac{-1}{x.re}\right)\right|\right)}{t_1}\\ \mathbf{elif}\;x.re \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \sin t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{{x.re}^{y.re} \cdot \sin \left(t_2 + y.im \cdot \log x.re\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (exp t_0))
        (t_2 (* y.re (atan2 x.im x.re))))
   (if (<= x.re -2.1e+173)
     (/ (sin (fabs (* y.im (log (/ -1.0 x.re))))) t_1)
     (if (<= x.re 1.7e-12)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
        (sin t_2))
       (/ (* (pow x.re y.re) (sin (+ t_2 (* y.im (log x.re))))) t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = exp(t_0);
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -2.1e+173) {
		tmp = sin(fabs((y_46_im * log((-1.0 / x_46_re))))) / t_1;
	} else if (x_46_re <= 1.7e-12) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin(t_2);
	} else {
		tmp = (pow(x_46_re, y_46_re) * sin((t_2 + (y_46_im * log(x_46_re))))) / t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y_46im * atan2(x_46im, x_46re)
    t_1 = exp(t_0)
    t_2 = y_46re * atan2(x_46im, x_46re)
    if (x_46re <= (-2.1d+173)) then
        tmp = sin(abs((y_46im * log(((-1.0d0) / x_46re))))) / t_1
    else if (x_46re <= 1.7d-12) then
        tmp = exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - t_0)) * sin(t_2)
    else
        tmp = ((x_46re ** y_46re) * sin((t_2 + (y_46im * log(x_46re))))) / t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.exp(t_0);
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -2.1e+173) {
		tmp = Math.sin(Math.abs((y_46_im * Math.log((-1.0 / x_46_re))))) / t_1;
	} else if (x_46_re <= 1.7e-12) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * Math.sin(t_2);
	} else {
		tmp = (Math.pow(x_46_re, y_46_re) * Math.sin((t_2 + (y_46_im * Math.log(x_46_re))))) / t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = math.exp(t_0)
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if x_46_re <= -2.1e+173:
		tmp = math.sin(math.fabs((y_46_im * math.log((-1.0 / x_46_re))))) / t_1
	elif x_46_re <= 1.7e-12:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * math.sin(t_2)
	else:
		tmp = (math.pow(x_46_re, y_46_re) * math.sin((t_2 + (y_46_im * math.log(x_46_re))))) / t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = exp(t_0)
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= -2.1e+173)
		tmp = Float64(sin(abs(Float64(y_46_im * log(Float64(-1.0 / x_46_re))))) / t_1);
	elseif (x_46_re <= 1.7e-12)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * sin(t_2));
	else
		tmp = Float64(Float64((x_46_re ^ y_46_re) * sin(Float64(t_2 + Float64(y_46_im * log(x_46_re))))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = exp(t_0);
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (x_46_re <= -2.1e+173)
		tmp = sin(abs((y_46_im * log((-1.0 / x_46_re))))) / t_1;
	elseif (x_46_re <= 1.7e-12)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin(t_2);
	else
		tmp = ((x_46_re ^ y_46_re) * sin((t_2 + (y_46_im * log(x_46_re))))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -2.1e+173], N[(N[Sin[N[Abs[N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$46$re, 1.7e-12], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * N[Sin[N[(t$95$2 + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.1e173

   1. Initial program 0.0%

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

      1. exp-diff 0.0%

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

      2. +-rgt-identity 0.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 0.0%

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

      4. exp-to-pow 0.0%

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

      5. hypot-def 0.0%

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

      6. *-commutative 0.0%

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

      7. exp-prod 0.0%

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

      8. fma-def 0.0%

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

      9. hypot-def 54.2%

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

      10. *-commutative 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in y.re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. hypot-udef 38.3%

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

      5. pow1 38.3%

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

      6. metadata-eval 38.3%

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

      7. pow-pow 0.0%

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

      8. *-commutative 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. pow2 0.0%

          \[\leadsto \frac{\sin \left(\sqrt{\color{blue}{{\left(\log \left({\left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot y.im\right)}^{2}}}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Applied egg-rr 5.2%

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

      1. unpow2 5.2%

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

      2. rem-sqrt-square 5.2%

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

      3. log-pow 50.9%

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

      4. hypot-def 0.0%

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

      5. unpow2 0.0%

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

      6. unpow2 0.0%

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

      7. +-commutative 0.0%

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

      8. unpow2 0.0%

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

      9. unpow2 0.0%

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

      10. hypot-def 50.9%

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

   8. Simplified 50.9%

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

   9. Taylor expanded in x.re around -inf 50.9%

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

       1. associate-*r* 50.9%

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

       2. neg-mul-1 50.9%

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

   11. Simplified 50.9%

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

   if -2.1e173 < x.re < 1.7e-12

   1. Initial program 44.4%

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

   2. Taylor expanded in y.im around 0 61.2%

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

   if 1.7e-12 < x.re

   1. Initial program 26.7%

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

      1. exp-diff 22.7%

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

      2. +-rgt-identity 22.7%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 22.7%

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

      4. exp-to-pow 22.7%

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

      5. hypot-def 22.7%

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

      6. *-commutative 22.7%

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

      7. exp-prod 20.5%

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

      8. fma-def 20.5%

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

      9. hypot-def 54.9%

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

      10. *-commutative 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in x.im around 0 55.6%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.1 \cdot 10^{+173}:\\ \;\;\;\;\frac{\sin \left(\left|y.im \cdot \log \left(\frac{-1}{x.re}\right)\right|\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.re \leq 1.7 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{x.re}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 10: 60.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq -4.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{\sin \left(\left|y.im \cdot \log \left(\frac{-1}{x.re}\right)\right|\right)}{e^{t_0}}\\ \mathbf{elif}\;x.re \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(t_1 + y.im \cdot \log x.re\right)}{e^{t_0 - y.re \cdot \log x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re))) (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= x.re -4.5e+173)
     (/ (sin (fabs (* y.im (log (/ -1.0 x.re))))) (exp t_0))
     (if (<= x.re 2.3e-12)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
        (sin t_1))
       (/
        (sin (+ t_1 (* y.im (log x.re))))
        (exp (- t_0 (* y.re (log x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -4.5e+173) {
		tmp = sin(fabs((y_46_im * log((-1.0 / x_46_re))))) / exp(t_0);
	} else if (x_46_re <= 2.3e-12) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin(t_1);
	} else {
		tmp = sin((t_1 + (y_46_im * log(x_46_re)))) / exp((t_0 - (y_46_re * log(x_46_re))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y_46im * atan2(x_46im, x_46re)
    t_1 = y_46re * atan2(x_46im, x_46re)
    if (x_46re <= (-4.5d+173)) then
        tmp = sin(abs((y_46im * log(((-1.0d0) / x_46re))))) / exp(t_0)
    else if (x_46re <= 2.3d-12) then
        tmp = exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - t_0)) * sin(t_1)
    else
        tmp = sin((t_1 + (y_46im * log(x_46re)))) / exp((t_0 - (y_46re * log(x_46re))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -4.5e+173) {
		tmp = Math.sin(Math.abs((y_46_im * Math.log((-1.0 / x_46_re))))) / Math.exp(t_0);
	} else if (x_46_re <= 2.3e-12) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * Math.sin(t_1);
	} else {
		tmp = Math.sin((t_1 + (y_46_im * Math.log(x_46_re)))) / Math.exp((t_0 - (y_46_re * Math.log(x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if x_46_re <= -4.5e+173:
		tmp = math.sin(math.fabs((y_46_im * math.log((-1.0 / x_46_re))))) / math.exp(t_0)
	elif x_46_re <= 2.3e-12:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * math.sin(t_1)
	else:
		tmp = math.sin((t_1 + (y_46_im * math.log(x_46_re)))) / math.exp((t_0 - (y_46_re * math.log(x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= -4.5e+173)
		tmp = Float64(sin(abs(Float64(y_46_im * log(Float64(-1.0 / x_46_re))))) / exp(t_0));
	elseif (x_46_re <= 2.3e-12)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * sin(t_1));
	else
		tmp = Float64(sin(Float64(t_1 + Float64(y_46_im * log(x_46_re)))) / exp(Float64(t_0 - Float64(y_46_re * log(x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (x_46_re <= -4.5e+173)
		tmp = sin(abs((y_46_im * log((-1.0 / x_46_re))))) / exp(t_0);
	elseif (x_46_re <= 2.3e-12)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin(t_1);
	else
		tmp = sin((t_1 + (y_46_im * log(x_46_re)))) / exp((t_0 - (y_46_re * log(x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -4.5e+173], N[(N[Sin[N[Abs[N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.3e-12], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$0 - N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -4.5000000000000002e173

