powComplex, imaginary part

Percentage Accurate: 40.5% → 79.7%

Time: 27.6s

Alternatives: 26

Speedup: 2.6×

• 36.4% 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: 79.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := t\_0 \cdot y.im\\ t_2 := y.re \cdot t\_0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq 5 \cdot 10^{+18}:\\ \;\;\;\;e^{t\_2} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t\_1\right)\right)\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+242}:\\ \;\;\;\;e^{\sqrt[3]{{t\_2}^{3}}} \cdot \sin t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* t_0 y.im))
        (t_2 (- (* y.re t_0) (* y.im (atan2 x.im x.re)))))
   (if (<= y.re 5e+18)
     (* (exp t_2) (sin (fma y.re (atan2 x.im x.re) t_1)))
     (if (<= y.re 3.5e+242)
       (* (exp (cbrt (pow t_2 3.0))) (sin t_1))
       (*
        (sin (* y.re (pow (cbrt (atan2 x.im x.re)) 3.0)))
        (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) {
	double t_0 = log(hypot(x_46_im, x_46_re));
	double t_1 = t_0 * y_46_im;
	double t_2 = (y_46_re * t_0) - (y_46_im * atan2(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= 5e+18) {
		tmp = exp(t_2) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), t_1));
	} else if (y_46_re <= 3.5e+242) {
		tmp = exp(cbrt(pow(t_2, 3.0))) * sin(t_1);
	} else {
		tmp = sin((y_46_re * pow(cbrt(atan2(x_46_im, x_46_re)), 3.0))) * pow(hypot(x_46_im, 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 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(t_0 * y_46_im)
	t_2 = Float64(Float64(y_46_re * t_0) - Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= 5e+18)
		tmp = Float64(exp(t_2) * sin(fma(y_46_re, atan(x_46_im, x_46_re), t_1)));
	elseif (y_46_re <= 3.5e+242)
		tmp = Float64(exp(cbrt((t_2 ^ 3.0))) * sin(t_1));
	else
		tmp = Float64(sin(Float64(y_46_re * (cbrt(atan(x_46_im, x_46_re)) ^ 3.0))) * (hypot(x_46_im, x_46_re) ^ y_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$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, 5e+18], N[(N[Exp[t$95$2], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.5e+242], N[(N[Exp[N[Power[N[Power[t$95$2, 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$re * N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < 5e18

   1. Initial program 44.1%

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

      1. fma-define 44.1%

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

      2. hypot-define 66.0%

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

      3. *-commutative 66.0%

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

      4. add-cbrt-cube 57.2%

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

      5. pow3 57.2%

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

   4. Applied egg-rr 57.2%

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

      1. add-exp-log 47.8%

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

      2. rem-cbrt-cube 51.3%

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

      3. log-prod 34.7%

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

   6. Applied egg-rr 42.1%

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

      1. exp-sum 42.1%

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

      2. rem-exp-log 86.2%

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

   8. Simplified 86.2%

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

   if 5e18 < y.re < 3.4999999999999999e242

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0 33.3%

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

      1. *-commutative 33.3%

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

      2. unpow2 33.3%

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

      3. unpow2 33.3%

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

      4. hypot-undefine 68.5%

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

   5. Simplified 68.5%

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

      1. add-cbrt-cube 68.5%

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

      2. pow3 68.5%

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

      3. *-commutative 68.5%

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

      4. +-commutative 68.5%

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

      5. hypot-undefine 73.7%

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

      6. *-commutative 73.7%

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

   7. Applied egg-rr 73.7%

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

   if 3.4999999999999999e242 < 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. fmm-def 33.3%

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

      2. hypot-define 33.3%

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

      3. distribute-rgt-neg-out 33.3%

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

      4. fma-define 33.3%

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

      5. hypot-define 66.7%

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

      6. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in y.im around 0 73.3%

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

      1. unpow2 73.3%

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

      2. unpow2 73.3%

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

      3. hypot-undefine 73.3%

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

   7. Simplified 73.3%

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

      1. add-cube-cbrt 73.3%

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

      2. pow3 86.7%

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

   9. Applied egg-rr 86.7%

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

Alternative 2: 28.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{y.re \cdot t\_1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(t\_0 + y.im \cdot t\_1\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-274} \lor \neg \left(t\_2 \leq 10^{-196}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(t\_0\right)} + -1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2
         (*
          (exp (- (* y.re t_1) (* y.im (atan2 x.im x.re))))
          (sin (+ t_0 (* y.im t_1))))))
   (if (or (<= t_2 -5e-274) (not (<= t_2 1e-196)))
     (log1p (expm1 t_0))
     (+ (exp (log1p t_0)) -1.0))))

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 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((y_46_re * t_1) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin((t_0 + (y_46_im * t_1)));
	double tmp;
	if ((t_2 <= -5e-274) || !(t_2 <= 1e-196)) {
		tmp = log1p(expm1(t_0));
	} else {
		tmp = exp(log1p(t_0)) + -1.0;
	}
	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.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = Math.exp(((y_46_re * t_1) - (y_46_im * Math.atan2(x_46_im, x_46_re)))) * Math.sin((t_0 + (y_46_im * t_1)));
	double tmp;
	if ((t_2 <= -5e-274) || !(t_2 <= 1e-196)) {
		tmp = Math.log1p(Math.expm1(t_0));
	} else {
		tmp = Math.exp(Math.log1p(t_0)) + -1.0;
	}
	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.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	t_2 = math.exp(((y_46_re * t_1) - (y_46_im * math.atan2(x_46_im, x_46_re)))) * math.sin((t_0 + (y_46_im * t_1)))
	tmp = 0
	if (t_2 <= -5e-274) or not (t_2 <= 1e-196):
		tmp = math.log1p(math.expm1(t_0))
	else:
		tmp = math.exp(math.log1p(t_0)) + -1.0
	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 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = Float64(exp(Float64(Float64(y_46_re * t_1) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(t_0 + Float64(y_46_im * t_1))))
	tmp = 0.0
	if ((t_2 <= -5e-274) || !(t_2 <= 1e-196))
		tmp = log1p(expm1(t_0));
	else
		tmp = Float64(exp(log1p(t_0)) + -1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(N[(y$46$re * t$95$1), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 + N[(y$46$im * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e-274], N[Not[LessEqual[t$95$2, 1e-196]], $MachinePrecision]], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[Log[1 + t$95$0], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -5e-274 or 1e-196 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 24.9%

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

      1. fmm-def 24.9%

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

      2. hypot-define 24.9%

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

      3. distribute-rgt-neg-out 24.9%

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

      4. fma-define 24.9%

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

      5. hypot-define 73.1%

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

      6. *-commutative 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in y.im around 0 40.7%

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

      1. unpow2 40.7%

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

      2. unpow2 40.7%

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

      3. hypot-undefine 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in y.re around 0 14.0%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 28.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   10. Applied egg-rr 28.7%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   if -5e-274 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1e-196

   1. Initial program 100.0%

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

      1. fmm-def 100.0%

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

      2. hypot-define 100.0%

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

      3. distribute-rgt-neg-out 100.0%

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

      4. fma-define 100.0%

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

      5. hypot-define 100.0%

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

      6. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y.im around 0 58.3%

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

      1. unpow2 58.3%

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

      2. unpow2 58.3%

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

      3. hypot-undefine 58.3%

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

   7. Simplified 58.3%

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

   8. Taylor expanded in y.re around 0 5.7%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 5.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      2. expm1-undefine 39.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} - 1} \]

   10. Applied egg-rr 39.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} - 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right) \leq -5 \cdot 10^{-274} \lor \neg \left(e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right) \leq 10^{-196}\right):\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{log1p}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := t\_0 \cdot y.im\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{-15} \lor \neg \left(y.im \leq 6.5 \cdot 10^{-74}\right):\\ \;\;\;\;e^{\sqrt[3]{{\left(y.re \cdot t\_0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \cdot \sin t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t\_1\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re))) (t_1 (* t_0 y.im)))
   (if (or (<= y.im -6e-15) (not (<= y.im 6.5e-74)))
     (*
      (exp (cbrt (pow (- (* y.re t_0) (* y.im (atan2 x.im x.re))) 3.0)))
      (sin t_1))
     (* (sin (fma y.re (atan2 x.im x.re) t_1)) (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) {
	double t_0 = log(hypot(x_46_im, x_46_re));
	double t_1 = t_0 * y_46_im;
	double tmp;
	if ((y_46_im <= -6e-15) || !(y_46_im <= 6.5e-74)) {
		tmp = exp(cbrt(pow(((y_46_re * t_0) - (y_46_im * atan2(x_46_im, x_46_re))), 3.0))) * sin(t_1);
	} else {
		tmp = sin(fma(y_46_re, atan2(x_46_im, x_46_re), t_1)) * pow(hypot(x_46_im, 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 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(t_0 * y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6e-15) || !(y_46_im <= 6.5e-74))
		tmp = Float64(exp(cbrt((Float64(Float64(y_46_re * t_0) - Float64(y_46_im * atan(x_46_im, x_46_re))) ^ 3.0))) * sin(t_1));
	else
		tmp = Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), t_1)) * (hypot(x_46_im, x_46_re) ^ y_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$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * y$46$im), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -6e-15], N[Not[LessEqual[y$46$im, 6.5e-74]], $MachinePrecision]], N[(N[Exp[N[Power[N[Power[N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6e-15 or 6.5000000000000002e-74 < y.im

   1. Initial program 34.8%

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

   3. Taylor expanded in y.re around 0 36.2%

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

      1. *-commutative 36.2%

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

      2. unpow2 36.2%

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

      3. unpow2 36.2%

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

      4. hypot-undefine 63.2%

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

   5. Simplified 63.2%

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

      1. add-cbrt-cube 63.2%

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

      2. pow3 63.2%

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

      3. *-commutative 63.2%

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

      4. +-commutative 63.2%

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

      5. hypot-undefine 76.2%

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

      6. *-commutative 76.2%

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

   7. Applied egg-rr 76.2%

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

   if -6e-15 < y.im < 6.5000000000000002e-74

   1. Initial program 45.2%

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

      1. fma-define 45.2%

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

      2. hypot-define 69.8%

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

      3. *-commutative 69.8%

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

      4. add-cbrt-cube 54.3%

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

      5. pow3 54.3%

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

   4. Applied egg-rr 54.3%

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

      1. add-exp-log 42.1%

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

      2. rem-cbrt-cube 48.5%

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

      3. log-prod 37.9%

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

   6. Applied egg-rr 45.4%

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

      1. exp-sum 45.5%

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

      2. rem-exp-log 88.7%

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

   8. Simplified 88.7%

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

      1. add-cube-cbrt 64.7%

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

      2. pow3 67.3%

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

   10. Applied egg-rr 88.7%

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

   11. Taylor expanded in y.im around 0 69.8%

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

       1. unpow2 69.8%

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

       2. unpow2 69.8%

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

       3. hypot-undefine 88.7%

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

   13. Simplified 88.7%

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

4. Final simplification 81.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{-15} \lor \neg \left(y.im \leq 6.5 \cdot 10^{-74}\right):\\ \;\;\;\;e^{\sqrt[3]{{\left(y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.6:\\ \;\;\;\;e^{y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{-y.re}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 1.76 \cdot 10^{+242}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (sin (* y.re (pow (cbrt (atan2 x.im x.re)) 3.0)))
          (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -9.5e-7)
     t_0
     (if (<= y.re 1.6)
       (*
        (exp (* y.re (* y.im (/ (atan2 x.im x.re) (- y.re)))))
        (sin (fma (log (hypot x.re x.im)) y.im (* y.re (atan2 x.im x.re)))))
       (if (<= y.re 1.76e+242)
         (*
          (sin (* (log (hypot x.im x.re)) y.im))
          (exp
           (-
            (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
            (* y.im (atan2 x.im x.re)))))
         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 * pow(cbrt(atan2(x_46_im, x_46_re)), 3.0))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -9.5e-7) {
		tmp = t_0;
	} else if (y_46_re <= 1.6) {
		tmp = exp((y_46_re * (y_46_im * (atan2(x_46_im, x_46_re) / -y_46_re)))) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	} else if (y_46_re <= 1.76e+242) {
		tmp = sin((log(hypot(x_46_im, x_46_re)) * y_46_im)) * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(y_46_re * (cbrt(atan(x_46_im, x_46_re)) ^ 3.0))) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -9.5e-7)
		tmp = t_0;
	elseif (y_46_re <= 1.6)
		tmp = Float64(exp(Float64(y_46_re * Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / Float64(-y_46_re))))) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	elseif (y_46_re <= 1.76e+242)
		tmp = Float64(sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)) * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(y$46$re * N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.5e-7], t$95$0, If[LessEqual[y$46$re, 1.6], N[(N[Exp[N[(y$46$re * N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.76e+242], N[(N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * 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] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.5000000000000001e-7 or 1.76e242 < y.re

   1. Initial program 41.3%

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

      1. fmm-def 41.3%

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

      2. hypot-define 41.3%

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

      3. distribute-rgt-neg-out 41.3%

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

      4. fma-define 41.3%

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

      5. hypot-define 86.6%

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

      6. *-commutative 86.6%

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

   3. Simplified 86.6%

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

   5. Taylor expanded in y.im around 0 84.1%

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

      1. unpow2 84.1%

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

      2. unpow2 84.1%

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

      3. hypot-undefine 85.3%

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

   7. Simplified 85.3%

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

      1. add-cube-cbrt 85.3%

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

      2. pow3 89.3%

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

   9. Applied egg-rr 89.3%

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

   if -9.5000000000000001e-7 < y.re < 1.6000000000000001

   1. Initial program 45.1%

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

      1. fmm-def 45.1%

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

      2. hypot-define 45.1%

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

      3. distribute-rgt-neg-out 45.1%

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

      4. fma-define 45.1%

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

      5. hypot-define 83.6%

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

      6. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in y.re around inf 53.7%

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

      1. mul-1-neg 53.7%

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

      2. unsub-neg 53.7%

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

      3. unpow2 53.7%

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

      4. unpow2 53.7%

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

      5. hypot-undefine 83.6%

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

      6. associate-/l* 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in y.im around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. associate-*r/ 82.2%

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

      3. distribute-rgt-neg-in 82.2%

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

      4. distribute-neg-frac2 82.2%

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

   10. Simplified 82.2%

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

   if 1.6000000000000001 < y.re < 1.76e242

   1. Initial program 27.0%

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

   3. Taylor expanded in y.re around 0 31.7%

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

      1. *-commutative 31.7%

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

      2. unpow2 31.7%

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

      3. unpow2 31.7%

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

      4. hypot-undefine 68.3%

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

   5. Simplified 68.3%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.6:\\ \;\;\;\;e^{y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{-y.re}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 1.76 \cdot 10^{+242}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.55 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9:\\ \;\;\;\;e^{y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{-y.re}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+239}:\\ \;\;\;\;\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (sin (* y.re (pow (cbrt (atan2 x.im x.re)) 3.0)))
          (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -2.55e-6)
     t_0
     (if (<= y.re 9.0)
       (*
        (exp (* y.re (* y.im (/ (atan2 x.im x.re) (- y.re)))))
        (sin (fma (log (hypot x.re x.im)) y.im (* y.re (atan2 x.im x.re)))))
       (if (<= y.re 2.3e+239)
         (*
          (* (log (hypot x.im x.re)) y.im)
          (exp
           (-
            (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
            (* y.im (atan2 x.im x.re)))))
         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 * pow(cbrt(atan2(x_46_im, x_46_re)), 3.0))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.55e-6) {
		tmp = t_0;
	} else if (y_46_re <= 9.0) {
		tmp = exp((y_46_re * (y_46_im * (atan2(x_46_im, x_46_re) / -y_46_re)))) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	} else if (y_46_re <= 2.3e+239) {
		tmp = (log(hypot(x_46_im, x_46_re)) * y_46_im) * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(y_46_re * (cbrt(atan(x_46_im, x_46_re)) ^ 3.0))) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.55e-6)
		tmp = t_0;
	elseif (y_46_re <= 9.0)
		tmp = Float64(exp(Float64(y_46_re * Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / Float64(-y_46_re))))) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	elseif (y_46_re <= 2.3e+239)
		tmp = Float64(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im) * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(y$46$re * N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.55e-6], t$95$0, If[LessEqual[y$46$re, 9.0], N[(N[Exp[N[(y$46$re * N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.3e+239], N[(N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] * 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] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.5500000000000001e-6 or 2.3000000000000002e239 < y.re

   1. Initial program 40.8%

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

      1. fmm-def 40.8%

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

      2. hypot-define 40.8%

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

      3. distribute-rgt-neg-out 40.8%

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

      4. fma-define 40.8%

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

      5. hypot-define 85.5%

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

      6. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in y.im around 0 83.0%

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

      1. unpow2 83.0%

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

      2. unpow2 83.0%

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

      3. hypot-undefine 84.2%

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

   7. Simplified 84.2%

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

      1. add-cube-cbrt 84.2%

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

      2. pow3 88.2%

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

   9. Applied egg-rr 88.2%

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

   if -2.5500000000000001e-6 < y.re < 9

   1. Initial program 45.1%

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

      1. fmm-def 45.1%

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

      2. hypot-define 45.1%

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

      3. distribute-rgt-neg-out 45.1%

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

      4. fma-define 45.1%

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

      5. hypot-define 83.6%

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

      6. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in y.re around inf 53.7%

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

      1. mul-1-neg 53.7%

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

      2. unsub-neg 53.7%

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

      3. unpow2 53.7%

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

      4. unpow2 53.7%

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

      5. hypot-undefine 83.6%

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

      6. associate-/l* 83.6%

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

   7. Simplified 83.6%

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

   8. Taylor expanded in y.im around inf 82.2%

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

      1. mul-1-neg 82.2%

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

      2. associate-*r/ 82.2%

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

      3. distribute-rgt-neg-in 82.2%

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

      4. distribute-neg-frac2 82.2%

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

   10. Simplified 82.2%

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

   if 9 < y.re < 2.3000000000000002e239

   1. Initial program 27.4%

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

   3. Taylor expanded in y.re around 0 30.6%

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

      1. *-commutative 30.6%

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

      2. unpow2 30.6%

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

      3. unpow2 30.6%

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

      4. hypot-undefine 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in y.im around 0 59.8%

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

      1. unpow2 59.8%

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

      2. unpow2 59.8%

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

      3. hypot-undefine 64.6%

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

   8. Simplified 64.6%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.55 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 9:\\ \;\;\;\;e^{y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{-y.re}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+239}:\\ \;\;\;\;\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{-10} \lor \neg \left(y.im \leq 1.8 \cdot 10^{-14}\right):\\ \;\;\;\;\sin t\_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t\_0\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (log (hypot x.im x.re)) y.im)))
   (if (or (<= y.im -8e-10) (not (<= y.im 1.8e-14)))
     (* (sin t_0) (exp (* (atan2 x.im x.re) (- y.im))))
     (* (sin (fma y.re (atan2 x.im x.re) t_0)) (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) {
	double t_0 = log(hypot(x_46_im, x_46_re)) * y_46_im;
	double tmp;
	if ((y_46_im <= -8e-10) || !(y_46_im <= 1.8e-14)) {
		tmp = sin(t_0) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = sin(fma(y_46_re, atan2(x_46_im, x_46_re), t_0)) * pow(hypot(x_46_im, 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(log(hypot(x_46_im, x_46_re)) * y_46_im)
	tmp = 0.0
	if ((y_46_im <= -8e-10) || !(y_46_im <= 1.8e-14))
		tmp = Float64(sin(t_0) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), t_0)) * (hypot(x_46_im, x_46_re) ^ y_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[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -8e-10], N[Not[LessEqual[y$46$im, 1.8e-14]], $MachinePrecision]], N[(N[Sin[t$95$0], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.00000000000000029e-10 or 1.7999999999999999e-14 < y.im

   1. Initial program 35.4%

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

      1. fmm-def 35.4%

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

      2. hypot-define 35.4%

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

      3. distribute-rgt-neg-out 35.4%

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

      4. fma-define 35.4%

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

      5. hypot-define 67.9%

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

      6. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in y.re around 0 30.9%

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

      1. neg-mul-1 30.9%

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

      2. distribute-lft-neg-in 30.9%

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

      3. *-commutative 30.9%

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

      4. unpow2 30.9%

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

      5. unpow2 30.9%

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

      6. hypot-undefine 61.2%

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

   7. Simplified 61.2%

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

   if -8.00000000000000029e-10 < y.im < 1.7999999999999999e-14

   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. Add Preprocessing
   3. Step-by-step derivation

      1. fma-define 43.6%

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

      2. hypot-define 70.1%

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

      3. *-commutative 70.1%

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

      4. add-cbrt-cube 55.9%

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

      5. pow3 55.9%

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

   4. Applied egg-rr 55.9%

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

      1. add-exp-log 44.7%

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

      2. rem-cbrt-cube 50.6%

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

      3. log-prod 39.3%

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

   6. Applied egg-rr 46.9%

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

      1. exp-sum 47.0%

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

      2. rem-exp-log 88.9%

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

   8. Simplified 88.9%

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

      1. add-cube-cbrt 65.4%

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

      2. pow3 67.8%

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

   10. Applied egg-rr 88.9%

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

   11. Taylor expanded in y.im around 0 70.1%

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

       1. unpow2 70.1%

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

       2. unpow2 70.1%

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

       3. hypot-undefine 88.9%

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

   13. Simplified 88.9%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{-10} \lor \neg \left(y.im \leq 1.8 \cdot 10^{-14}\right):\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-50}:\\ \;\;\;\;\sin t\_2 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;t\_0 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+238}:\\ \;\;\;\;t\_2 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* y.im (atan2 x.im x.re)))))
        (t_1
         (*
          (sin (* y.re (pow (cbrt (atan2 x.im x.re)) 3.0)))
          (pow (hypot x.im x.re) y.re)))
        (t_2 (* (log (hypot x.im x.re)) y.im)))
   (if (<= y.re -2.5e-9)
     t_1
     (if (<= y.re 4.5e-50)
       (* (sin t_2) (exp (* (atan2 x.im x.re) (- y.im))))
       (if (<= y.re 1.8e+26)
         (* t_0 (sin (* y.re (atan2 x.im x.re))))
         (if (<= y.re 2e+238) (* t_2 t_0) t_1))))))

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_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_1 = sin((y_46_re * pow(cbrt(atan2(x_46_im, x_46_re)), 3.0))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = log(hypot(x_46_im, x_46_re)) * y_46_im;
	double tmp;
	if (y_46_re <= -2.5e-9) {
		tmp = t_1;
	} else if (y_46_re <= 4.5e-50) {
		tmp = sin(t_2) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (y_46_re <= 1.8e+26) {
		tmp = t_0 * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 2e+238) {
		tmp = t_2 * t_0;
	} else {
		tmp = 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 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	double t_1 = Math.sin((y_46_re * Math.pow(Math.cbrt(Math.atan2(x_46_im, x_46_re)), 3.0))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im;
	double tmp;
	if (y_46_re <= -2.5e-9) {
		tmp = t_1;
	} else if (y_46_re <= 4.5e-50) {
		tmp = Math.sin(t_2) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (y_46_re <= 1.8e+26) {
		tmp = t_0 * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 2e+238) {
		tmp = t_2 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	t_1 = Float64(sin(Float64(y_46_re * (cbrt(atan(x_46_im, x_46_re)) ^ 3.0))) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	t_2 = Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.5e-9)
		tmp = t_1;
	elseif (y_46_re <= 4.5e-50)
		tmp = Float64(sin(t_2) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	elseif (y_46_re <= 1.8e+26)
		tmp = Float64(t_0 * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 2e+238)
		tmp = Float64(t_2 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = 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] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[N[(y$46$re * N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5e-9], t$95$1, If[LessEqual[y$46$re, 4.5e-50], N[(N[Sin[t$95$2], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.8e+26], N[(t$95$0 * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2e+238], N[(t$95$2 * t$95$0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.5000000000000001e-9 or 2.0000000000000001e238 < y.re