   1. Initial program 0.0%

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

      1. exp-diff 0.0%

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

      2. +-rgt-identity 0.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 0.0%

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

      4. exp-to-pow 0.0%

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

      5. hypot-def 0.0%

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

      6. *-commutative 0.0%

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

      7. exp-prod 0.0%

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

      8. fma-def 0.0%

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

      9. hypot-def 54.2%

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

      10. *-commutative 54.2%

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

   3. Simplified 54.2%

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

   4. Taylor expanded in y.re around 0 0.0%

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

      1. +-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. hypot-udef 38.3%

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

      5. pow1 38.3%

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

      6. metadata-eval 38.3%

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

      7. pow-pow 0.0%

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

      8. *-commutative 0.0%

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

      9. add-sqr-sqrt 0.0%

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

      10. sqrt-unprod 0.0%

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

      11. pow2 0.0%

          \[\leadsto \frac{\sin \left(\sqrt{\color{blue}{{\left(\log \left({\left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{3}\right)}^{0.3333333333333333}\right) \cdot y.im\right)}^{2}}}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Applied egg-rr 5.2%

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

      1. unpow2 5.2%

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

      2. rem-sqrt-square 5.2%

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

      3. log-pow 50.9%

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

      4. hypot-def 0.0%

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

      5. unpow2 0.0%

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

      6. unpow2 0.0%

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

      7. +-commutative 0.0%

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

      8. unpow2 0.0%

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

      9. unpow2 0.0%

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

      10. hypot-def 50.9%

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

   8. Simplified 50.9%

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

   9. Taylor expanded in x.re around -inf 50.9%

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

       1. associate-*r* 50.9%

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

       2. neg-mul-1 50.9%

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

   11. Simplified 50.9%

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

   if -4.5000000000000002e173 < x.re < 2.29999999999999989e-12

   1. Initial program 44.4%

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

   2. Taylor expanded in y.im around 0 61.2%

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

   if 2.29999999999999989e-12 < x.re

   1. Initial program 26.7%

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

      1. exp-diff 22.7%

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

      2. +-rgt-identity 22.7%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 22.7%

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

      4. exp-to-pow 22.7%

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

      5. hypot-def 22.7%

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

      6. *-commutative 22.7%

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

      7. exp-prod 20.5%

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

      8. fma-def 20.5%

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

      9. hypot-def 54.9%

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

      10. *-commutative 54.9%

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

   3. Simplified 54.9%

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

   4. Taylor expanded in x.re around inf 55.6%

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

      1. associate-/l* 55.6%

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

      2. +-commutative 55.6%

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

      3. mul-1-neg 55.6%

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

      4. unsub-neg 55.6%

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

      5. *-commutative 55.6%

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

      6. log-rec 55.6%

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

      7. *-commutative 55.6%

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

      8. *-commutative 55.6%

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

      9. div-exp 61.6%

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

      10. *-commutative 61.6%

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

   6. Simplified 61.6%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -4.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{\sin \left(\left|y.im \cdot \log \left(\frac{-1}{x.re}\right)\right|\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.re \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.re \cdot \log x.re}}\\ \end{array} \]

Alternative 11: 50.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.im \leq -1.3 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{t_1}\\ \mathbf{elif}\;x.im \leq -8.5 \cdot 10^{-123}:\\ \;\;\;\;\sin t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(t_0 + y.im \cdot \log x.im\right) \cdot {x.im}^{y.re}}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (exp (* y.im (atan2 x.im x.re)))))
   (if (<= x.im -1.3e+187)
     (/ (sin (* y.im (log (- x.im)))) t_1)
     (if (<= x.im -8.5e-123)
       (* (sin t_0) (pow (hypot x.im x.re) y.re))
       (if (<= x.im 5.2e-213)
         (/
          (sin (* y.im (log (hypot x.im x.re))))
          (pow (exp y.im) (atan2 x.im x.re)))
         (/ (* (sin (+ t_0 (* y.im (log x.im)))) (pow x.im y.re)) t_1))))))

C

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 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -1.3e+187) {
		tmp = sin((y_46_im * log(-x_46_im))) / t_1;
	} else if (x_46_im <= -8.5e-123) {
		tmp = sin(t_0) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (x_46_im <= 5.2e-213) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	} else {
		tmp = (sin((t_0 + (y_46_im * log(x_46_im)))) * pow(x_46_im, y_46_re)) / t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -1.3e+187) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_im))) / t_1;
	} else if (x_46_im <= -8.5e-123) {
		tmp = Math.sin(t_0) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (x_46_im <= 5.2e-213) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = (Math.sin((t_0 + (y_46_im * Math.log(x_46_im)))) * Math.pow(x_46_im, y_46_re)) / t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -1.3e+187:
		tmp = math.sin((y_46_im * math.log(-x_46_im))) / t_1
	elif x_46_im <= -8.5e-123:
		tmp = math.sin(t_0) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif x_46_im <= 5.2e-213:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re))
	else:
		tmp = (math.sin((t_0 + (y_46_im * math.log(x_46_im)))) * math.pow(x_46_im, y_46_re)) / t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -1.3e+187)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_im)))) / t_1);
	elseif (x_46_im <= -8.5e-123)
		tmp = Float64(sin(t_0) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (x_46_im <= 5.2e-213)
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	else
		tmp = Float64(Float64(sin(Float64(t_0 + Float64(y_46_im * log(x_46_im)))) * (x_46_im ^ y_46_re)) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -1.3e+187)
		tmp = sin((y_46_im * log(-x_46_im))) / t_1;
	elseif (x_46_im <= -8.5e-123)
		tmp = sin(t_0) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (x_46_im <= 5.2e-213)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp(y_46_im) ^ atan2(x_46_im, x_46_re));
	else
		tmp = (sin((t_0 + (y_46_im * log(x_46_im)))) * (x_46_im ^ y_46_re)) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -1.3e+187], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[x$46$im, -8.5e-123], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 5.2e-213], N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[N[(t$95$0 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -1.2999999999999999e187