   1. Initial program 40.8%

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

      1. fmm-def 40.8%

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

      2. hypot-define 40.8%

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

      3. distribute-rgt-neg-out 40.8%

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

      4. fma-define 40.8%

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

      5. hypot-define 85.5%

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

      6. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in y.im around 0 83.0%

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

      1. unpow2 83.0%

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

      2. unpow2 83.0%

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

      3. hypot-undefine 84.2%

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

   7. Simplified 84.2%

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

      1. add-cube-cbrt 84.2%

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

      2. pow3 88.2%

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

   9. Applied egg-rr 88.2%

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

   if -2.5000000000000001e-9 < y.re < 4.49999999999999962e-50

   1. Initial program 41.9%

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

      1. fmm-def 41.9%

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

      2. hypot-define 41.9%

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

      3. distribute-rgt-neg-out 41.9%

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

      4. fma-define 41.9%

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

      5. hypot-define 84.0%

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

      6. *-commutative 84.0%

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

   3. Simplified 84.0%

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

   5. Taylor expanded in y.re around 0 35.8%

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

      1. neg-mul-1 35.8%

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

      2. distribute-lft-neg-in 35.8%

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

      3. *-commutative 35.8%

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

      4. unpow2 35.8%

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

      5. unpow2 35.8%

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

      6. hypot-undefine 68.3%

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

   7. Simplified 68.3%

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

   if 4.49999999999999962e-50 < y.re < 1.80000000000000012e26

   1. Initial program 61.0%

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

   3. Taylor expanded in y.im around 0 83.3%

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

   if 1.80000000000000012e26 < y.re < 2.0000000000000001e238

   1. Initial program 25.9%

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

   3. Taylor expanded in y.re around 0 31.5%

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

      1. *-commutative 31.5%

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

      2. unpow2 31.5%

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

      3. unpow2 31.5%

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

      4. hypot-undefine 68.6%

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

   5. Simplified 68.6%

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

   6. Taylor expanded in y.im around 0 59.3%

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

      1. unpow2 59.3%

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

      2. unpow2 59.3%

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

      3. hypot-undefine 64.9%

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

   8. Simplified 64.9%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-9}:\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-50}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+26}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+238}:\\ \;\;\;\;\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.3 \cdot 10^{-8} \lor \neg \left(y.re \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4.3e-8) (not (<= y.re 8.2e-5)))
   (*
    (sin (* y.re (pow (cbrt (atan2 x.im x.re)) 3.0)))
    (pow (hypot x.im x.re) y.re))
   (*
    (sin (* (log (hypot x.im x.re)) y.im))
    (exp (* (atan2 x.im x.re) (- y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4.3e-8) || !(y_46_re <= 8.2e-5)) {
		tmp = sin((y_46_re * pow(cbrt(atan2(x_46_im, x_46_re)), 3.0))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = sin((log(hypot(x_46_im, x_46_re)) * y_46_im)) * exp((atan2(x_46_im, x_46_re) * -y_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 tmp;
	if ((y_46_re <= -4.3e-8) || !(y_46_re <= 8.2e-5)) {
		tmp = Math.sin((y_46_re * Math.pow(Math.cbrt(Math.atan2(x_46_im, x_46_re)), 3.0))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im)) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4.3e-8) || !(y_46_re <= 8.2e-5))
		tmp = Float64(sin(Float64(y_46_re * (cbrt(atan(x_46_im, x_46_re)) ^ 3.0))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4.3e-8], N[Not[LessEqual[y$46$re, 8.2e-5]], $MachinePrecision]], N[(N[Sin[N[(y$46$re * N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.3000000000000001e-8 or 8.20000000000000009e-5 < y.re

   1. Initial program 35.2%

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

      1. fmm-def 35.2%

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

      2. hypot-define 35.2%

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

      3. distribute-rgt-neg-out 35.2%

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

      4. fma-define 35.2%

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

      5. hypot-define 74.1%

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

      6. *-commutative 74.1%

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y.im around 0 66.3%

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

      1. unpow2 66.3%

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

      2. unpow2 66.3%

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

      3. hypot-undefine 67.0%

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

   7. Simplified 67.0%

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

      1. add-cube-cbrt 71.3%

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

      2. pow3 72.7%

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

   9. Applied egg-rr 72.7%

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

   if -4.3000000000000001e-8 < y.re < 8.20000000000000009e-5

   1. Initial program 44.6%

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

      1. fmm-def 44.6%

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

      2. hypot-define 44.6%

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

      3. distribute-rgt-neg-out 44.6%

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

      4. fma-define 44.6%

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

      5. hypot-define 83.5%

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

      6. *-commutative 83.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in y.re around 0 37.4%

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

      1. neg-mul-1 37.4%

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

      2. distribute-lft-neg-in 37.4%

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

      3. *-commutative 37.4%

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

      4. unpow2 37.4%

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

      5. unpow2 37.4%

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

      6. hypot-undefine 67.4%

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

   7. Simplified 67.4%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.3 \cdot 10^{-8} \lor \neg \left(y.re \leq 8.2 \cdot 10^{-5}\right):\\ \;\;\;\;\sin \left(y.re \cdot {\left(\sqrt[3]{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 55.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot t\_0\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := {\log \left(e^{{t\_2}^{3}}\right)}^{0.3333333333333333}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+58}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-217}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{-224}:\\ \;\;\;\;t\_0 \cdot \sin t\_2\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+183}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (* y.im (* (log (hypot x.im x.re)) t_0)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (pow (log (exp (pow t_2 3.0))) 0.3333333333333333)))
   (if (<= y.im -1e+58)
     t_3
     (if (<= y.im -4.2e-217)
       t_1
       (if (<= y.im 5.8e-224)
         (* t_0 (sin t_2))
         (if (<= y.im 5.8e+183) t_1 t_3))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = pow(log(exp(pow(t_2, 3.0))), 0.3333333333333333);
	double tmp;
	if (y_46_im <= -1e+58) {
		tmp = t_3;
	} else if (y_46_im <= -4.2e-217) {
		tmp = t_1;
	} else if (y_46_im <= 5.8e-224) {
		tmp = t_0 * sin(t_2);
	} else if (y_46_im <= 5.8e+183) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * t_0);
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.pow(Math.log(Math.exp(Math.pow(t_2, 3.0))), 0.3333333333333333);
	double tmp;
	if (y_46_im <= -1e+58) {
		tmp = t_3;
	} else if (y_46_im <= -4.2e-217) {
		tmp = t_1;
	} else if (y_46_im <= 5.8e-224) {
		tmp = t_0 * Math.sin(t_2);
	} else if (y_46_im <= 5.8e+183) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * t_0)
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = math.pow(math.log(math.exp(math.pow(t_2, 3.0))), 0.3333333333333333)
	tmp = 0
	if y_46_im <= -1e+58:
		tmp = t_3
	elif y_46_im <= -4.2e-217:
		tmp = t_1
	elif y_46_im <= 5.8e-224:
		tmp = t_0 * math.sin(t_2)
	elif y_46_im <= 5.8e+183:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * t_0))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = log(exp((t_2 ^ 3.0))) ^ 0.3333333333333333
	tmp = 0.0
	if (y_46_im <= -1e+58)
		tmp = t_3;
	elseif (y_46_im <= -4.2e-217)
		tmp = t_1;
	elseif (y_46_im <= 5.8e-224)
		tmp = Float64(t_0 * sin(t_2));
	elseif (y_46_im <= 5.8e+183)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_1 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = log(exp((t_2 ^ 3.0))) ^ 0.3333333333333333;
	tmp = 0.0;
	if (y_46_im <= -1e+58)
		tmp = t_3;
	elseif (y_46_im <= -4.2e-217)
		tmp = t_1;
	elseif (y_46_im <= 5.8e-224)
		tmp = t_0 * sin(t_2);
	elseif (y_46_im <= 5.8e+183)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * t$95$0), $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[Log[N[Exp[N[Power[t$95$2, 3.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 0.3333333333333333], $MachinePrecision]}, If[LessEqual[y$46$im, -1e+58], t$95$3, If[LessEqual[y$46$im, -4.2e-217], t$95$1, If[LessEqual[y$46$im, 5.8e-224], N[(t$95$0 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5.8e+183], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -9.99999999999999944e57 or 5.8000000000000001e183 < y.im

   1. Initial program 38.4%

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

      1. fmm-def 38.4%

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

      2. hypot-define 38.4%

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

      3. distribute-rgt-neg-out 38.4%

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

      4. fma-define 38.4%

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

      5. hypot-define 66.3%

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

      6. *-commutative 66.3%

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

   3. Simplified 66.3%

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

   5. Taylor expanded in y.im around 0 33.0%

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

      1. unpow2 33.0%

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

      2. unpow2 33.0%

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

      3. hypot-undefine 22.5%

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

   7. Simplified 22.5%

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

   8. Taylor expanded in y.re around 0 3.1%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 23.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]

      2. pow1/3 26.6%

         \[\leadsto \color{blue}{{\left(\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 26.6%

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

   10. Applied egg-rr 26.6%

       \[\leadsto \color{blue}{{\left({\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. add-log-exp 42.6%

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

       2. *-commutative 42.6%

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

   12. Applied egg-rr 42.6%

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

   if -9.99999999999999944e57 < y.im < -4.2e-217 or 5.8000000000000001e-224 < y.im < 5.8000000000000001e183

   1. Initial program 37.0%

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

   3. Taylor expanded in y.re around 0 35.7%

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

      1. *-commutative 35.7%

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

      2. unpow2 35.7%

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

      3. unpow2 35.7%

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

      4. hypot-undefine 65.1%

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

   5. Simplified 65.1%

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

      1. log1p-expm1-u 65.1%

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

   7. Applied egg-rr 65.1%

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

   8. Taylor expanded in y.im around 0 41.0%

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. unpow2 41.0%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      2. unpow2 41.0%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      3. hypot-undefine 59.9%

         \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      4. unpow2 59.9%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right) \]

      5. unpow2 59.9%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}\right) \]

      6. hypot-undefine 68.1%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right) \]

   10. Simplified 68.1%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if -4.2e-217 < y.im < 5.8000000000000001e-224

   1. Initial program 49.9%

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

      1. fmm-def 49.9%

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

      2. hypot-define 49.9%

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

      3. distribute-rgt-neg-out 49.9%

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

      4. fma-define 49.9%

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

      5. hypot-define 90.3%

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

      6. *-commutative 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in y.im around 0 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. hypot-undefine 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+58}:\\ \;\;\;\;{\log \left(e^{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}\right)}^{0.3333333333333333}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-217}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{-224}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{+183}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;{\log \left(e^{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}\right)}^{0.3333333333333333}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := y.im \cdot \left(t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{if}\;y.re \leq -19500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 5.5:\\ \;\;\;\;\sin \left(t\_0 \cdot y.im\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+243}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* y.im (* t_0 (pow (hypot x.im x.re) y.re)))))
   (if (<= y.re -19500000.0)
     t_1
     (if (<= y.re 5.5)
       (* (sin (* t_0 y.im)) (exp (* (atan2 x.im x.re) (- y.im))))
       (if (<= y.re 5e+243)
         t_1
         (*
          (sin (* y.re (atan2 x.im x.re)))
          (pow
           (* x.re (+ 1.0 (* 0.5 (* (/ x.im x.re) (/ 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(hypot(x_46_im, x_46_re));
	double t_1 = y_46_im * (t_0 * pow(hypot(x_46_im, x_46_re), y_46_re));
	double tmp;
	if (y_46_re <= -19500000.0) {
		tmp = t_1;
	} else if (y_46_re <= 5.5) {
		tmp = sin((t_0 * y_46_im)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (y_46_re <= 5e+243) {
		tmp = t_1;
	} else {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / 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 = Math.log(Math.hypot(x_46_im, x_46_re));
	double t_1 = y_46_im * (t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re));
	double tmp;
	if (y_46_re <= -19500000.0) {
		tmp = t_1;
	} else if (y_46_re <= 5.5) {
		tmp = Math.sin((t_0 * y_46_im)) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (y_46_re <= 5e+243) {
		tmp = t_1;
	} else {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / 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.log(math.hypot(x_46_im, x_46_re))
	t_1 = y_46_im * (t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re))
	tmp = 0
	if y_46_re <= -19500000.0:
		tmp = t_1
	elif y_46_re <= 5.5:
		tmp = math.sin((t_0 * y_46_im)) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	elif y_46_re <= 5e+243:
		tmp = t_1
	else:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / 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 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(y_46_im * Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re)))
	tmp = 0.0
	if (y_46_re <= -19500000.0)
		tmp = t_1;
	elseif (y_46_re <= 5.5)
		tmp = Float64(sin(Float64(t_0 * y_46_im)) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	elseif (y_46_re <= 5e+243)
		tmp = t_1;
	else
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (Float64(x_46_re * Float64(1.0 + Float64(0.5 * Float64(Float64(x_46_im / x_46_re) * Float64(x_46_im / 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 = log(hypot(x_46_im, x_46_re));
	t_1 = y_46_im * (t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	tmp = 0.0;
	if (y_46_re <= -19500000.0)
		tmp = t_1;
	elseif (y_46_re <= 5.5)
		tmp = sin((t_0 * y_46_im)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	elseif (y_46_re <= 5e+243)
		tmp = t_1;
	else
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * ((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / 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[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -19500000.0], t$95$1, If[LessEqual[y$46$re, 5.5], N[(N[Sin[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5e+243], t$95$1, N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(x$46$re * N[(1.0 + N[(0.5 * N[(N[(x$46$im / x$46$re), $MachinePrecision] * N[(x$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.95e7 or 5.5 < y.re < 5.00000000000000037e243

   1. Initial program 34.7%

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

   3. Taylor expanded in y.re around 0 37.2%

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

      1. *-commutative 37.2%

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

      2. unpow2 37.2%

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

      3. unpow2 37.2%

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

      4. hypot-undefine 80.2%

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

   5. Simplified 80.2%

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

      1. log1p-expm1-u 80.2%

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

   7. Applied egg-rr 80.2%

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

   8. Taylor expanded in y.im around 0 48.9%

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. unpow2 48.9%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      2. unpow2 48.9%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      3. hypot-undefine 74.5%