   1. Initial program 0.0%

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

      1. exp-diff 0.0%

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

      2. +-rgt-identity 0.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 0.0%

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

      4. exp-to-pow 0.0%

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

      5. hypot-def 0.0%

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

      6. *-commutative 0.0%

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

      7. exp-prod 0.0%

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

      8. fma-def 0.0%

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

      9. hypot-def 44.3%

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

      10. *-commutative 44.3%

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

   3. Simplified 44.3%

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

   4. Taylor expanded in y.re around 0 0.0%

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

   5. Taylor expanded in x.im around -inf 49.0%

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

      1. mul-1-neg 49.0%

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

   7. Simplified 49.0%

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

   if -1.2999999999999999e187 < x.im < -8.4999999999999995e-123

   1. Initial program 54.9%

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

      1. exp-diff 49.0%

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

      2. +-rgt-identity 49.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 49.0%

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

      4. exp-to-pow 49.0%

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

      5. hypot-def 49.0%

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

      6. *-commutative 49.0%

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

      7. exp-prod 47.0%

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

      8. fma-def 47.0%

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

      9. hypot-def 68.6%

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

      10. *-commutative 68.6%

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

   3. Simplified 68.6%

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

   4. Taylor expanded in y.im around 0 57.1%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

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

      4. hypot-def 59.2%

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

   6. Simplified 59.2%

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

   if -8.4999999999999995e-123 < x.im < 5.2000000000000003e-213

   1. Initial program 32.7%

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

      1. exp-diff 31.0%

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

      2. +-rgt-identity 31.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 31.0%

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

      4. exp-to-pow 31.0%

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

      5. hypot-def 31.0%

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

      6. *-commutative 31.0%

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

      7. exp-prod 30.8%

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

      8. fma-def 30.8%

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

      9. hypot-def 68.1%

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

      10. *-commutative 68.1%

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

   3. Simplified 68.1%

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

   4. Taylor expanded in y.re around 0 28.1%

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

      1. unpow2 28.1%

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

      2. unpow2 28.1%

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

      3. hypot-def 54.6%

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

      4. *-commutative 54.6%

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

      5. exp-prod 54.1%

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

   6. Simplified 54.1%

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

   if 5.2000000000000003e-213 < x.im

   1. Initial program 39.3%

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

      1. exp-diff 33.4%

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

      2. +-rgt-identity 33.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.4%

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

      4. exp-to-pow 33.5%

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

      5. hypot-def 33.5%

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

      6. *-commutative 33.5%

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

      7. exp-prod 32.7%

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

      8. fma-def 32.7%

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

      9. hypot-def 65.6%

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

      10. *-commutative 65.6%

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

   3. Simplified 65.6%

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

   4. Taylor expanded in x.re around 0 58.9%

      \[\leadsto \color{blue}{\frac{\sin \left(\log x.im \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.3 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.im \leq -8.5 \cdot 10^{-123}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{-213}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right) \cdot {x.im}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 12: 61.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_1 := \frac{t_0}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_4 := \sin t_2 \cdot t_3\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{-74}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-281}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-250}:\\ \;\;\;\;\frac{\left|t_0\right|}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+207}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+287}:\\ \;\;\;\;t_3 \cdot \sin \left(\left|t_2\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_3 \cdot \sin \left(e^{\log t_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.im (log (hypot x.im x.re)))))
        (t_1 (/ t_0 (pow (exp y.im) (atan2 x.im x.re))))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (pow (hypot x.im x.re) y.re))
        (t_4 (* (sin t_2) t_3)))
   (if (<= y.re -4e-74)
     t_4
     (if (<= y.re -6.2e-281)
       t_1
       (if (<= y.re 5.8e-250)
         (/ (fabs t_0) (exp (* y.im (atan2 x.im x.re))))
         (if (<= y.re 7.8e-82)
           t_1
           (if (<= y.re 5e+207)
             t_4
             (if (<= y.re 3e+287)
               (* t_3 (sin (fabs t_2)))
               (* t_3 (sin (exp (log t_2))))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_1 = t_0 / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_4 = sin(t_2) * t_3;
	double tmp;
	if (y_46_re <= -4e-74) {
		tmp = t_4;
	} else if (y_46_re <= -6.2e-281) {
		tmp = t_1;
	} else if (y_46_re <= 5.8e-250) {
		tmp = fabs(t_0) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 7.8e-82) {
		tmp = t_1;
	} else if (y_46_re <= 5e+207) {
		tmp = t_4;
	} else if (y_46_re <= 3e+287) {
		tmp = t_3 * sin(fabs(t_2));
	} else {
		tmp = t_3 * sin(exp(log(t_2)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double t_1 = t_0 / Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re));
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_4 = Math.sin(t_2) * t_3;
	double tmp;
	if (y_46_re <= -4e-74) {
		tmp = t_4;
	} else if (y_46_re <= -6.2e-281) {
		tmp = t_1;
	} else if (y_46_re <= 5.8e-250) {
		tmp = Math.abs(t_0) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 7.8e-82) {
		tmp = t_1;
	} else if (y_46_re <= 5e+207) {
		tmp = t_4;
	} else if (y_46_re <= 3e+287) {
		tmp = t_3 * Math.sin(Math.abs(t_2));
	} else {
		tmp = t_3 * Math.sin(Math.exp(Math.log(t_2)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	t_1 = t_0 / math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re))
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_4 = math.sin(t_2) * t_3
	tmp = 0
	if y_46_re <= -4e-74:
		tmp = t_4
	elif y_46_re <= -6.2e-281:
		tmp = t_1
	elif y_46_re <= 5.8e-250:
		tmp = math.fabs(t_0) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 7.8e-82:
		tmp = t_1
	elif y_46_re <= 5e+207:
		tmp = t_4
	elif y_46_re <= 3e+287:
		tmp = t_3 * math.sin(math.fabs(t_2))
	else:
		tmp = t_3 * math.sin(math.exp(math.log(t_2)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	t_1 = Float64(t_0 / (exp(y_46_im) ^ atan(x_46_im, x_46_re)))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_4 = Float64(sin(t_2) * t_3)
	tmp = 0.0
	if (y_46_re <= -4e-74)
		tmp = t_4;
	elseif (y_46_re <= -6.2e-281)
		tmp = t_1;
	elseif (y_46_re <= 5.8e-250)
		tmp = Float64(abs(t_0) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 7.8e-82)
		tmp = t_1;
	elseif (y_46_re <= 5e+207)
		tmp = t_4;
	elseif (y_46_re <= 3e+287)
		tmp = Float64(t_3 * sin(abs(t_2)));
	else
		tmp = Float64(t_3 * sin(exp(log(t_2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	t_1 = t_0 / (exp(y_46_im) ^ atan2(x_46_im, x_46_re));
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_4 = sin(t_2) * t_3;
	tmp = 0.0;
	if (y_46_re <= -4e-74)
		tmp = t_4;
	elseif (y_46_re <= -6.2e-281)
		tmp = t_1;
	elseif (y_46_re <= 5.8e-250)
		tmp = abs(t_0) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 7.8e-82)
		tmp = t_1;
	elseif (y_46_re <= 5e+207)
		tmp = t_4;
	elseif (y_46_re <= 3e+287)
		tmp = t_3 * sin(abs(t_2));
	else
		tmp = t_3 * sin(exp(log(t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[t$95$2], $MachinePrecision] * t$95$3), $MachinePrecision]}, If[LessEqual[y$46$re, -4e-74], t$95$4, If[LessEqual[y$46$re, -6.2e-281], t$95$1, If[LessEqual[y$46$re, 5.8e-250], N[(N[Abs[t$95$0], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e-82], t$95$1, If[LessEqual[y$46$re, 5e+207], t$95$4, If[LessEqual[y$46$re, 3e+287], N[(t$95$3 * N[Sin[N[Abs[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[Sin[N[Exp[N[Log[t$95$2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.99999999999999983e-74 or 7.79999999999999947e-82 < y.re < 4.9999999999999999e207

   1. Initial program 32.6%

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

      1. exp-diff 25.6%

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

      2. +-rgt-identity 25.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 25.6%

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

      4. exp-to-pow 25.7%

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

      5. hypot-def 25.7%

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

      6. *-commutative 25.7%

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

      7. exp-prod 24.9%

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

      8. fma-def 24.9%

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

      9. hypot-def 59.2%

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

      10. *-commutative 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in y.im around 0 60.8%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 60.8%

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

      2. unpow2 60.8%

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

      3. unpow2 60.8%

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

      4. hypot-def 63.0%

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

   6. Simplified 63.0%

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

   if -3.99999999999999983e-74 < y.re < -6.2000000000000004e-281 or 5.8000000000000004e-250 < y.re < 7.79999999999999947e-82

   1. Initial program 47.1%

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

      1. exp-diff 47.1%

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

      2. +-rgt-identity 47.1%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 47.1%

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

      4. exp-to-pow 47.1%

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

      5. hypot-def 47.1%

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

      6. *-commutative 47.1%

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

      7. exp-prod 47.0%

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

      8. fma-def 47.0%

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

      9. hypot-def 85.5%

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

      10. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in y.re around 0 33.6%

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

      1. unpow2 33.6%

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

      2. unpow2 33.6%

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

      3. hypot-def 63.2%

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

      4. *-commutative 63.2%

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

      5. exp-prod 63.0%

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

   6. Simplified 63.0%

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

   if -6.2000000000000004e-281 < y.re < 5.8000000000000004e-250

   1. Initial program 27.6%

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

      1. exp-diff 27.6%

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

      2. +-rgt-identity 27.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 27.6%

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

      4. exp-to-pow 27.6%

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

      5. hypot-def 27.6%

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

      6. *-commutative 27.6%

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

      7. exp-prod 27.6%

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

      8. fma-def 27.6%

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

      9. hypot-def 55.6%

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

      10. *-commutative 55.6%

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

   3. Simplified 55.6%

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

   4. Taylor expanded in y.re around 0 26.1%

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

      1. add-sqr-sqrt 20.3%

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

      2. sqrt-unprod 28.8%

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

      3. pow2 28.8%

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

      4. add-log-exp 6.7%

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

      5. *-commutative 6.7%

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

      6. exp-prod 6.7%

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

      7. add-exp-log 6.7%

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

      8. +-commutative 6.7%

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

      9. unpow2 6.7%

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

      10. unpow2 6.7%

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

      11. hypot-udef 7.3%

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

   6. Applied egg-rr 7.3%

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

      1. unpow2 7.3%

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

      2. rem-sqrt-square 7.3%

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

      3. log-pow 82.7%

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

      4. hypot-def 42.7%

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

      5. unpow2 42.7%

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

      6. unpow2 42.7%

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

      7. +-commutative 42.7%

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

      8. unpow2 42.7%

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

      9. unpow2 42.7%

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

      10. hypot-def 82.7%

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

   8. Simplified 82.7%

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

   if 4.9999999999999999e207 < y.re < 2.9999999999999999e287

   1. Initial program 30.0%

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

      1. exp-diff 20.0%

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

      2. +-rgt-identity 20.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 20.0%

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

      4. exp-to-pow 20.0%

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

      5. hypot-def 20.0%

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

      6. *-commutative 20.0%

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

      7. exp-prod 15.0%

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

      8. fma-def 15.0%

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

      9. hypot-def 25.0%

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

      10. *-commutative 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in y.im around 0 40.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 40.0%

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

      2. unpow2 40.0%

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

      3. unpow2 40.0%

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

      4. hypot-def 40.0%

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

   6. Simplified 40.0%

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

      1. *-commutative 25.5%

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

      2. add-sqr-sqrt 5.2%

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

      3. sqrt-unprod 10.0%

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

      4. pow2 10.0%

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

   8. Applied egg-rr 10.0%

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

      1. unpow2 10.0%

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

      2. rem-sqrt-square 30.5%

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

   10. Simplified 65.0%

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

   if 2.9999999999999999e287 < y.re

   1. Initial program 33.3%

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

      1. exp-diff 33.3%

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

      2. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.3%

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

      4. exp-to-pow 33.3%

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

      5. hypot-def 33.3%

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

      6. *-commutative 33.3%

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

      7. exp-prod 33.3%

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

      8. fma-def 33.3%

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

      9. hypot-def 50.0%

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

      10. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in y.im around 0 50.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 50.0%

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

      2. unpow2 50.0%

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

      3. unpow2 50.0%

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

      4. hypot-def 50.0%

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

   6. Simplified 50.0%

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

      1. *-commutative 50.0%

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

      2. add-exp-log 66.7%

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

   8. Applied egg-rr 66.7%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{-74}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-281}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-250}:\\ \;\;\;\;\frac{\left|\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right|}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+287}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \end{array} \]

Alternative 13: 48.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := \sin t_0 \cdot t_1\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+207}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+291}:\\ \;\;\;\;t_1 \cdot \sin \left(\left|t_0\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(e^{\log t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (pow (hypot x.im x.re) y.re))
        (t_2 (* (sin t_0) t_1)))
   (if (<= y.re -2.15e-197)
     t_2
     (if (<= y.re 4.6e-219)
       (/ (sin (* y.im (log (- x.re)))) (exp (* y.im (atan2 x.im x.re))))
       (if (<= y.re 5e+207)
         t_2
         (if (<= y.re 1.45e+291)
           (* t_1 (sin (fabs t_0)))
           (* t_1 (sin (exp (log t_0))))))))))

C

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 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = sin(t_0) * t_1;
	double tmp;
	if (y_46_re <= -2.15e-197) {
		tmp = t_2;
	} else if (y_46_re <= 4.6e-219) {
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 5e+207) {
		tmp = t_2;
	} else if (y_46_re <= 1.45e+291) {
		tmp = t_1 * sin(fabs(t_0));
	} else {
		tmp = t_1 * sin(exp(log(t_0)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = Math.sin(t_0) * t_1;
	double tmp;
	if (y_46_re <= -2.15e-197) {
		tmp = t_2;
	} else if (y_46_re <= 4.6e-219) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_re))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 5e+207) {
		tmp = t_2;
	} else if (y_46_re <= 1.45e+291) {
		tmp = t_1 * Math.sin(Math.abs(t_0));
	} else {
		tmp = t_1 * Math.sin(Math.exp(Math.log(t_0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_2 = math.sin(t_0) * t_1
	tmp = 0
	if y_46_re <= -2.15e-197:
		tmp = t_2
	elif y_46_re <= 4.6e-219:
		tmp = math.sin((y_46_im * math.log(-x_46_re))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 5e+207:
		tmp = t_2
	elif y_46_re <= 1.45e+291:
		tmp = t_1 * math.sin(math.fabs(t_0))
	else:
		tmp = t_1 * math.sin(math.exp(math.log(t_0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_2 = Float64(sin(t_0) * t_1)
	tmp = 0.0
	if (y_46_re <= -2.15e-197)
		tmp = t_2;
	elseif (y_46_re <= 4.6e-219)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_re)))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 5e+207)
		tmp = t_2;
	elseif (y_46_re <= 1.45e+291)
		tmp = Float64(t_1 * sin(abs(t_0)));
	else
		tmp = Float64(t_1 * sin(exp(log(t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_2 = sin(t_0) * t_1;
	tmp = 0.0;
	if (y_46_re <= -2.15e-197)
		tmp = t_2;
	elseif (y_46_re <= 4.6e-219)
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 5e+207)
		tmp = t_2;
	elseif (y_46_re <= 1.45e+291)
		tmp = t_1 * sin(abs(t_0));
	else
		tmp = t_1 * sin(exp(log(t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -2.15e-197], t$95$2, If[LessEqual[y$46$re, 4.6e-219], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5e+207], t$95$2, If[LessEqual[y$46$re, 1.45e+291], N[(t$95$1 * N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[N[Exp[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.15e-197 or 4.59999999999999977e-219 < y.re < 4.9999999999999999e207

   1. Initial program 38.2%

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

      1. exp-diff 33.3%

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

      2. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.3%

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

      4. exp-to-pow 33.4%

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

      5. hypot-def 33.4%

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

      6. *-commutative 33.4%

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

      7. exp-prod 32.8%

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

      8. fma-def 32.9%

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

      9. hypot-def 67.9%

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

      10. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in y.im around 0 52.6%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 52.6%

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

      2. unpow2 52.6%

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

      3. unpow2 52.6%

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

      4. hypot-def 53.8%

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

   6. Simplified 53.8%

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

   if -2.15e-197 < y.re < 4.59999999999999977e-219

   1. Initial program 34.4%

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

      1. exp-diff 34.4%

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

      2. +-rgt-identity 34.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 34.4%

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

      4. exp-to-pow 34.4%

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

      5. hypot-def 34.4%

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

      6. *-commutative 34.4%

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

      7. exp-prod 34.0%

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

      8. fma-def 34.1%

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

      9. hypot-def 70.5%

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

      10. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in y.re around 0 31.8%

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

   5. Taylor expanded in x.re around -inf 52.1%

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

      1. mul-1-neg 52.1%

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

   7. Simplified 52.1%

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

   if 4.9999999999999999e207 < y.re < 1.4500000000000001e291

   1. Initial program 30.0%

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

      1. exp-diff 20.0%

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

      2. +-rgt-identity 20.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 20.0%

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

      4. exp-to-pow 20.0%

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

      5. hypot-def 20.0%

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

      6. *-commutative 20.0%

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

      7. exp-prod 15.0%

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

      8. fma-def 15.0%

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

      9. hypot-def 25.0%

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

      10. *-commutative 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in y.im around 0 40.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 40.0%

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

      2. unpow2 40.0%

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

      3. unpow2 40.0%

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

      4. hypot-def 40.0%

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

   6. Simplified 40.0%

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

      1. *-commutative 25.5%

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

      2. add-sqr-sqrt 5.2%

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

      3. sqrt-unprod 10.0%

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

      4. pow2 10.0%

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

   8. Applied egg-rr 10.0%

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

      1. unpow2 10.0%

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

      2. rem-sqrt-square 30.5%

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

   10. Simplified 65.0%

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

   if 1.4500000000000001e291 < y.re

   1. Initial program 33.3%

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

      1. exp-diff 33.3%

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

      2. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.3%

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

      4. exp-to-pow 33.3%

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

      5. hypot-def 33.3%

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

      6. *-commutative 33.3%

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

      7. exp-prod 33.3%

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

      8. fma-def 33.3%

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

      9. hypot-def 50.0%

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

      10. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in y.im around 0 50.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 50.0%

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

      2. unpow2 50.0%

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

      3. unpow2 50.0%

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

      4. hypot-def 50.0%

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

   6. Simplified 50.0%

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

      1. *-commutative 50.0%

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

      2. add-exp-log 66.7%

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

   8. Applied egg-rr 66.7%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-197}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+207}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+291}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \end{array} \]

Alternative 14: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := \sin t_0 \cdot t_1\\ \mathbf{if}\;y.re \leq -3.7 \cdot 10^{-74}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+209}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+291}:\\ \;\;\;\;t_1 \cdot \sin \left(\left|t_0\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \sin \left(e^{\log t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (pow (hypot x.im x.re) y.re))
        (t_2 (* (sin t_0) t_1)))
   (if (<= y.re -3.7e-74)
     t_2
     (if (<= y.re 1.25e-57)
       (/
        (sin (* y.im (log (hypot x.im x.re))))
        (pow (exp y.im) (atan2 x.im x.re)))
       (if (<= y.re 4e+209)
         t_2
         (if (<= y.re 1.7e+291)
           (* t_1 (sin (fabs t_0)))
           (* t_1 (sin (exp (log t_0))))))))))