         \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      4. unpow2 74.5%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right) \]

      5. unpow2 74.5%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}\right) \]

      6. hypot-undefine 74.5%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right) \]

   10. Simplified 74.5%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if -1.95e7 < y.re < 5.5

   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

      1. fmm-def 44.4%

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

      2. hypot-define 44.4%

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

      3. distribute-rgt-neg-out 44.4%

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

      4. fma-define 44.4%

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

      5. hypot-define 84.2%

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

      6. *-commutative 84.2%

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

   3. Simplified 84.2%

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

   5. Taylor expanded in y.re around 0 35.9%

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

      1. neg-mul-1 35.9%

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

      2. distribute-lft-neg-in 35.9%

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

      3. *-commutative 35.9%

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

      4. unpow2 35.9%

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

      5. unpow2 35.9%

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

      6. hypot-undefine 65.6%

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

   7. Simplified 65.6%

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

   if 5.00000000000000037e243 < y.re

   1. Initial program 38.5%

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

      1. fmm-def 38.5%

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

      2. hypot-define 38.5%

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

      3. distribute-rgt-neg-out 38.5%

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

      4. fma-define 38.5%

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

      5. hypot-define 69.2%

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

      6. *-commutative 69.2%

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

   3. Simplified 69.2%

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

   5. Taylor expanded in y.im around 0 76.9%

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

      1. unpow2 76.9%

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

      2. unpow2 76.9%

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

      3. hypot-undefine 76.9%

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

   7. Simplified 76.9%

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

   8. Taylor expanded in x.re around inf 61.5%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.re \cdot \left(1 + 0.5 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right)}}^{y.re} \]
   9. Step-by-step derivation

      1. pow2 61.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}\right)\right)}^{y.re} \]

      2. pow2 61.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}\right)\right)}^{y.re} \]

      3. times-frac 76.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   10. Applied egg-rr 76.9%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -19500000:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{elif}\;y.re \leq 5.5:\\ \;\;\;\;\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+243}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot t\_0\right)\\ t_2 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+108}:\\ \;\;\;\;t\_2 \cdot {\left(x.im + 0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -98000000000:\\ \;\;\;\;t\_2 \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(x.im \cdot \frac{x.im}{{x.re}^{2}}\right)\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -2.6 \cdot 10^{-216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-222}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{elif}\;y.im \leq 3.15 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log x.im\right) \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (* y.im (* (log (hypot x.im x.re)) t_0)))
        (t_2 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.im -2.1e+108)
     (* t_2 (pow (+ x.im (* 0.5 (/ (pow x.re 2.0) x.im))) y.re))
     (if (<= y.im -98000000000.0)
       (*
        t_2
        (pow (* x.re (+ 1.0 (* 0.5 (* x.im (/ x.im (pow x.re 2.0)))))) y.re))
       (if (<= y.im -2.6e-216)
         t_1
         (if (<= y.im 1.1e-222)
           (* t_0 t_2)
           (if (<= y.im 3.15e+187)
             t_1
             (*
              (* y.im (log x.im))
              (exp (- (* y.re (log x.im)) (* 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 t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	double t_2 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.1e+108) {
		tmp = t_2 * pow((x_46_im + (0.5 * (pow(x_46_re, 2.0) / x_46_im))), y_46_re);
	} else if (y_46_im <= -98000000000.0) {
		tmp = t_2 * pow((x_46_re * (1.0 + (0.5 * (x_46_im * (x_46_im / pow(x_46_re, 2.0)))))), y_46_re);
	} else if (y_46_im <= -2.6e-216) {
		tmp = t_1;
	} else if (y_46_im <= 1.1e-222) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 3.15e+187) {
		tmp = t_1;
	} else {
		tmp = (y_46_im * log(x_46_im)) * exp(((y_46_re * log(x_46_im)) - (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 t_0 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * t_0);
	double t_2 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.1e+108) {
		tmp = t_2 * Math.pow((x_46_im + (0.5 * (Math.pow(x_46_re, 2.0) / x_46_im))), y_46_re);
	} else if (y_46_im <= -98000000000.0) {
		tmp = t_2 * Math.pow((x_46_re * (1.0 + (0.5 * (x_46_im * (x_46_im / Math.pow(x_46_re, 2.0)))))), y_46_re);
	} else if (y_46_im <= -2.6e-216) {
		tmp = t_1;
	} else if (y_46_im <= 1.1e-222) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 3.15e+187) {
		tmp = t_1;
	} else {
		tmp = (y_46_im * Math.log(x_46_im)) * Math.exp(((y_46_re * Math.log(x_46_im)) - (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):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * t_0)
	t_2 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -2.1e+108:
		tmp = t_2 * math.pow((x_46_im + (0.5 * (math.pow(x_46_re, 2.0) / x_46_im))), y_46_re)
	elif y_46_im <= -98000000000.0:
		tmp = t_2 * math.pow((x_46_re * (1.0 + (0.5 * (x_46_im * (x_46_im / math.pow(x_46_re, 2.0)))))), y_46_re)
	elif y_46_im <= -2.6e-216:
		tmp = t_1
	elif y_46_im <= 1.1e-222:
		tmp = t_0 * t_2
	elif y_46_im <= 3.15e+187:
		tmp = t_1
	else:
		tmp = (y_46_im * math.log(x_46_im)) * math.exp(((y_46_re * math.log(x_46_im)) - (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)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * t_0))
	t_2 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -2.1e+108)
		tmp = Float64(t_2 * (Float64(x_46_im + Float64(0.5 * Float64((x_46_re ^ 2.0) / x_46_im))) ^ y_46_re));
	elseif (y_46_im <= -98000000000.0)
		tmp = Float64(t_2 * (Float64(x_46_re * Float64(1.0 + Float64(0.5 * Float64(x_46_im * Float64(x_46_im / (x_46_re ^ 2.0)))))) ^ y_46_re));
	elseif (y_46_im <= -2.6e-216)
		tmp = t_1;
	elseif (y_46_im <= 1.1e-222)
		tmp = Float64(t_0 * t_2);
	elseif (y_46_im <= 3.15e+187)
		tmp = t_1;
	else
		tmp = Float64(Float64(y_46_im * log(x_46_im)) * exp(Float64(Float64(y_46_re * log(x_46_im)) - 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)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_1 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	t_2 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -2.1e+108)
		tmp = t_2 * ((x_46_im + (0.5 * ((x_46_re ^ 2.0) / x_46_im))) ^ y_46_re);
	elseif (y_46_im <= -98000000000.0)
		tmp = t_2 * ((x_46_re * (1.0 + (0.5 * (x_46_im * (x_46_im / (x_46_re ^ 2.0)))))) ^ y_46_re);
	elseif (y_46_im <= -2.6e-216)
		tmp = t_1;
	elseif (y_46_im <= 1.1e-222)
		tmp = t_0 * t_2;
	elseif (y_46_im <= 3.15e+187)
		tmp = t_1;
	else
		tmp = (y_46_im * log(x_46_im)) * exp(((y_46_re * log(x_46_im)) - (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_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -2.1e+108], N[(t$95$2 * N[Power[N[(x$46$im + N[(0.5 * N[(N[Power[x$46$re, 2.0], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -98000000000.0], N[(t$95$2 * N[Power[N[(x$46$re * N[(1.0 + N[(0.5 * N[(x$46$im * N[(x$46$im / N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.6e-216], t$95$1, If[LessEqual[y$46$im, 1.1e-222], N[(t$95$0 * t$95$2), $MachinePrecision], If[LessEqual[y$46$im, 3.15e+187], t$95$1, N[(N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.1000000000000001e108

   1. Initial program 31.6%

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

      1. fmm-def 31.6%

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

      2. hypot-define 31.6%

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

      3. distribute-rgt-neg-out 31.6%

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

      4. fma-define 31.6%

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

      5. hypot-define 58.3%

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

      6. *-commutative 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in y.im around 0 41.2%

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

      1. unpow2 41.2%

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

      2. unpow2 41.2%

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

      3. hypot-undefine 28.6%

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

   7. Simplified 28.6%

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

   8. Taylor expanded in x.re around 0 38.5%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.im + 0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)}}^{y.re} \]

   if -2.1000000000000001e108 < y.im < -9.8e10

   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

      1. fmm-def 31.3%

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

      2. hypot-define 31.3%

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

      3. distribute-rgt-neg-out 31.3%

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

      4. fma-define 31.3%

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

      5. hypot-define 75.1%

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

      6. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in y.im around 0 45.7%

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

      1. unpow2 45.7%

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

      2. unpow2 45.7%

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

      3. hypot-undefine 28.8%

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

   7. Simplified 28.8%

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

   8. Taylor expanded in x.re around inf 56.5%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.re \cdot \left(1 + 0.5 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right)}}^{y.re} \]
   9. Step-by-step derivation

      1. pow2 56.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}\right)\right)}^{y.re} \]

      2. *-un-lft-identity 56.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{x.im \cdot x.im}{\color{blue}{1 \cdot {x.re}^{2}}}\right)\right)}^{y.re} \]

      3. pow2 56.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{x.im \cdot x.im}{1 \cdot \color{blue}{\left(x.re \cdot x.re\right)}}\right)\right)}^{y.re} \]

      4. times-frac 63.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{1} \cdot \frac{x.im}{x.re \cdot x.re}\right)}\right)\right)}^{y.re} \]

      5. pow2 63.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{1} \cdot \frac{x.im}{\color{blue}{{x.re}^{2}}}\right)\right)\right)}^{y.re} \]

   10. Applied egg-rr 63.0%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{1} \cdot \frac{x.im}{{x.re}^{2}}\right)}\right)\right)}^{y.re} \]

   if -9.8e10 < y.im < -2.5999999999999999e-216 or 1.1e-222 < y.im < 3.15000000000000003e187

   1. Initial program 35.6%

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

   3. Taylor expanded in y.re around 0 34.2%

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

      1. *-commutative 34.2%

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

      2. unpow2 34.2%

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

      3. unpow2 34.2%

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

      4. hypot-undefine 63.5%

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

   5. Simplified 63.5%

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

      1. log1p-expm1-u 63.5%

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

   7. Applied egg-rr 63.5%

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

   8. Taylor expanded in y.im around 0 42.0%

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. unpow2 42.0%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      2. unpow2 42.0%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      3. hypot-undefine 60.3%

         \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      4. unpow2 60.3%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right) \]

      5. unpow2 60.3%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}\right) \]

      6. hypot-undefine 68.8%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right) \]

   10. Simplified 68.8%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if -2.5999999999999999e-216 < y.im < 1.1e-222

   1. Initial program 49.9%

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

      1. fmm-def 49.9%

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

      2. hypot-define 49.9%

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

      3. distribute-rgt-neg-out 49.9%

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

      4. fma-define 49.9%

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

      5. hypot-define 90.3%

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

      6. *-commutative 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in y.im around 0 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. hypot-undefine 79.7%

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

   7. Simplified 79.7%

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

   if 3.15000000000000003e187 < y.im

   1. Initial program 63.1%

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

   3. Taylor expanded in y.re around 0 63.1%

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

      1. *-commutative 63.1%

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

      2. unpow2 63.1%

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

      3. unpow2 63.1%

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

      4. hypot-undefine 75.8%

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

   5. Simplified 75.8%

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

      1. log1p-expm1-u 75.8%

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

   7. Applied egg-rr 75.8%

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

   8. Taylor expanded in x.re around 0 42.4%

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

      1. *-commutative 42.4%

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

      2. *-commutative 42.4%

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

      3. *-commutative 42.4%

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

   10. Simplified 42.4%

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

   11. Taylor expanded in y.im around 0 41.7%

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

4. Final simplification 62.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+108}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -98000000000:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(x.im \cdot \frac{x.im}{{x.re}^{2}}\right)\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -2.6 \cdot 10^{-216}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-222}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 3.15 \cdot 10^{+187}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log x.im\right) \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot t\_0\right)\\ t_2 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+108}:\\ \;\;\;\;t\_2 \cdot {\left(x.im + 0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -450:\\ \;\;\;\;t\_2 \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-216}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-220}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log x.im\right) \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (* y.im (* (log (hypot x.im x.re)) t_0)))
        (t_2 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.im -2.3e+108)
     (* t_2 (pow (+ x.im (* 0.5 (/ (pow x.re 2.0) x.im))) y.re))
     (if (<= y.im -450.0)
       (*
        t_2
        (pow (* x.re (+ 1.0 (* 0.5 (* (/ x.im x.re) (/ x.im x.re))))) y.re))
       (if (<= y.im -3e-216)
         t_1
         (if (<= y.im 1.5e-220)
           (* t_0 t_2)
           (if (<= y.im 5.4e+187)
             t_1
             (*
              (* y.im (log x.im))
              (exp (- (* y.re (log x.im)) (* 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 t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	double t_2 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.3e+108) {
		tmp = t_2 * pow((x_46_im + (0.5 * (pow(x_46_re, 2.0) / x_46_im))), y_46_re);
	} else if (y_46_im <= -450.0) {
		tmp = t_2 * pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else if (y_46_im <= -3e-216) {
		tmp = t_1;
	} else if (y_46_im <= 1.5e-220) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 5.4e+187) {
		tmp = t_1;
	} else {
		tmp = (y_46_im * log(x_46_im)) * exp(((y_46_re * log(x_46_im)) - (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 t_0 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * t_0);
	double t_2 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.3e+108) {
		tmp = t_2 * Math.pow((x_46_im + (0.5 * (Math.pow(x_46_re, 2.0) / x_46_im))), y_46_re);
	} else if (y_46_im <= -450.0) {
		tmp = t_2 * Math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else if (y_46_im <= -3e-216) {
		tmp = t_1;
	} else if (y_46_im <= 1.5e-220) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 5.4e+187) {
		tmp = t_1;
	} else {
		tmp = (y_46_im * Math.log(x_46_im)) * Math.exp(((y_46_re * Math.log(x_46_im)) - (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):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * t_0)
	t_2 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -2.3e+108:
		tmp = t_2 * math.pow((x_46_im + (0.5 * (math.pow(x_46_re, 2.0) / x_46_im))), y_46_re)
	elif y_46_im <= -450.0:
		tmp = t_2 * math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re)
	elif y_46_im <= -3e-216:
		tmp = t_1
	elif y_46_im <= 1.5e-220:
		tmp = t_0 * t_2
	elif y_46_im <= 5.4e+187:
		tmp = t_1
	else:
		tmp = (y_46_im * math.log(x_46_im)) * math.exp(((y_46_re * math.log(x_46_im)) - (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)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * t_0))
	t_2 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -2.3e+108)
		tmp = Float64(t_2 * (Float64(x_46_im + Float64(0.5 * Float64((x_46_re ^ 2.0) / x_46_im))) ^ y_46_re));
	elseif (y_46_im <= -450.0)
		tmp = Float64(t_2 * (Float64(x_46_re * Float64(1.0 + Float64(0.5 * Float64(Float64(x_46_im / x_46_re) * Float64(x_46_im / x_46_re))))) ^ y_46_re));
	elseif (y_46_im <= -3e-216)
		tmp = t_1;
	elseif (y_46_im <= 1.5e-220)
		tmp = Float64(t_0 * t_2);
	elseif (y_46_im <= 5.4e+187)
		tmp = t_1;
	else
		tmp = Float64(Float64(y_46_im * log(x_46_im)) * exp(Float64(Float64(y_46_re * log(x_46_im)) - 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)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_1 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	t_2 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -2.3e+108)
		tmp = t_2 * ((x_46_im + (0.5 * ((x_46_re ^ 2.0) / x_46_im))) ^ y_46_re);
	elseif (y_46_im <= -450.0)
		tmp = t_2 * ((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))) ^ y_46_re);
	elseif (y_46_im <= -3e-216)
		tmp = t_1;
	elseif (y_46_im <= 1.5e-220)
		tmp = t_0 * t_2;
	elseif (y_46_im <= 5.4e+187)
		tmp = t_1;
	else
		tmp = (y_46_im * log(x_46_im)) * exp(((y_46_re * log(x_46_im)) - (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_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -2.3e+108], N[(t$95$2 * N[Power[N[(x$46$im + N[(0.5 * N[(N[Power[x$46$re, 2.0], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -450.0], N[(t$95$2 * N[Power[N[(x$46$re * N[(1.0 + N[(0.5 * N[(N[(x$46$im / x$46$re), $MachinePrecision] * N[(x$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3e-216], t$95$1, If[LessEqual[y$46$im, 1.5e-220], N[(t$95$0 * t$95$2), $MachinePrecision], If[LessEqual[y$46$im, 5.4e+187], t$95$1, N[(N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.2999999999999999e108

   1. Initial program 31.6%

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

      1. fmm-def 31.6%

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

      2. hypot-define 31.6%

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

      3. distribute-rgt-neg-out 31.6%

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

      4. fma-define 31.6%

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

      5. hypot-define 58.3%

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

      6. *-commutative 58.3%

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

   3. Simplified 58.3%

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

   5. Taylor expanded in y.im around 0 41.2%

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

      1. unpow2 41.2%

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

      2. unpow2 41.2%

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

      3. hypot-undefine 28.6%

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

   7. Simplified 28.6%

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

   8. Taylor expanded in x.re around 0 38.5%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.im + 0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)}}^{y.re} \]

   if -2.2999999999999999e108 < y.im < -450

   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

      1. fmm-def 31.3%

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

      2. hypot-define 31.3%

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

      3. distribute-rgt-neg-out 31.3%

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

      4. fma-define 31.3%

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

      5. hypot-define 75.1%

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

      6. *-commutative 75.1%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in y.im around 0 45.7%

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

      1. unpow2 45.7%

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

      2. unpow2 45.7%

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

      3. hypot-undefine 28.8%

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

   7. Simplified 28.8%

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

   8. Taylor expanded in x.re around inf 56.5%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.re \cdot \left(1 + 0.5 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right)}}^{y.re} \]
   9. Step-by-step derivation

      1. pow2 56.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}\right)\right)}^{y.re} \]

      2. pow2 56.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}\right)\right)}^{y.re} \]

      3. times-frac 56.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   10. Applied egg-rr 56.6%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   if -450 < y.im < -3.00000000000000013e-216 or 1.50000000000000009e-220 < y.im < 5.40000000000000016e187

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0 34.5%

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

      1. *-commutative 34.5%

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

      2. unpow2 34.5%

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

      3. unpow2 34.5%

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

      4. hypot-undefine 64.0%

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

   5. Simplified 64.0%

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

      1. log1p-expm1-u 64.0%

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

   7. Applied egg-rr 64.0%

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

   8. Taylor expanded in y.im around 0 42.4%

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. unpow2 42.4%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      2. unpow2 42.4%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      3. hypot-undefine 60.7%

         \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      4. unpow2 60.7%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right) \]

      5. unpow2 60.7%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}\right) \]

      6. hypot-undefine 69.4%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right) \]

   10. Simplified 69.4%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if -3.00000000000000013e-216 < y.im < 1.50000000000000009e-220

   1. Initial program 48.7%

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

      1. fmm-def 48.7%

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

      2. hypot-define 48.7%

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

      3. distribute-rgt-neg-out 48.7%

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

      4. fma-define 48.7%

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

      5. hypot-define 88.2%

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

      6. *-commutative 88.2%

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

   3. Simplified 88.2%

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

   5. Taylor expanded in y.im around 0 63.9%

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. hypot-undefine 77.8%

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

   7. Simplified 77.8%

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

   if 5.40000000000000016e187 < y.im

   1. Initial program 63.1%

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

   3. Taylor expanded in y.re around 0 63.1%

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

      1. *-commutative 63.1%

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

      2. unpow2 63.1%

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

      3. unpow2 63.1%

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

      4. hypot-undefine 75.8%

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

   5. Simplified 75.8%

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

      1. log1p-expm1-u 75.8%

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

   7. Applied egg-rr 75.8%

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

   8. Taylor expanded in x.re around 0 42.4%

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

      1. *-commutative 42.4%

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

      2. *-commutative 42.4%

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

      3. *-commutative 42.4%

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

   10. Simplified 42.4%

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

   11. Taylor expanded in y.im around 0 41.7%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+108}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -450:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-216}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-220}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{+187}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log x.im\right) \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -190000000:\\ \;\;\;\;t\_1 \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-216} \lor \neg \left(y.im \leq 1.9 \cdot 10^{-225}\right):\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.im -190000000.0)
     (*
      t_1
      (pow (* x.re (+ 1.0 (* 0.5 (* (/ x.im x.re) (/ x.im x.re))))) y.re))
     (if (or (<= y.im -1.65e-216) (not (<= y.im 1.9e-225)))
       (* y.im (* (log (hypot x.im x.re)) t_0))
       (* t_0 t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -190000000.0) {
		tmp = t_1 * pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else if ((y_46_im <= -1.65e-216) || !(y_46_im <= 1.9e-225)) {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	} else {
		tmp = t_0 * 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 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -190000000.0) {
		tmp = t_1 * Math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else if ((y_46_im <= -1.65e-216) || !(y_46_im <= 1.9e-225)) {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * t_0);
	} else {
		tmp = t_0 * t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -190000000.0:
		tmp = t_1 * math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re)
	elif (y_46_im <= -1.65e-216) or not (y_46_im <= 1.9e-225):
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * t_0)
	else:
		tmp = t_0 * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -190000000.0)
		tmp = Float64(t_1 * (Float64(x_46_re * Float64(1.0 + Float64(0.5 * Float64(Float64(x_46_im / x_46_re) * Float64(x_46_im / x_46_re))))) ^ y_46_re));
	elseif ((y_46_im <= -1.65e-216) || !(y_46_im <= 1.9e-225))
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * t_0));
	else
		tmp = Float64(t_0 * 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 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -190000000.0)
		tmp = t_1 * ((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))) ^ y_46_re);
	elseif ((y_46_im <= -1.65e-216) || ~((y_46_im <= 1.9e-225)))
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_0);
	else
		tmp = t_0 * 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -190000000.0], N[(t$95$1 * N[Power[N[(x$46$re * N[(1.0 + N[(0.5 * N[(N[(x$46$im / x$46$re), $MachinePrecision] * N[(x$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$im, -1.65e-216], N[Not[LessEqual[y$46$im, 1.9e-225]], $MachinePrecision]], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.9e8

   1. Initial program 31.5%

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

      1. fmm-def 31.5%

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

      2. hypot-define 31.5%

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

      3. distribute-rgt-neg-out 31.5%

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

      4. fma-define 31.5%

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

      5. hypot-define 62.7%

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

      6. *-commutative 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in y.im around 0 42.4%

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

      1. unpow2 42.4%

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

      2. unpow2 42.4%

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

      3. hypot-undefine 28.6%

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

   7. Simplified 28.6%

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

   8. Taylor expanded in x.re around inf 28.5%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.re \cdot \left(1 + 0.5 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right)}}^{y.re} \]
   9. Step-by-step derivation

      1. pow2 28.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}\right)\right)}^{y.re} \]

      2. pow2 28.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}\right)\right)}^{y.re} \]

      3. times-frac 35.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   10. Applied egg-rr 35.1%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   if -1.9e8 < y.im < -1.64999999999999984e-216 or 1.9000000000000001e-225 < y.im

   1. Initial program 39.9%

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

   3. Taylor expanded in y.re around 0 38.8%

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

      1. *-commutative 38.8%

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

      2. unpow2 38.8%

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

      3. unpow2 38.8%

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

      4. hypot-undefine 65.4%

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

   5. Simplified 65.4%

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

      1. log1p-expm1-u 65.4%

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

   7. Applied egg-rr 65.4%

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

   8. Taylor expanded in y.im around 0 38.9%

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. unpow2 38.9%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      2. unpow2 38.9%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      3. hypot-undefine 54.3%

         \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      4. unpow2 54.3%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right) \]

      5. unpow2 54.3%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}\right) \]

      6. hypot-undefine 61.5%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right) \]

   10. Simplified 61.5%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if -1.64999999999999984e-216 < y.im < 1.9000000000000001e-225

   1. Initial program 49.9%

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

      1. fmm-def 49.9%

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

      2. hypot-define 49.9%

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

      3. distribute-rgt-neg-out 49.9%

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

      4. fma-define 49.9%

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

      5. hypot-define 90.3%

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

      6. *-commutative 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in y.im around 0 65.4%

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

      1. unpow2 65.4%

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

      2. unpow2 65.4%

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

      3. hypot-undefine 79.7%

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

   7. Simplified 79.7%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -190000000:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-216} \lor \neg \left(y.im \leq 1.9 \cdot 10^{-225}\right):\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 55.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{+19}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log x.im\right) \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im 1.75e+19)
   (* y.im (* (log (hypot x.im x.re)) (pow (hypot x.im x.re) y.re)))
   (*
    (* y.im (log x.im))
    (exp (- (* y.re (log x.im)) (* 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 (x_46_im <= 1.75e+19) {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re));
	} else {
		tmp = (y_46_im * log(x_46_im)) * exp(((y_46_re * log(x_46_im)) - (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 (x_46_im <= 1.75e+19) {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re));
	} else {
		tmp = (y_46_im * Math.log(x_46_im)) * Math.exp(((y_46_re * Math.log(x_46_im)) - (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 x_46_im <= 1.75e+19:
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re))
	else:
		tmp = (y_46_im * math.log(x_46_im)) * math.exp(((y_46_re * math.log(x_46_im)) - (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 (x_46_im <= 1.75e+19)
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re)));
	else
		tmp = Float64(Float64(y_46_im * log(x_46_im)) * exp(Float64(Float64(y_46_re * log(x_46_im)) - 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 (x_46_im <= 1.75e+19)
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = (y_46_im * log(x_46_im)) * exp(((y_46_re * log(x_46_im)) - (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[x$46$im, 1.75e+19], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.75e19