C

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 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = sin(t_0) * t_1;
	double tmp;
	if (y_46_re <= -3.7e-74) {
		tmp = t_2;
	} else if (y_46_re <= 1.25e-57) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 4e+209) {
		tmp = t_2;
	} else if (y_46_re <= 1.7e+291) {
		tmp = t_1 * sin(fabs(t_0));
	} else {
		tmp = t_1 * sin(exp(log(t_0)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = Math.sin(t_0) * t_1;
	double tmp;
	if (y_46_re <= -3.7e-74) {
		tmp = t_2;
	} else if (y_46_re <= 1.25e-57) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 4e+209) {
		tmp = t_2;
	} else if (y_46_re <= 1.7e+291) {
		tmp = t_1 * Math.sin(Math.abs(t_0));
	} else {
		tmp = t_1 * Math.sin(Math.exp(Math.log(t_0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_2 = math.sin(t_0) * t_1
	tmp = 0
	if y_46_re <= -3.7e-74:
		tmp = t_2
	elif y_46_re <= 1.25e-57:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re))
	elif y_46_re <= 4e+209:
		tmp = t_2
	elif y_46_re <= 1.7e+291:
		tmp = t_1 * math.sin(math.fabs(t_0))
	else:
		tmp = t_1 * math.sin(math.exp(math.log(t_0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_2 = Float64(sin(t_0) * t_1)
	tmp = 0.0
	if (y_46_re <= -3.7e-74)
		tmp = t_2;
	elseif (y_46_re <= 1.25e-57)
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 4e+209)
		tmp = t_2;
	elseif (y_46_re <= 1.7e+291)
		tmp = Float64(t_1 * sin(abs(t_0)));
	else
		tmp = Float64(t_1 * sin(exp(log(t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_2 = sin(t_0) * t_1;
	tmp = 0.0;
	if (y_46_re <= -3.7e-74)
		tmp = t_2;
	elseif (y_46_re <= 1.25e-57)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp(y_46_im) ^ atan2(x_46_im, x_46_re));
	elseif (y_46_re <= 4e+209)
		tmp = t_2;
	elseif (y_46_re <= 1.7e+291)
		tmp = t_1 * sin(abs(t_0));
	else
		tmp = t_1 * sin(exp(log(t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -3.7e-74], t$95$2, If[LessEqual[y$46$re, 1.25e-57], N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4e+209], t$95$2, If[LessEqual[y$46$re, 1.7e+291], N[(t$95$1 * N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Sin[N[Exp[N[Log[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.69999999999999994e-74 or 1.25e-57 < y.re < 4.0000000000000003e209

   1. Initial program 32.6%

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

      1. exp-diff 25.6%

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

      2. +-rgt-identity 25.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 25.6%

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

      4. exp-to-pow 25.7%

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

      5. hypot-def 25.7%

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

      6. *-commutative 25.7%

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

      7. exp-prod 24.9%

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

      8. fma-def 24.9%

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

      9. hypot-def 59.2%

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

      10. *-commutative 59.2%

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

   3. Simplified 59.2%

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

   4. Taylor expanded in y.im around 0 60.8%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 60.8%

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

      2. unpow2 60.8%

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

      3. unpow2 60.8%

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

      4. hypot-def 63.0%

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

   6. Simplified 63.0%

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

   if -3.69999999999999994e-74 < y.re < 1.25e-57

   1. Initial program 43.6%

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

      1. exp-diff 43.6%

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

      2. +-rgt-identity 43.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 43.6%

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

      4. exp-to-pow 43.6%

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

      5. hypot-def 43.6%

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

      6. *-commutative 43.6%

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

      7. exp-prod 43.5%

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

      8. fma-def 43.5%

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

      9. hypot-def 80.2%

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

      10. *-commutative 80.2%

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

   3. Simplified 80.2%

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

   4. Taylor expanded in y.re around 0 32.2%

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

      1. unpow2 32.2%

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

      2. unpow2 32.2%

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

      3. hypot-def 61.7%

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

      4. *-commutative 61.7%

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

      5. exp-prod 61.4%

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

   6. Simplified 61.4%

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

   if 4.0000000000000003e209 < y.re < 1.70000000000000012e291

   1. Initial program 30.0%

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

      1. exp-diff 20.0%

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

      2. +-rgt-identity 20.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 20.0%

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

      4. exp-to-pow 20.0%

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

      5. hypot-def 20.0%

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

      6. *-commutative 20.0%

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

      7. exp-prod 15.0%

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

      8. fma-def 15.0%

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

      9. hypot-def 25.0%

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

      10. *-commutative 25.0%

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

   3. Simplified 25.0%

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

   4. Taylor expanded in y.im around 0 40.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 40.0%

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

      2. unpow2 40.0%

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

      3. unpow2 40.0%

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

      4. hypot-def 40.0%

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

   6. Simplified 40.0%

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

      1. *-commutative 25.5%

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

      2. add-sqr-sqrt 5.2%

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

      3. sqrt-unprod 10.0%

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

      4. pow2 10.0%

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

   8. Applied egg-rr 10.0%

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

      1. unpow2 10.0%

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

      2. rem-sqrt-square 30.5%

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

   10. Simplified 65.0%

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

   if 1.70000000000000012e291 < y.re

   1. Initial program 33.3%

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

      1. exp-diff 33.3%

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

      2. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.3%

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

      4. exp-to-pow 33.3%

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

      5. hypot-def 33.3%

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

      6. *-commutative 33.3%

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

      7. exp-prod 33.3%

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

      8. fma-def 33.3%

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

      9. hypot-def 50.0%

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

      10. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   4. Taylor expanded in y.im around 0 50.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 50.0%

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

      2. unpow2 50.0%

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

      3. unpow2 50.0%

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

      4. hypot-def 50.0%

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

   6. Simplified 50.0%

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

      1. *-commutative 50.0%

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

      2. add-exp-log 66.7%

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

   8. Applied egg-rr 66.7%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.7 \cdot 10^{-74}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+209}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+291}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(e^{\log \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \end{array} \]

Alternative 15: 48.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-196}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+212} \lor \neg \left(y.re \leq 4 \cdot 10^{+261}\right):\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left|t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (pow (hypot x.im x.re) y.re))
        (t_2 (* t_0 t_1)))
   (if (<= y.re -6.2e-196)
     t_2
     (if (<= y.re 4.8e-219)
       (/ (sin (* y.im (log (- x.re)))) (exp (* y.im (atan2 x.im x.re))))
       (if (or (<= y.re 2e+212) (not (<= y.re 4e+261)))
         t_2
         (* t_1 (fabs t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = t_0 * t_1;
	double tmp;
	if (y_46_re <= -6.2e-196) {
		tmp = t_2;
	} else if (y_46_re <= 4.8e-219) {
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else if ((y_46_re <= 2e+212) || !(y_46_re <= 4e+261)) {
		tmp = t_2;
	} else {
		tmp = t_1 * fabs(t_0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = t_0 * t_1;
	double tmp;
	if (y_46_re <= -6.2e-196) {
		tmp = t_2;
	} else if (y_46_re <= 4.8e-219) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_re))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if ((y_46_re <= 2e+212) || !(y_46_re <= 4e+261)) {
		tmp = t_2;
	} else {
		tmp = t_1 * Math.abs(t_0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_2 = t_0 * t_1
	tmp = 0
	if y_46_re <= -6.2e-196:
		tmp = t_2
	elif y_46_re <= 4.8e-219:
		tmp = math.sin((y_46_im * math.log(-x_46_re))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	elif (y_46_re <= 2e+212) or not (y_46_re <= 4e+261):
		tmp = t_2
	else:
		tmp = t_1 * math.fabs(t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (y_46_re <= -6.2e-196)
		tmp = t_2;
	elseif (y_46_re <= 4.8e-219)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_re)))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	elseif ((y_46_re <= 2e+212) || !(y_46_re <= 4e+261))
		tmp = t_2;
	else
		tmp = Float64(t_1 * abs(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (y_46_re <= -6.2e-196)
		tmp = t_2;
	elseif (y_46_re <= 4.8e-219)
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	elseif ((y_46_re <= 2e+212) || ~((y_46_re <= 4e+261)))
		tmp = t_2;
	else
		tmp = t_1 * abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -6.2e-196], t$95$2, If[LessEqual[y$46$re, 4.8e-219], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 2e+212], N[Not[LessEqual[y$46$re, 4e+261]], $MachinePrecision]], t$95$2, N[(t$95$1 * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.19999999999999986e-196 or 4.80000000000000028e-219 < y.re < 1.9999999999999998e212 or 3.9999999999999997e261 < y.re