   1. Initial program 43.0%

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

   3. Taylor expanded in y.re around 0 38.1%

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

      1. *-commutative 38.1%

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

      2. unpow2 38.1%

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

      3. unpow2 38.1%

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

      4. hypot-undefine 62.1%

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

   5. Simplified 62.1%

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

      1. log1p-expm1-u 62.1%

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

   7. Applied egg-rr 62.1%

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

   8. Taylor expanded in y.im around 0 35.8%

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. unpow2 35.8%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      2. unpow2 35.8%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      3. hypot-undefine 50.9%

         \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      4. unpow2 50.9%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right) \]

      5. unpow2 50.9%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}\right) \]

      6. hypot-undefine 52.5%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right) \]

   10. Simplified 52.5%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if 1.75e19 < x.im

   1. Initial program 28.3%

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

   3. Taylor expanded in y.re around 0 28.5%

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

      1. *-commutative 28.5%

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

      2. unpow2 28.5%

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

      3. unpow2 28.5%

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

      4. hypot-undefine 59.1%

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

   5. Simplified 59.1%

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

      1. log1p-expm1-u 59.1%

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

   7. Applied egg-rr 59.1%

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

   8. Taylor expanded in x.re around 0 71.8%

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

      1. *-commutative 71.8%

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

      2. *-commutative 71.8%

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

      3. *-commutative 71.8%

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

   10. Simplified 71.8%

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

   11. Taylor expanded in y.im around 0 66.2%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.75 \cdot 10^{+19}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log x.im\right) \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -620000000:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -620000000.0)
   (*
    (sin (* y.re (atan2 x.im x.re)))
    (pow (* x.re (+ 1.0 (* 0.5 (* (/ x.im x.re) (/ x.im x.re))))) y.re))
   (* y.im (* (log (hypot 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) {
	double tmp;
	if (y_46_im <= -620000000.0) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * pow(hypot(x_46_im, 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 tmp;
	if (y_46_im <= -620000000.0) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_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 <= -620000000.0:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re)
	else:
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_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 <= -620000000.0)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (Float64(x_46_re * Float64(1.0 + Float64(0.5 * Float64(Float64(x_46_im / x_46_re) * Float64(x_46_im / x_46_re))))) ^ y_46_re));
	else
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (hypot(x_46_im, 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)
	tmp = 0.0;
	if (y_46_im <= -620000000.0)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * ((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))) ^ y_46_re);
	else
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * (hypot(x_46_im, 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_] := If[LessEqual[y$46$im, -620000000.0], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(x$46$re * N[(1.0 + N[(0.5 * N[(N[(x$46$im / x$46$re), $MachinePrecision] * N[(x$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.2e8

   1. Initial program 31.5%

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

      1. fmm-def 31.5%

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

      2. hypot-define 31.5%

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

      3. distribute-rgt-neg-out 31.5%

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

      4. fma-define 31.5%

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

      5. hypot-define 62.7%

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

      6. *-commutative 62.7%

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

   3. Simplified 62.7%

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

   5. Taylor expanded in y.im around 0 42.4%

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

      1. unpow2 42.4%

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

      2. unpow2 42.4%

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

      3. hypot-undefine 28.6%

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

   7. Simplified 28.6%

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

   8. Taylor expanded in x.re around inf 28.5%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.re \cdot \left(1 + 0.5 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right)}}^{y.re} \]
   9. Step-by-step derivation

      1. pow2 28.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}\right)\right)}^{y.re} \]

      2. pow2 28.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}\right)\right)}^{y.re} \]

      3. times-frac 35.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   10. Applied egg-rr 35.1%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   if -6.2e8 < y.im

   1. Initial program 42.0%

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

   3. Taylor expanded in y.re around 0 36.7%

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

      1. *-commutative 36.7%

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

      2. unpow2 36.7%

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

      3. unpow2 36.7%

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

      4. hypot-undefine 61.4%

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

   5. Simplified 61.4%

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

      1. log1p-expm1-u 61.4%

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

   7. Applied egg-rr 61.4%

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

   8. Taylor expanded in y.im around 0 39.4%

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. unpow2 39.4%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      2. unpow2 39.4%

         \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      3. hypot-undefine 52.6%

         \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) \]

      4. unpow2 52.6%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re}\right) \]

      5. unpow2 52.6%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re}\right) \]

      6. hypot-undefine 58.8%

         \[\leadsto y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re}\right) \]

   10. Simplified 58.8%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 45.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.re \leq -1.3 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot {\left(-x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq -2.8 \cdot 10^{-260}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;x.re \leq 51:\\ \;\;\;\;t\_0 \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.re -1.3e-5)
     (* t_0 (pow (- x.re) y.re))
     (if (<= x.re -2.8e-260)
       (* t_0 (pow x.im y.re))
       (if (<= x.re 51.0)
         (*
          t_0
          (pow (* x.re (+ 1.0 (* 0.5 (* (/ x.im x.re) (/ x.im x.re))))) y.re))
         (* y.im (* (log (hypot x.im x.re)) (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 <= -1.3e-5) {
		tmp = t_0 * pow(-x_46_re, y_46_re);
	} else if (x_46_re <= -2.8e-260) {
		tmp = t_0 * pow(x_46_im, y_46_re);
	} else if (x_46_re <= 51.0) {
		tmp = t_0 * pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * 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 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -1.3e-5) {
		tmp = t_0 * Math.pow(-x_46_re, y_46_re);
	} else if (x_46_re <= -2.8e-260) {
		tmp = t_0 * Math.pow(x_46_im, y_46_re);
	} else if (x_46_re <= 51.0) {
		tmp = t_0 * Math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re);
	} else {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * 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 <= -1.3e-5:
		tmp = t_0 * math.pow(-x_46_re, y_46_re)
	elif x_46_re <= -2.8e-260:
		tmp = t_0 * math.pow(x_46_im, y_46_re)
	elif x_46_re <= 51.0:
		tmp = t_0 * math.pow((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))), y_46_re)
	else:
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * 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 <= -1.3e-5)
		tmp = Float64(t_0 * (Float64(-x_46_re) ^ y_46_re));
	elseif (x_46_re <= -2.8e-260)
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	elseif (x_46_re <= 51.0)
		tmp = Float64(t_0 * (Float64(x_46_re * Float64(1.0 + Float64(0.5 * Float64(Float64(x_46_im / x_46_re) * Float64(x_46_im / x_46_re))))) ^ y_46_re));
	else
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (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 <= -1.3e-5)
		tmp = t_0 * (-x_46_re ^ y_46_re);
	elseif (x_46_re <= -2.8e-260)
		tmp = t_0 * (x_46_im ^ y_46_re);
	elseif (x_46_re <= 51.0)
		tmp = t_0 * ((x_46_re * (1.0 + (0.5 * ((x_46_im / x_46_re) * (x_46_im / x_46_re))))) ^ y_46_re);
	else
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * (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, -1.3e-5], N[(t$95$0 * N[Power[(-x$46$re), y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -2.8e-260], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 51.0], N[(t$95$0 * N[Power[N[(x$46$re * N[(1.0 + N[(0.5 * N[(N[(x$46$im / x$46$re), $MachinePrecision] * N[(x$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.29999999999999992e-5

   1. Initial program 29.4%

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

      1. fmm-def 29.4%

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

      2. hypot-define 29.4%

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

      3. distribute-rgt-neg-out 29.4%

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

      4. fma-define 29.4%

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

      5. hypot-define 75.3%

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

      6. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in y.im around 0 44.0%

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

      1. unpow2 44.0%

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

      2. unpow2 44.0%

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

      3. hypot-undefine 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in x.re around -inf 41.0%

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

      1. mul-1-neg 41.0%

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

   10. Simplified 41.0%

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

   if -1.29999999999999992e-5 < x.re < -2.7999999999999998e-260

   1. Initial program 56.6%

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

      1. fmm-def 56.6%

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

      2. hypot-define 56.6%

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

      3. distribute-rgt-neg-out 56.6%

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

      4. fma-define 56.6%

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

      5. hypot-define 86.3%

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

      6. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y.im around 0 43.1%

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

      1. unpow2 43.1%

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

      2. unpow2 43.1%

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

      3. hypot-undefine 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in x.im around inf 39.8%

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

   if -2.7999999999999998e-260 < x.re < 51

   1. Initial program 42.0%

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

      1. fmm-def 42.0%

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

      2. hypot-define 42.0%

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

      3. distribute-rgt-neg-out 42.0%

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

      4. fma-define 42.0%

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

      5. hypot-define 81.6%

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

      6. *-commutative 81.6%

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

   3. Simplified 81.6%

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

   5. Taylor expanded in y.im around 0 44.7%

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

      1. unpow2 44.7%

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

      2. unpow2 44.7%

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

      3. hypot-undefine 46.7%

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

   7. Simplified 46.7%

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

   8. Taylor expanded in x.re around inf 43.8%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.re \cdot \left(1 + 0.5 \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)\right)}}^{y.re} \]
   9. Step-by-step derivation

      1. pow2 43.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}\right)\right)}^{y.re} \]

      2. pow2 43.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}\right)\right)}^{y.re} \]

      3. times-frac 51.4%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   10. Applied egg-rr 51.4%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot \left(1 + 0.5 \cdot \color{blue}{\left(\frac{x.im}{x.re} \cdot \frac{x.im}{x.re}\right)}\right)\right)}^{y.re} \]

   if 51 < x.re

   1. Initial program 31.7%

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

   3. Taylor expanded in y.re around 0 29.1%

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

      1. *-commutative 29.1%

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

      2. unpow2 29.1%

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

      3. unpow2 29.1%

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

      4. hypot-undefine 58.8%

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

   5. Simplified 58.8%

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

   6. Taylor expanded in x.re around inf 69.1%

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

      1. mul-1-neg 69.1%

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

      2. log-rec 69.1%

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

      3. remove-double-neg 69.1%

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

   8. Simplified 69.1%

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

      1. expm1-log1p-u 74.5%

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

      2. expm1-undefine 73.1%

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

   10. Applied egg-rr 73.1%

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

       1. expm1-define 74.5%

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

   12. Simplified 74.5%

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

   13. Taylor expanded in y.im around 0 45.1%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right)} \]
   14. Step-by-step derivation

       1. unpow2 45.1%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right) \]

       2. unpow2 45.1%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {x.re}^{y.re}\right) \]

       3. hypot-undefine 67.4%

          \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {x.re}^{y.re}\right) \]

   15. Simplified 67.4%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 38.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}\\ t_1 := y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)\\ \mathbf{if}\;y.re \leq -19500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-282}:\\ \;\;\;\;\sqrt[3]{t\_0}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+237}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{0.3333333333333333}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (* y.re (atan2 x.im x.re)) 3.0))
        (t_1 (* y.im (* (log (hypot x.im x.re)) (pow x.re y.re)))))
   (if (<= y.re -19500000.0)
     t_1
     (if (<= y.re 1.3e-282)
       (cbrt t_0)
       (if (<= y.re 6e+237) t_1 (pow t_0 0.3333333333333333))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow((y_46_re * atan2(x_46_im, x_46_re)), 3.0);
	double t_1 = y_46_im * (log(hypot(x_46_im, x_46_re)) * pow(x_46_re, y_46_re));
	double tmp;
	if (y_46_re <= -19500000.0) {
		tmp = t_1;
	} else if (y_46_re <= 1.3e-282) {
		tmp = cbrt(t_0);
	} else if (y_46_re <= 6e+237) {
		tmp = t_1;
	} else {
		tmp = pow(t_0, 0.3333333333333333);
	}
	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.pow((y_46_re * Math.atan2(x_46_im, x_46_re)), 3.0);
	double t_1 = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.pow(x_46_re, y_46_re));
	double tmp;
	if (y_46_re <= -19500000.0) {
		tmp = t_1;
	} else if (y_46_re <= 1.3e-282) {
		tmp = Math.cbrt(t_0);
	} else if (y_46_re <= 6e+237) {
		tmp = t_1;
	} else {
		tmp = Math.pow(t_0, 0.3333333333333333);
	}
	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)) ^ 3.0
	t_1 = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (x_46_re ^ y_46_re)))
	tmp = 0.0
	if (y_46_re <= -19500000.0)
		tmp = t_1;
	elseif (y_46_re <= 1.3e-282)
		tmp = cbrt(t_0);
	elseif (y_46_re <= 6e+237)
		tmp = t_1;
	else
		tmp = t_0 ^ 0.3333333333333333;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -19500000.0], t$95$1, If[LessEqual[y$46$re, 1.3e-282], N[Power[t$95$0, 1/3], $MachinePrecision], If[LessEqual[y$46$re, 6e+237], t$95$1, N[Power[t$95$0, 0.3333333333333333], $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.95e7 or 1.30000000000000006e-282 < y.re < 6e237