   1. Initial program 38.8%

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

      1. exp-diff 33.3%

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

      2. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.3%

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

      4. exp-to-pow 33.4%

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

      5. hypot-def 33.4%

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

      6. *-commutative 33.4%

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

      7. exp-prod 32.9%

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

      8. fma-def 32.9%

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

      9. hypot-def 66.3%

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

      10. *-commutative 66.3%

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

   3. Simplified 66.3%

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

   4. Taylor expanded in y.im around 0 52.7%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 52.7%

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

      2. unpow2 52.7%

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

      3. unpow2 52.7%

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

      4. hypot-def 53.8%

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

   6. Simplified 53.8%

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

   if -6.19999999999999986e-196 < y.re < 4.80000000000000028e-219

   1. Initial program 34.4%

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

      1. exp-diff 34.4%

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

      2. +-rgt-identity 34.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 34.4%

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

      4. exp-to-pow 34.4%

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

      5. hypot-def 34.4%

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

      6. *-commutative 34.4%

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

      7. exp-prod 34.0%

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

      8. fma-def 34.1%

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

      9. hypot-def 70.5%

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

      10. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in y.re around 0 31.8%

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

   5. Taylor expanded in x.re around -inf 52.1%

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

      1. mul-1-neg 52.1%

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

   7. Simplified 52.1%

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

   if 1.9999999999999998e212 < y.re < 3.9999999999999997e261

   1. Initial program 9.1%

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

      1. exp-diff 9.1%

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

      2. +-rgt-identity 9.1%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 9.1%

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

      4. exp-to-pow 9.1%

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

      5. hypot-def 9.1%

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

      6. *-commutative 9.1%

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

      7. exp-prod 0.0%

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

      8. fma-def 0.0%

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

      9. hypot-def 9.1%

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

      10. *-commutative 9.1%

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

   3. Simplified 9.1%

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

   4. Taylor expanded in y.im around 0 27.3%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 27.3%

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

      2. unpow2 27.3%

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

      3. unpow2 27.3%

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

      4. hypot-def 27.3%

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

   6. Simplified 27.3%

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

      1. add-sqr-sqrt 27.4%

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

      2. sqrt-unprod 63.9%

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

      3. pow2 63.9%

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

      4. *-commutative 63.9%

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

   8. Applied egg-rr 72.7%

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

      1. unpow2 63.9%

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

      2. rem-sqrt-square 63.9%

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

   10. Simplified 72.7%

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

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-196}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+212} \lor \neg \left(y.re \leq 4 \cdot 10^{+261}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left|\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right|\\ \end{array} \]

Alternative 16: 48.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := t_1 \cdot t_2\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-198}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{+207}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+283}:\\ \;\;\;\;t_2 \cdot \sin \left(\left|t_0\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (pow (hypot x.im x.re) y.re))
        (t_3 (* t_1 t_2)))
   (if (<= y.re -2.5e-198)
     t_3
     (if (<= y.re 6.4e-224)
       (/ (sin (* y.im (log (- x.re)))) (exp (* y.im (atan2 x.im x.re))))
       (if (<= y.re 8.6e+207)
         t_3
         (if (<= y.re 6e+283)
           (* t_2 (sin (fabs t_0)))
           (* t_1 (pow x.re y.re))))))))

C

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 = sin(t_0);
	double t_2 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_1 * t_2;
	double tmp;
	if (y_46_re <= -2.5e-198) {
		tmp = t_3;
	} else if (y_46_re <= 6.4e-224) {
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 8.6e+207) {
		tmp = t_3;
	} else if (y_46_re <= 6e+283) {
		tmp = t_2 * sin(fabs(t_0));
	} else {
		tmp = t_1 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_1 * t_2;
	double tmp;
	if (y_46_re <= -2.5e-198) {
		tmp = t_3;
	} else if (y_46_re <= 6.4e-224) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_re))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 8.6e+207) {
		tmp = t_3;
	} else if (y_46_re <= 6e+283) {
		tmp = t_2 * Math.sin(Math.abs(t_0));
	} else {
		tmp = t_1 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin(t_0)
	t_2 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_3 = t_1 * t_2
	tmp = 0
	if y_46_re <= -2.5e-198:
		tmp = t_3
	elif y_46_re <= 6.4e-224:
		tmp = math.sin((y_46_im * math.log(-x_46_re))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 8.6e+207:
		tmp = t_3
	elif y_46_re <= 6e+283:
		tmp = t_2 * math.sin(math.fabs(t_0))
	else:
		tmp = t_1 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_3 = Float64(t_1 * t_2)
	tmp = 0.0
	if (y_46_re <= -2.5e-198)
		tmp = t_3;
	elseif (y_46_re <= 6.4e-224)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_re)))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 8.6e+207)
		tmp = t_3;
	elseif (y_46_re <= 6e+283)
		tmp = Float64(t_2 * sin(abs(t_0)));
	else
		tmp = Float64(t_1 * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_3 = t_1 * t_2;
	tmp = 0.0;
	if (y_46_re <= -2.5e-198)
		tmp = t_3;
	elseif (y_46_re <= 6.4e-224)
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 8.6e+207)
		tmp = t_3;
	elseif (y_46_re <= 6e+283)
		tmp = t_2 * sin(abs(t_0));
	else
		tmp = t_1 * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5e-198], t$95$3, If[LessEqual[y$46$re, 6.4e-224], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.6e+207], t$95$3, If[LessEqual[y$46$re, 6e+283], N[(t$95$2 * N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.5e-198 or 6.3999999999999997e-224 < y.re < 8.5999999999999995e207

   1. Initial program 38.2%

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

      1. exp-diff 33.3%

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

      2. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.3%

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

      4. exp-to-pow 33.4%

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

      5. hypot-def 33.4%

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

      6. *-commutative 33.4%

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

      7. exp-prod 32.8%

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

      8. fma-def 32.9%

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

      9. hypot-def 67.9%

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

      10. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   4. Taylor expanded in y.im around 0 52.6%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 52.6%

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

      2. unpow2 52.6%

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

      3. unpow2 52.6%

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

      4. hypot-def 53.8%

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

   6. Simplified 53.8%

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

   if -2.5e-198 < y.re < 6.3999999999999997e-224

   1. Initial program 34.4%

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

      1. exp-diff 34.4%

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

      2. +-rgt-identity 34.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 34.4%

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

      4. exp-to-pow 34.4%

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

      5. hypot-def 34.4%

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

      6. *-commutative 34.4%

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

      7. exp-prod 34.0%

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

      8. fma-def 34.1%

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

      9. hypot-def 70.5%

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

      10. *-commutative 70.5%

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

   3. Simplified 70.5%

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

   4. Taylor expanded in y.re around 0 31.8%

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

   5. Taylor expanded in x.re around -inf 52.1%

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

      1. mul-1-neg 52.1%

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

   7. Simplified 52.1%

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

   if 8.5999999999999995e207 < y.re < 6.0000000000000001e283

   1. Initial program 26.3%

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

      1. exp-diff 15.8%

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

      2. +-rgt-identity 15.8%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 15.8%

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

      4. exp-to-pow 15.8%

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

      5. hypot-def 15.8%

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

      6. *-commutative 15.8%

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

      7. exp-prod 10.5%

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

      8. fma-def 10.5%

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

      9. hypot-def 21.1%

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

      10. *-commutative 21.1%

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

   3. Simplified 21.1%

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

   4. Taylor expanded in y.im around 0 36.8%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 36.8%

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

      2. unpow2 36.8%

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

      3. unpow2 36.8%

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

      4. hypot-def 36.8%

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

   6. Simplified 36.8%

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

      1. *-commutative 21.6%

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

      2. add-sqr-sqrt 5.5%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re} \cdot \sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)} \cdot {x.re}^{y.re} \]

      3. sqrt-unprod 5.3%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {x.re}^{y.re} \]

      4. pow2 5.3%

         \[\leadsto \sin \left(\sqrt{\color{blue}{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}}\right) \cdot {x.re}^{y.re} \]

   8. Applied egg-rr 5.3%

      \[\leadsto \sin \color{blue}{\left(\sqrt{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]
   9. Step-by-step derivation

      1. unpow2 5.3%

         \[\leadsto \sin \left(\sqrt{\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \cdot {x.re}^{y.re} \]

      2. rem-sqrt-square 26.9%

         \[\leadsto \sin \color{blue}{\left(\left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right|\right)} \cdot {x.re}^{y.re} \]

   10. Simplified 63.2%

       \[\leadsto \sin \color{blue}{\left(\left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right|\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if 6.0000000000000001e283 < y.re

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.9%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 42.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 42.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 42.9%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 57.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 57.1%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 57.1%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 57.1%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 57.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.im around 0 57.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-198}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{+207}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+283}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 17: 48.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+212} \lor \neg \left(y.re \leq 1.3 \cdot 10^{+262}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \left|t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* t_0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -5.5e-197)
     t_1
     (if (<= y.re 1.7e-219)
       (/ (sin (* y.im (log (- x.re)))) (exp (* y.im (atan2 x.im x.re))))
       (if (or (<= y.re 3e+212) (not (<= y.re 1.3e+262)))
         t_1
         (* (pow x.im y.re) (fabs t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -5.5e-197) {
		tmp = t_1;
	} else if (y_46_re <= 1.7e-219) {
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else if ((y_46_re <= 3e+212) || !(y_46_re <= 1.3e+262)) {
		tmp = t_1;
	} else {
		tmp = pow(x_46_im, y_46_re) * fabs(t_0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -5.5e-197) {
		tmp = t_1;
	} else if (y_46_re <= 1.7e-219) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_re))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if ((y_46_re <= 3e+212) || !(y_46_re <= 1.3e+262)) {
		tmp = t_1;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * Math.abs(t_0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -5.5e-197:
		tmp = t_1
	elif y_46_re <= 1.7e-219:
		tmp = math.sin((y_46_im * math.log(-x_46_re))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	elif (y_46_re <= 3e+212) or not (y_46_re <= 1.3e+262):
		tmp = t_1
	else:
		tmp = math.pow(x_46_im, y_46_re) * math.fabs(t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -5.5e-197)
		tmp = t_1;
	elseif (y_46_re <= 1.7e-219)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_re)))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	elseif ((y_46_re <= 3e+212) || !(y_46_re <= 1.3e+262))
		tmp = t_1;
	else
		tmp = Float64((x_46_im ^ y_46_re) * abs(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -5.5e-197)
		tmp = t_1;
	elseif (y_46_re <= 1.7e-219)
		tmp = sin((y_46_im * log(-x_46_re))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	elseif ((y_46_re <= 3e+212) || ~((y_46_re <= 1.3e+262)))
		tmp = t_1;
	else
		tmp = (x_46_im ^ y_46_re) * abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.5e-197], t$95$1, If[LessEqual[y$46$re, 1.7e-219], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 3e+212], N[Not[LessEqual[y$46$re, 1.3e+262]], $MachinePrecision]], t$95$1, N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.50000000000000037e-197 or 1.6999999999999999e-219 < y.re < 3e212 or 1.3e262 < y.re

   1. Initial program 38.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 33.3%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 33.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 33.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 33.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 32.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 32.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 66.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 66.3%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 66.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 52.7%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 52.7%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 52.7%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 52.7%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 53.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 53.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   if -5.50000000000000037e-197 < y.re < 1.6999999999999999e-219

   1. Initial program 34.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 34.4%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 34.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 34.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 34.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 34.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 34.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 34.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 34.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 70.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 70.5%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 70.5%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.re around 0 31.8%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   5. Taylor expanded in x.re around -inf 52.1%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-1 \cdot x.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
   6. Step-by-step derivation

      1. mul-1-neg 52.1%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   7. Simplified 52.1%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   if 3e212 < y.re < 1.3e262

   1. Initial program 9.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 9.1%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 9.1%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 9.1%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 9.1%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 9.1%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 9.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 0.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 0.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 9.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 9.1%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 9.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 27.3%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 27.3%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 27.3%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 27.3%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 27.3%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 27.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.re around 0 27.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 27.4%

         \[\leadsto \color{blue}{\left(\sqrt{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot {x.im}^{y.re} \]

      2. sqrt-unprod 63.9%

         \[\leadsto \color{blue}{\sqrt{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot {x.im}^{y.re} \]

      3. pow2 63.9%

         \[\leadsto \sqrt{\color{blue}{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}} \cdot {x.im}^{y.re} \]

      4. *-commutative 63.9%

         \[\leadsto \sqrt{{\sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{2}} \cdot {x.im}^{y.re} \]

   9. Applied egg-rr 63.9%

      \[\leadsto \color{blue}{\sqrt{{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}} \cdot {x.im}^{y.re} \]
   10. Step-by-step derivation

       1. unpow2 63.9%

          \[\leadsto \sqrt{\color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}} \cdot {x.im}^{y.re} \]

       2. rem-sqrt-square 63.9%

          \[\leadsto \color{blue}{\left|\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right|} \cdot {x.im}^{y.re} \]

   11. Simplified 63.9%

       \[\leadsto \color{blue}{\left|\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right|} \cdot {x.im}^{y.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-197}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+212} \lor \neg \left(y.re \leq 1.3 \cdot 10^{+262}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \left|\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right|\\ \end{array} \]

Alternative 18: 35.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ t_2 := {x.re}^{y.re} \cdot \sin \left(\left|t_0\right|\right)\\ \mathbf{if}\;x.re \leq -3.8 \cdot 10^{+137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x.re \leq -6.2 \cdot 10^{+104}:\\ \;\;\;\;{x.im}^{y.re} \cdot \left|t_1\right|\\ \mathbf{elif}\;x.re \leq 1.75 \cdot 10^{+52}:\\ \;\;\;\;t_1 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (* (pow x.re y.re) (sin (fabs t_0)))))
   (if (<= x.re -3.8e+137)
     t_2
     (if (<= x.re -6.2e+104)
       (* (pow x.im y.re) (fabs t_1))
       (if (<= x.re 1.75e+52) (* t_1 (pow x.im y.re)) t_2)))))

C

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 = sin(t_0);
	double t_2 = pow(x_46_re, y_46_re) * sin(fabs(t_0));
	double tmp;
	if (x_46_re <= -3.8e+137) {
		tmp = t_2;
	} else if (x_46_re <= -6.2e+104) {
		tmp = pow(x_46_im, y_46_re) * fabs(t_1);
	} else if (x_46_re <= 1.75e+52) {
		tmp = t_1 * pow(x_46_im, y_46_re);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    t_1 = sin(t_0)
    t_2 = (x_46re ** y_46re) * sin(abs(t_0))
    if (x_46re <= (-3.8d+137)) then
        tmp = t_2
    else if (x_46re <= (-6.2d+104)) then
        tmp = (x_46im ** y_46re) * abs(t_1)
    else if (x_46re <= 1.75d+52) then
        tmp = t_1 * (x_46im ** y_46re)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.pow(x_46_re, y_46_re) * Math.sin(Math.abs(t_0));
	double tmp;
	if (x_46_re <= -3.8e+137) {
		tmp = t_2;
	} else if (x_46_re <= -6.2e+104) {
		tmp = Math.pow(x_46_im, y_46_re) * Math.abs(t_1);
	} else if (x_46_re <= 1.75e+52) {
		tmp = t_1 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin(t_0)
	t_2 = math.pow(x_46_re, y_46_re) * math.sin(math.fabs(t_0))
	tmp = 0
	if x_46_re <= -3.8e+137:
		tmp = t_2
	elif x_46_re <= -6.2e+104:
		tmp = math.pow(x_46_im, y_46_re) * math.fabs(t_1)
	elif x_46_re <= 1.75e+52:
		tmp = t_1 * math.pow(x_46_im, y_46_re)
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64((x_46_re ^ y_46_re) * sin(abs(t_0)))
	tmp = 0.0
	if (x_46_re <= -3.8e+137)
		tmp = t_2;
	elseif (x_46_re <= -6.2e+104)
		tmp = Float64((x_46_im ^ y_46_re) * abs(t_1));
	elseif (x_46_re <= 1.75e+52)
		tmp = Float64(t_1 * (x_46_im ^ y_46_re));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	t_2 = (x_46_re ^ y_46_re) * sin(abs(t_0));
	tmp = 0.0;
	if (x_46_re <= -3.8e+137)
		tmp = t_2;
	elseif (x_46_re <= -6.2e+104)
		tmp = (x_46_im ^ y_46_re) * abs(t_1);
	elseif (x_46_re <= 1.75e+52)
		tmp = t_1 * (x_46_im ^ y_46_re);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -3.8e+137], t$95$2, If[LessEqual[x$46$re, -6.2e+104], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.75e+52], N[(t$95$1 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -3.79999999999999963e137 or 1.75e52 < x.re

   1. Initial program 16.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 14.2%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 14.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 14.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 14.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 14.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 14.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 14.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 14.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 58.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 58.1%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 58.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 39.7%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 39.7%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 39.7%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 39.7%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 38.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 38.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.im around 0 33.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
   8. Step-by-step derivation

      1. *-commutative 33.9%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {x.re}^{y.re} \]

      2. add-sqr-sqrt 18.6%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re} \cdot \sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)} \cdot {x.re}^{y.re} \]

      3. sqrt-unprod 24.4%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {x.re}^{y.re} \]

      4. pow2 24.4%

         \[\leadsto \sin \left(\sqrt{\color{blue}{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}}\right) \cdot {x.re}^{y.re} \]

   9. Applied egg-rr 24.4%

      \[\leadsto \sin \color{blue}{\left(\sqrt{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}\right)} \cdot {x.re}^{y.re} \]
   10. Step-by-step derivation

       1. unpow2 24.4%

          \[\leadsto \sin \left(\sqrt{\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \cdot {x.re}^{y.re} \]

       2. rem-sqrt-square 36.7%

          \[\leadsto \sin \color{blue}{\left(\left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right|\right)} \cdot {x.re}^{y.re} \]

   11. Simplified 36.7%

       \[\leadsto \sin \color{blue}{\left(\left|\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right|\right)} \cdot {x.re}^{y.re} \]

   if -3.79999999999999963e137 < x.re < -6.20000000000000033e104

   1. Initial program 75.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 50.0%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 50.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 50.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 50.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 50.0%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 50.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 50.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 50.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 50.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 50.0%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 50.0%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 34.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 34.0%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 34.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 34.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 34.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 34.0%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.re around 0 50.0%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 12.5%

         \[\leadsto \color{blue}{\left(\sqrt{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sqrt{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \cdot {x.im}^{y.re} \]