   1. Initial program 37.9%

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

   3. Taylor expanded in y.re around 0 34.7%

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

      1. *-commutative 34.7%

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

      2. unpow2 34.7%

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

      3. unpow2 34.7%

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

      4. hypot-undefine 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in x.re around inf 33.1%

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

      1. mul-1-neg 33.1%

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

      2. log-rec 33.1%

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

      3. remove-double-neg 33.1%

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

   8. Simplified 33.1%

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

      1. expm1-log1p-u 26.3%

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

      2. expm1-undefine 26.8%

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

   10. Applied egg-rr 26.8%

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

       1. expm1-define 26.3%

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

   12. Simplified 26.3%

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

   13. Taylor expanded in y.im around 0 29.8%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right)} \]
   14. Step-by-step derivation

       1. unpow2 29.8%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right) \]

       2. unpow2 29.8%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {x.re}^{y.re}\right) \]

       3. hypot-undefine 46.2%

          \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {x.re}^{y.re}\right) \]

   15. Simplified 46.2%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)} \]

   if -1.95e7 < y.re < 1.30000000000000006e-282

   1. Initial program 44.7%

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

      1. fmm-def 44.7%

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

      2. hypot-define 44.7%

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

      3. distribute-rgt-neg-out 44.7%

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

      4. fma-define 44.7%

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

      5. hypot-define 84.3%

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

      6. *-commutative 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in y.im around 0 18.1%

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

      1. unpow2 18.1%

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

      2. unpow2 18.1%

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

      3. hypot-undefine 20.1%

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

   7. Simplified 20.1%

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

   8. Taylor expanded in y.re around 0 16.3%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 29.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]

      2. pow3 29.1%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \]

   10. Applied egg-rr 29.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \]

   if 6e237 < y.re

   1. Initial program 35.3%

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

      1. fmm-def 35.3%

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

      2. hypot-define 35.3%

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

      3. distribute-rgt-neg-out 35.3%

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

      4. fma-define 35.3%

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

      5. hypot-define 64.7%

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

      6. *-commutative 64.7%

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

   3. Simplified 64.7%

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

   5. Taylor expanded in y.im around 0 64.8%

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

      1. unpow2 64.8%

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

      2. unpow2 64.8%

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

      3. hypot-undefine 64.8%

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

   7. Simplified 64.8%

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

   8. Taylor expanded in y.re around 0 3.6%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 47.5%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]

      2. pow1/3 59.2%

         \[\leadsto \color{blue}{{\left(\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}^{0.3333333333333333}} \]

      3. pow3 59.2%

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

   10. Applied egg-rr 59.2%

       \[\leadsto \color{blue}{{\left({\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}\right)}^{0.3333333333333333}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 42.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.7 \cdot 10^{+51} \lor \neg \left(x.re \leq 7 \cdot 10^{-43}\right):\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= x.re -2.7e+51) (not (<= x.re 7e-43)))
   (* y.im (* (log (hypot x.im x.re)) (pow x.re y.re)))
   (* (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) {
	double tmp;
	if ((x_46_re <= -2.7e+51) || !(x_46_re <= 7e-43)) {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * pow(x_46_re, y_46_re));
	} else {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_im, 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 tmp;
	if ((x_46_re <= -2.7e+51) || !(x_46_re <= 7e-43)) {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.pow(x_46_re, y_46_re));
	} else {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (x_46_re <= -2.7e+51) or not (x_46_re <= 7e-43):
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.pow(x_46_re, y_46_re))
	else:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((x_46_re <= -2.7e+51) || !(x_46_re <= 7e-43))
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (x_46_re ^ y_46_re)));
	else
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_im ^ y_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 ((x_46_re <= -2.7e+51) || ~((x_46_re <= 7e-43)))
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * (x_46_re ^ y_46_re));
	else
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[x$46$re, -2.7e+51], N[Not[LessEqual[x$46$re, 7e-43]], $MachinePrecision]], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if x.re < -2.69999999999999992e51 or 6.99999999999999994e-43 < x.re

   1. Initial program 28.1%

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

   3. Taylor expanded in y.re around 0 25.2%

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

      1. *-commutative 25.2%

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

      2. unpow2 25.2%

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

      3. unpow2 25.2%

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

      4. hypot-undefine 56.2%

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

   5. Simplified 56.2%

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

   6. Taylor expanded in x.re around inf 41.0%

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

      1. mul-1-neg 41.0%

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

      2. log-rec 41.0%

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

      3. remove-double-neg 41.0%

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

   8. Simplified 41.0%

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

      1. expm1-log1p-u 42.5%

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

      2. expm1-undefine 41.7%

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

   10. Applied egg-rr 41.7%

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

       1. expm1-define 42.5%

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

   12. Simplified 42.5%

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

   13. Taylor expanded in y.im around 0 34.8%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right)} \]
   14. Step-by-step derivation

       1. unpow2 34.8%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right) \]

       2. unpow2 34.8%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {x.re}^{y.re}\right) \]

       3. hypot-undefine 52.5%

          \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {x.re}^{y.re}\right) \]

   15. Simplified 52.5%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)} \]

   if -2.69999999999999992e51 < x.re < 6.99999999999999994e-43

   1. Initial program 51.5%

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

      1. fmm-def 51.5%

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

      2. hypot-define 51.5%

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

      3. distribute-rgt-neg-out 51.5%

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

      4. fma-define 51.5%

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

      5. hypot-define 85.0%

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

      6. *-commutative 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in y.im around 0 43.2%

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

      1. unpow2 43.2%

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

      2. unpow2 43.2%

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

      3. hypot-undefine 45.2%

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

   7. Simplified 45.2%

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

   8. Taylor expanded in x.im around inf 37.2%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.7 \cdot 10^{+51} \lor \neg \left(x.re \leq 7 \cdot 10^{-43}\right):\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 45.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.re \leq -0.0037:\\ \;\;\;\;t\_0 \cdot {\left(-x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 4.1 \cdot 10^{-44}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.re -0.0037)
     (* t_0 (pow (- x.re) y.re))
     (if (<= x.re 4.1e-44)
       (* t_0 (pow x.im y.re))
       (* y.im (* (log (hypot x.im x.re)) (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 <= -0.0037) {
		tmp = t_0 * pow(-x_46_re, y_46_re);
	} else if (x_46_re <= 4.1e-44) {
		tmp = t_0 * pow(x_46_im, y_46_re);
	} else {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * 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 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -0.0037) {
		tmp = t_0 * Math.pow(-x_46_re, y_46_re);
	} else if (x_46_re <= 4.1e-44) {
		tmp = t_0 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * 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 <= -0.0037:
		tmp = t_0 * math.pow(-x_46_re, y_46_re)
	elif x_46_re <= 4.1e-44:
		tmp = t_0 * math.pow(x_46_im, y_46_re)
	else:
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * 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 <= -0.0037)
		tmp = Float64(t_0 * (Float64(-x_46_re) ^ y_46_re));
	elseif (x_46_re <= 4.1e-44)
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	else
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (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 <= -0.0037)
		tmp = t_0 * (-x_46_re ^ y_46_re);
	elseif (x_46_re <= 4.1e-44)
		tmp = t_0 * (x_46_im ^ y_46_re);
	else
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * (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, -0.0037], N[(t$95$0 * N[Power[(-x$46$re), y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.1e-44], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -0.0037000000000000002

   1. Initial program 29.4%

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

      1. fmm-def 29.4%

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

      2. hypot-define 29.4%

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

      3. distribute-rgt-neg-out 29.4%

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

      4. fma-define 29.4%

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

      5. hypot-define 75.3%

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

      6. *-commutative 75.3%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in y.im around 0 44.0%

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

      1. unpow2 44.0%

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

      2. unpow2 44.0%

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

      3. hypot-undefine 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in x.re around -inf 41.0%

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

      1. mul-1-neg 41.0%

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

   10. Simplified 41.0%

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

   if -0.0037000000000000002 < x.re < 4.09999999999999992e-44

   1. Initial program 49.5%

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

      1. fmm-def 49.5%

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

      2. hypot-define 49.5%

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

      3. distribute-rgt-neg-out 49.5%

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

      4. fma-define 49.5%

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

      5. hypot-define 84.7%

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

      6. *-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in y.im around 0 44.7%

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

      1. unpow2 44.7%

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

      2. unpow2 44.7%

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

      3. hypot-undefine 46.8%

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

   7. Simplified 46.8%

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

   8. Taylor expanded in x.im around inf 39.2%

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

   if 4.09999999999999992e-44 < x.re

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0 29.1%

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

      1. *-commutative 29.1%

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

      2. unpow2 29.1%

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

      3. unpow2 29.1%

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

      4. hypot-undefine 56.1%

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

   5. Simplified 56.1%

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

   6. Taylor expanded in x.re around inf 68.1%

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

      1. mul-1-neg 68.1%

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

      2. log-rec 68.1%

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

      3. remove-double-neg 68.1%

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

   8. Simplified 68.1%

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

      1. expm1-log1p-u 70.4%

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

      2. expm1-undefine 69.1%

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

   10. Applied egg-rr 69.1%

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

       1. expm1-define 70.4%

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

   12. Simplified 70.4%

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

   13. Taylor expanded in y.im around 0 41.3%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right)} \]
   14. Step-by-step derivation

       1. unpow2 41.3%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right) \]

       2. unpow2 41.3%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {x.re}^{y.re}\right) \]

       3. hypot-undefine 62.9%

          \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {x.re}^{y.re}\right) \]

   15. Simplified 62.9%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 46.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -31000000000:\\ \;\;\;\;t\_0 \cdot {\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 0.056:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

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.im -31000000000.0)
     (* t_0 (pow (- x.im) y.re))
     (if (<= x.im 0.056)
       (* y.im (* (log (hypot x.im x.re)) (pow x.re y.re)))
       (* t_0 (pow x.im 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_im <= -31000000000.0) {
		tmp = t_0 * pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 0.056) {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * pow(x_46_re, y_46_re));
	} else {
		tmp = t_0 * pow(x_46_im, 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 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -31000000000.0) {
		tmp = t_0 * Math.pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 0.056) {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.pow(x_46_re, y_46_re));
	} else {
		tmp = t_0 * Math.pow(x_46_im, 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_im <= -31000000000.0:
		tmp = t_0 * math.pow(-x_46_im, y_46_re)
	elif x_46_im <= 0.056:
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.pow(x_46_re, y_46_re))
	else:
		tmp = t_0 * math.pow(x_46_im, 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_im <= -31000000000.0)
		tmp = Float64(t_0 * (Float64(-x_46_im) ^ y_46_re));
	elseif (x_46_im <= 0.056)
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (x_46_re ^ y_46_re)));
	else
		tmp = Float64(t_0 * (x_46_im ^ 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_im <= -31000000000.0)
		tmp = t_0 * (-x_46_im ^ y_46_re);
	elseif (x_46_im <= 0.056)
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * (x_46_re ^ y_46_re));
	else
		tmp = t_0 * (x_46_im ^ 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$46$im, -31000000000.0], N[(t$95$0 * N[Power[(-x$46$im), y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 0.056], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.1e10

   1. Initial program 30.9%

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

      1. fmm-def 30.9%

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

      2. hypot-define 30.9%

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

      3. distribute-rgt-neg-out 30.9%

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

      4. fma-define 30.9%

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

      5. hypot-define 83.6%

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

      6. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in y.im around 0 54.3%

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

      1. unpow2 54.3%

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

      2. unpow2 54.3%

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

      3. hypot-undefine 51.5%

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

   7. Simplified 51.5%

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

   8. Taylor expanded in x.im around -inf 51.5%

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

      1. mul-1-neg 51.5%

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

   10. Simplified 51.5%

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

   if -3.1e10 < x.im < 0.0560000000000000012

   1. Initial program 47.2%

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

   3. Taylor expanded in y.re around 0 42.2%

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

      1. *-commutative 42.2%

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

      2. unpow2 42.2%

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

      3. unpow2 42.2%

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

      4. hypot-undefine 62.1%

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

   5. Simplified 62.1%

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

   6. Taylor expanded in x.re around inf 36.0%

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

      1. mul-1-neg 36.0%

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

      2. log-rec 36.0%

         \[\leadsto e^{\left(-\color{blue}{\left(-\log x.re\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

      3. remove-double-neg 36.0%

         \[\leadsto e^{\color{blue}{\log x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

   8. Simplified 36.0%

      \[\leadsto e^{\color{blue}{\log x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 23.4%

         \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)} \cdot y.im\right) \]

      2. expm1-undefine 23.4%

         \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} - 1\right)} \cdot y.im\right) \]

   10. Applied egg-rr 23.4%

       \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} - 1\right)} \cdot y.im\right) \]
   11. Step-by-step derivation

       1. expm1-define 23.4%

          \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)} \cdot y.im\right) \]

   12. Simplified 23.4%

       \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)} \cdot y.im\right) \]

   13. Taylor expanded in y.im around 0 32.7%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right)} \]
   14. Step-by-step derivation

       1. unpow2 32.7%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right) \]

       2. unpow2 32.7%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {x.re}^{y.re}\right) \]

       3. hypot-undefine 46.2%

          \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {x.re}^{y.re}\right) \]

   15. Simplified 46.2%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)} \]

   if 0.0560000000000000012 < x.im

   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

      1. fmm-def 31.3%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 31.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 31.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 31.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 69.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 69.6%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 69.6%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 38.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 38.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 38.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 41.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 41.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in x.im around inf 41.6%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{x.im}}^{y.re} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 41.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.re \leq -2.8 \cdot 10^{+84}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \mathbf{elif}\;x.re \leq 2.6 \cdot 10^{-44}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.re -2.8e+84)
     (* t_0 (pow x.re y.re))
     (if (<= x.re 2.6e-44)
       (* t_0 (pow x.im y.re))
       (* y.im (* (log (hypot x.im x.re)) (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 <= -2.8e+84) {
		tmp = t_0 * pow(x_46_re, y_46_re);
	} else if (x_46_re <= 2.6e-44) {
		tmp = t_0 * pow(x_46_im, y_46_re);
	} else {
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * 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 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -2.8e+84) {
		tmp = t_0 * Math.pow(x_46_re, y_46_re);
	} else if (x_46_re <= 2.6e-44) {
		tmp = t_0 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * 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 <= -2.8e+84:
		tmp = t_0 * math.pow(x_46_re, y_46_re)
	elif x_46_re <= 2.6e-44:
		tmp = t_0 * math.pow(x_46_im, y_46_re)
	else:
		tmp = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * 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 <= -2.8e+84)
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	elseif (x_46_re <= 2.6e-44)
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	else
		tmp = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * (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 <= -2.8e+84)
		tmp = t_0 * (x_46_re ^ y_46_re);
	elseif (x_46_re <= 2.6e-44)
		tmp = t_0 * (x_46_im ^ y_46_re);
	else
		tmp = y_46_im * (log(hypot(x_46_im, x_46_re)) * (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, -2.8e+84], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.6e-44], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.79999999999999982e84

   1. Initial program 17.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. fmm-def 17.7%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 17.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 17.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 17.7%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 73.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 73.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 73.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 53.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 53.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 53.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 49.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 49.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in x.im around 0 40.1%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{x.re}}^{y.re} \]

   if -2.79999999999999982e84 < x.re < 2.5999999999999998e-44

   1. Initial program 51.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. fmm-def 51.1%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 51.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 51.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 51.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 84.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 84.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 84.2%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 41.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 41.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 41.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 43.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 43.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in x.im around inf 36.0%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{x.im}}^{y.re} \]

   if 2.5999999999999998e-44 < x.re

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0 29.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   4. Step-by-step derivation

      1. *-commutative 29.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot y.im\right)} \]

      2. unpow2 29.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot y.im\right) \]

      3. unpow2 29.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot y.im\right) \]

      4. hypot-undefine 56.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot y.im\right) \]

   5. Simplified 56.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)} \]

   6. Taylor expanded in x.re around inf 68.1%

      \[\leadsto e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{x.re}\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]
   7. Step-by-step derivation

      1. mul-1-neg 68.1%

         \[\leadsto e^{\color{blue}{\left(-\log \left(\frac{1}{x.re}\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

      2. log-rec 68.1%

         \[\leadsto e^{\left(-\color{blue}{\left(-\log x.re\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

      3. remove-double-neg 68.1%

         \[\leadsto e^{\color{blue}{\log x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]

   8. Simplified 68.1%

      \[\leadsto e^{\color{blue}{\log x.re} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \]
   9. Step-by-step derivation

      1. expm1-log1p-u 70.4%

         \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)} \cdot y.im\right) \]

      2. expm1-undefine 69.1%

         \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} - 1\right)} \cdot y.im\right) \]

   10. Applied egg-rr 69.1%

       \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(e^{\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} - 1\right)} \cdot y.im\right) \]
   11. Step-by-step derivation

       1. expm1-define 70.4%

          \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)} \cdot y.im\right) \]

   12. Simplified 70.4%

       \[\leadsto e^{\log x.re \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)} \cdot y.im\right) \]

   13. Taylor expanded in y.im around 0 41.3%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right)} \]
   14. Step-by-step derivation

       1. unpow2 41.3%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right) \cdot {x.re}^{y.re}\right) \]

       2. unpow2 41.3%

          \[\leadsto y.im \cdot \left(\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right) \cdot {x.re}^{y.re}\right) \]

       3. hypot-undefine 62.9%

          \[\leadsto y.im \cdot \left(\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {x.re}^{y.re}\right) \]

   15. Simplified 62.9%

       \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {x.re}^{y.re}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 25.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-137}:\\ \;\;\;\;y.re \cdot \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-123}:\\ \;\;\;\;\sqrt[3]{{t\_0}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -6.5e-137)
     (* y.re (log (exp (atan2 x.im x.re))))
     (if (<= y.re 9.5e-123) (cbrt (pow t_0 3.0)) (log1p (expm1 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 tmp;
	if (y_46_re <= -6.5e-137) {
		tmp = y_46_re * log(exp(atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9.5e-123) {
		tmp = cbrt(pow(t_0, 3.0));
	} else {
		tmp = log1p(expm1(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 tmp;
	if (y_46_re <= -6.5e-137) {
		tmp = y_46_re * Math.log(Math.exp(Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9.5e-123) {
		tmp = Math.cbrt(Math.pow(t_0, 3.0));
	} else {
		tmp = Math.log1p(Math.expm1(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))
	tmp = 0.0
	if (y_46_re <= -6.5e-137)
		tmp = Float64(y_46_re * log(exp(atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 9.5e-123)
		tmp = cbrt((t_0 ^ 3.0));
	else
		tmp = log1p(expm1(t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6.5e-137], N[(y$46$re * N[Log[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-123], N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -6.49999999999999991e-137

   1. Initial program 42.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. fmm-def 42.0%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 42.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 42.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 42.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 90.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 90.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 90.5%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 64.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 64.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 64.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 70.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 70.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in y.re around 0 19.5%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. add-log-exp 28.3%

         \[\leadsto y.re \cdot \color{blue}{\log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]

   10. Applied egg-rr 28.3%

       \[\leadsto y.re \cdot \color{blue}{\log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]

   if -6.49999999999999991e-137 < y.re < 9.5000000000000002e-123

   1. Initial program 43.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. fmm-def 43.1%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 43.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 43.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 43.1%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 80.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 80.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 80.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 19.0%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 19.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 19.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 8.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 8.0%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in y.re around 0 8.0%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. add-cbrt-cube 24.7%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \]

      2. pow3 24.7%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \]

   10. Applied egg-rr 24.7%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}} \]

   if 9.5000000000000002e-123 < y.re

   1. Initial program 35.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. fmm-def 35.2%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 35.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 35.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 35.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 67.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 67.0%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 67.0%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 42.4%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 42.4%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 42.4%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 46.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 46.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in y.re around 0 9.1%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 34.5%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   10. Applied egg-rr 34.5%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 23.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+51}:\\ \;\;\;\;y.re \cdot \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.4e+51)
   (* y.re (log (exp (atan2 x.im x.re))))
   (log1p (expm1 (* 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 tmp;
	if (y_46_re <= -5.4e+51) {
		tmp = y_46_re * log(exp(atan2(x_46_im, x_46_re)));
	} else {
		tmp = log1p(expm1((y_46_re * 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_re <= -5.4e+51) {
		tmp = y_46_re * Math.log(Math.exp(Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = Math.log1p(Math.expm1((y_46_re * 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_re <= -5.4e+51:
		tmp = y_46_re * math.log(math.exp(math.atan2(x_46_im, x_46_re)))
	else:
		tmp = math.log1p(math.expm1((y_46_re * 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_re <= -5.4e+51)
		tmp = Float64(y_46_re * log(exp(atan(x_46_im, x_46_re))));
	else
		tmp = log1p(expm1(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_] := If[LessEqual[y$46$re, -5.4e+51], N[(y$46$re * N[Log[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.39999999999999983e51

   1. Initial program 48.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. fmm-def 48.9%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 48.9%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 48.9%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 48.9%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 93.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 93.3%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 93.3%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 95.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 95.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 95.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 95.6%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 95.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in y.re around 0 15.1%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. add-log-exp 30.2%

         \[\leadsto y.re \cdot \color{blue}{\log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]

   10. Applied egg-rr 30.2%

       \[\leadsto y.re \cdot \color{blue}{\log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]

   if -5.39999999999999983e51 < y.re

   1. Initial program 37.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. fmm-def 37.5%

         \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. hypot-define 37.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. distribute-rgt-neg-out 37.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. fma-define 37.5%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      5. hypot-define 75.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-commutative 75.2%

         \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in y.im around 0 33.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. unpow2 33.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 33.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-undefine 34.0%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   7. Simplified 34.0%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   8. Taylor expanded in y.re around 0 11.8%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. log1p-expm1-u 26.1%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   10. Applied egg-rr 26.1%

       \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 22.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (log1p (expm1 (* 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) {
	return log1p(expm1((y_46_re * atan2(x_46_im, x_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.log1p(Math.expm1((y_46_re * Math.atan2(x_46_im, x_46_re))));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.log1p(math.expm1((y_46_re * math.atan2(x_46_im, x_46_re))))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return log1p(expm1(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_] := N[Log[1 + N[(Exp[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 39.5%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. fmm-def 39.5%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. hypot-define 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. distribute-rgt-neg-out 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. fma-define 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   5. hypot-define 78.4%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   6. *-commutative 78.4%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

3. Simplified 78.4%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in y.im around 0 44.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
6. Step-by-step derivation

   1. unpow2 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

   2. unpow2 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

   3. hypot-undefine 44.9%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

7. Simplified 44.9%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

8. Taylor expanded in y.re around 0 12.4%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Step-by-step derivation

   1. log1p-expm1-u 24.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

10. Applied egg-rr 24.1%

    \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
11. Add Preprocessing

Alternative 25: 13.5% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (sin (* 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) {
	return sin((y_46_re * atan2(x_46_im, x_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)))
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)));
}

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)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return sin(Float64(y_46_re * atan(x_46_im, x_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)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 39.5%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. fmm-def 39.5%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. hypot-define 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. distribute-rgt-neg-out 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. fma-define 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   5. hypot-define 78.4%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   6. *-commutative 78.4%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

3. Simplified 78.4%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in y.im around 0 44.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
6. Step-by-step derivation

   1. unpow2 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

   2. unpow2 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

   3. hypot-undefine 44.9%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

7. Simplified 44.9%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

8. Taylor expanded in y.re around 0 12.4%

   \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

9. Taylor expanded in y.re around inf 12.4%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
10. Step-by-step derivation

    1. *-commutative 12.4%

       \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

11. Simplified 12.4%

    \[\leadsto \color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

12. Final simplification 12.4%

    \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
13. Add Preprocessing

Alternative 26: 13.5% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* 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) {
	return y_46_re * atan2(x_46_im, x_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 = y_46re * atan2(x_46im, x_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 y_46_re * Math.atan2(x_46_im, x_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_re * math.atan2(x_46_im, x_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_re * atan(x_46_im, x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_re * atan2(x_46_im, x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.5%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. fmm-def 39.5%

      \[\leadsto e^{\color{blue}{\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. hypot-define 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.re, -\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. distribute-rgt-neg-out 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. fma-define 39.5%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   5. hypot-define 78.4%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   6. *-commutative 78.4%

      \[\leadsto e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

3. Simplified 78.4%

   \[\leadsto \color{blue}{e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in y.im around 0 44.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
6. Step-by-step derivation

   1. unpow2 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

   2. unpow2 44.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

   3. hypot-undefine 44.9%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

7. Simplified 44.9%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

8. Taylor expanded in y.re around 0 12.4%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024191 
(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)))))