      2. sqrt-unprod 63.1%

         \[\leadsto \color{blue}{\sqrt{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot {x.im}^{y.re} \]

      3. pow2 63.1%

         \[\leadsto \sqrt{\color{blue}{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}} \cdot {x.im}^{y.re} \]

      4. *-commutative 63.1%

         \[\leadsto \sqrt{{\sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{2}} \cdot {x.im}^{y.re} \]

   9. Applied egg-rr 63.1%

      \[\leadsto \color{blue}{\sqrt{{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}} \cdot {x.im}^{y.re} \]
   10. Step-by-step derivation

       1. unpow2 63.1%

          \[\leadsto \sqrt{\color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}} \cdot {x.im}^{y.re} \]

       2. rem-sqrt-square 62.9%

          \[\leadsto \color{blue}{\left|\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right|} \cdot {x.im}^{y.re} \]

   11. Simplified 62.9%

       \[\leadsto \color{blue}{\left|\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right|} \cdot {x.im}^{y.re} \]

   if -6.20000000000000033e104 < x.re < 1.75e52

   1. Initial program 43.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 39.3%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 39.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 39.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 39.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 39.3%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 39.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 38.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 38.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 67.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 67.9%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 67.9%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 47.8%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 47.8%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 47.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 47.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 47.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 47.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.re around 0 40.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.8 \cdot 10^{+137}:\\ \;\;\;\;{x.re}^{y.re} \cdot \sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)\\ \mathbf{elif}\;x.re \leq -6.2 \cdot 10^{+104}:\\ \;\;\;\;{x.im}^{y.re} \cdot \left|\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right|\\ \mathbf{elif}\;x.re \leq 1.75 \cdot 10^{+52}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot \sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)\\ \end{array} \]

Alternative 19: 49.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{t_0}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+141}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.im (atan2 x.im x.re)))))
   (if (<= y.im -1.05e+73)
     (/ (sin (* y.im (log (- x.im)))) t_0)
     (if (<= y.im 2.8e+141)
       (* (sin (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
       (/ (sin (* y.im (log x.im))) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -1.05e+73) {
		tmp = sin((y_46_im * log(-x_46_im))) / t_0;
	} else if (y_46_im <= 2.8e+141) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = sin((y_46_im * log(x_46_im))) / t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -1.05e+73) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_im))) / t_0;
	} else if (y_46_im <= 2.8e+141) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.sin((y_46_im * Math.log(x_46_im))) / t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -1.05e+73:
		tmp = math.sin((y_46_im * math.log(-x_46_im))) / t_0
	elif y_46_im <= 2.8e+141:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.sin((y_46_im * math.log(x_46_im))) / t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -1.05e+73)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_im)))) / t_0);
	elseif (y_46_im <= 2.8e+141)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(sin(Float64(y_46_im * log(x_46_im))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -1.05e+73)
		tmp = sin((y_46_im * log(-x_46_im))) / t_0;
	elseif (y_46_im <= 2.8e+141)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = sin((y_46_im * log(x_46_im))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e+73], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 2.8e+141], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.0500000000000001e73

   1. Initial program 27.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 21.2%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 21.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 21.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 21.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 21.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 21.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 18.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 18.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 45.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 45.7%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 45.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.re around 0 25.3%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   5. Taylor expanded in x.im around -inf 35.3%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-1 \cdot x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
   6. Step-by-step derivation

      1. mul-1-neg 35.3%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   7. Simplified 35.3%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   if -1.0500000000000001e73 < y.im < 2.79999999999999991e141

   1. Initial program 39.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 37.3%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 37.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 37.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 37.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 37.3%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 36.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 36.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 73.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 73.6%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 50.9%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 50.9%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 50.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 50.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 54.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 54.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   if 2.79999999999999991e141 < y.im

   1. Initial program 36.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 25.0%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 25.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 25.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 25.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 25.0%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 25.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 25.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 25.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 47.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 47.3%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.re around 0 36.2%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   5. Taylor expanded in x.re around 0 50.1%

      \[\leadsto \frac{\color{blue}{\sin \left(\log x.im \cdot y.im\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{+141}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 20: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 3.2 \cdot 10^{+141}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 3.2e+141)
   (* (sin (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
   (/ (sin (* y.im (log x.im))) (exp (* y.im (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 3.2e+141) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = sin((y_46_im * log(x_46_im))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 3.2e+141) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.sin((y_46_im * Math.log(x_46_im))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 3.2e+141:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.sin((y_46_im * math.log(x_46_im))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 3.2e+141)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(sin(Float64(y_46_im * log(x_46_im))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 3.2e+141)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = sin((y_46_im * log(x_46_im))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 3.2e+141], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 3.20000000000000019e141

   1. Initial program 36.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 33.7%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 33.7%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.7%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 33.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 33.7%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 33.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 32.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 32.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 67.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 67.4%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 67.4%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 47.2%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 47.2%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 47.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 47.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 47.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 47.0%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   if 3.20000000000000019e141 < y.im

   1. Initial program 36.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 25.0%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 25.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 25.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 25.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 25.0%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 25.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 25.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 25.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 47.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 47.3%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 47.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.re around 0 36.2%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   5. Taylor expanded in x.re around 0 50.1%

      \[\leadsto \frac{\color{blue}{\sin \left(\log x.im \cdot y.im\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 47.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 3.2 \cdot 10^{+141}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 21: 45.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (sin (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. exp-diff 32.5%

      \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. +-rgt-identity 32.5%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. +-rgt-identity 32.5%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. exp-to-pow 32.5%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. hypot-def 32.5%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. *-commutative 32.5%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   7. exp-prod 31.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. fma-def 31.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   9. hypot-def 64.6%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   10. *-commutative 64.6%

       \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

3. Simplified 64.6%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

4. Taylor expanded in y.im around 0 45.1%

   \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
5. Step-by-step derivation

   1. *-commutative 45.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

   2. unpow2 45.1%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

   3. unpow2 45.1%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

   4. hypot-def 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

6. Simplified 44.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

7. Final simplification 44.2%

   \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

Alternative 22: 37.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.re \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;t_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.re 5.8e-14) (* t_0 (pow x.im y.re)) (* t_0 (pow x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= 5.8e-14) {
		tmp = t_0 * pow(x_46_im, y_46_re);
	} else {
		tmp = t_0 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46re <= 5.8d-14) then
        tmp = t_0 * (x_46im ** y_46re)
    else
        tmp = t_0 * (x_46re ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= 5.8e-14) {
		tmp = t_0 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = t_0 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_re <= 5.8e-14:
		tmp = t_0 * math.pow(x_46_im, y_46_re)
	else:
		tmp = t_0 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_re <= 5.8e-14)
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	else
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_re <= 5.8e-14)
		tmp = t_0 * (x_46_im ^ y_46_re);
	else
		tmp = t_0 * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, 5.8e-14], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.8000000000000005e-14

   1. Initial program 39.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 35.0%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 35.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 35.0%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 35.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 35.0%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 35.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 34.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 34.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 67.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 67.2%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 67.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 45.7%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 45.7%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 45.7%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 45.7%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 45.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 45.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.re around 0 37.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]

   if 5.8000000000000005e-14 < x.re

   1. Initial program 26.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 22.3%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 22.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 22.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 22.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 22.3%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 22.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 20.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. fma-def 20.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      9. hypot-def 53.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-commutative 53.8%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 53.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   4. Taylor expanded in y.im around 0 42.5%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   5. Step-by-step derivation

      1. *-commutative 42.5%

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. unpow2 42.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      3. unpow2 42.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      4. hypot-def 40.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 40.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.im around 0 40.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5.8 \cdot 10^{-14}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]

Alternative 23: 31.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (sin (* y.re (atan2 x.im x.re))) (pow x.im y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_im, y_46_re);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = sin((y_46re * atan2(x_46im, x_46re))) * (x_46im ** y_46re)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(x_46_im, y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(x_46_im, y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_im ^ y_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (x_46_im ^ y_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 36.8%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. exp-diff 32.5%

      \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. +-rgt-identity 32.5%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. +-rgt-identity 32.5%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. exp-to-pow 32.5%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. hypot-def 32.5%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. *-commutative 32.5%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   7. exp-prod 31.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. fma-def 31.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   9. hypot-def 64.6%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   10. *-commutative 64.6%

       \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

3. Simplified 64.6%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

4. Taylor expanded in y.im around 0 45.1%

   \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
5. Step-by-step derivation

   1. *-commutative 45.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

   2. unpow2 45.1%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

   3. unpow2 45.1%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

   4. hypot-def 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

6. Simplified 44.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

7. Taylor expanded in x.re around 0 35.7%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]

8. Final simplification 35.7%

   \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re} \]

Reproduce

herbie shell --seed 2023208 
(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)))))