powComplex, imaginary part

Percentage Accurate: 41.0% → 75.8%

Time: 25.8s

Alternatives: 15

Speedup: 2.7×

• 35.3% 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: 41.0% 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: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;\left|\sin t\_1 \cdot {t\_0}^{\left(\frac{y.re}{2}\right)}\right|\\ \mathbf{elif}\;y.re \leq 48000000000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{t\_0}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(0.5 \cdot \log t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im))) (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -7.5e+54)
     (fabs (* (sin t_1) (pow t_0 (/ y.re 2.0))))
     (if (<= y.re 48000000000.0)
       (*
        (/ (pow (hypot x.re x.im) y.re) (pow (exp y.im) (atan2 x.im x.re)))
        (sin (fma (log (hypot x.re x.im)) y.im t_1)))
       (*
        (exp (- (* y.re (log (sqrt t_0))) (* (atan2 x.im x.re) y.im)))
        (* y.im (* 0.5 (log t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -7.5e+54) {
		tmp = fabs((sin(t_1) * pow(t_0, (y_46_re / 2.0))));
	} else if (y_46_re <= 48000000000.0) {
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / pow(exp(y_46_im), atan2(x_46_im, x_46_re))) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_1));
	} else {
		tmp = exp(((y_46_re * log(sqrt(t_0))) - (atan2(x_46_im, x_46_re) * y_46_im))) * (y_46_im * (0.5 * log(t_0)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -7.5e+54)
		tmp = abs(Float64(sin(t_1) * (t_0 ^ Float64(y_46_re / 2.0))));
	elseif (y_46_re <= 48000000000.0)
		tmp = Float64(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / (exp(y_46_im) ^ atan(x_46_im, x_46_re))) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_1)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(t_0))) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * Float64(y_46_im * Float64(0.5 * log(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[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e+54], N[Abs[N[(N[Sin[t$95$1], $MachinePrecision] * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 48000000000.0], N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y$46$im * N[(0.5 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.50000000000000042e54

   1. Initial program 46.7%

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

      1. exp-diff 35.6%

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

      2. exp-to-pow 35.6%

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

      3. hypot-define 35.6%

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

      4. *-commutative 35.6%

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

      5. exp-prod 28.9%

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

      6. fma-define 28.9%

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

      7. hypot-define 66.7%

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

      8. *-commutative 66.7%

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

   3. Simplified 66.7%

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

   5. Taylor expanded in y.im around 0 82.4%

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

      1. unpow2 82.4%

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

      2. unpow2 82.4%

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

      3. hypot-undefine 82.4%

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

   7. Simplified 82.4%

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

   8. Taylor expanded in y.im around 0 82.4%

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

      1. add-sqr-sqrt 80.2%

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

      2. sqrt-unprod 86.8%

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

      3. pow2 86.8%

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

      4. *-commutative 86.8%

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

      5. *-commutative 86.8%

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

      6. hypot-undefine 86.8%

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

      7. +-commutative 86.8%

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

      8. sqrt-pow2 86.8%

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

   10. Applied egg-rr 86.8%

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

       1. unpow2 86.8%

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

       2. rem-sqrt-square 86.8%

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

       3. *-commutative 86.8%

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

   12. Simplified 86.8%

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

   if -7.50000000000000042e54 < y.re < 4.8e10

   1. Initial program 39.2%

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

      1. exp-diff 39.2%

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

      2. exp-to-pow 39.2%

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

      3. hypot-define 39.2%

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

      4. *-commutative 39.2%

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

      5. exp-prod 39.0%

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

      6. fma-define 39.0%

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

      7. hypot-define 80.5%

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

      8. *-commutative 80.5%

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

   3. Simplified 80.5%

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

   if 4.8e10 < y.re

   1. Initial program 29.0%

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

   3. Taylor expanded in y.re around 0 32.3%

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

      1. +-commutative 32.3%

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

      2. unpow2 32.3%

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

      3. unpow2 32.3%

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

      4. unpow1/2 32.3%

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

      5. exp-to-pow 32.3%

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

      6. *-commutative 32.3%

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

      7. rem-log-exp 32.3%

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

      8. associate-*r* 32.3%

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

   5. Simplified 32.3%

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

   6. Taylor expanded in y.im around 0 67.8%

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

      1. *-commutative 67.8%

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

      2. +-commutative 67.8%

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

      3. unpow2 67.8%

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

      4. unpow2 67.8%

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

      5. associate-*l* 67.8%

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

      6. *-commutative 67.8%

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

   8. Simplified 67.8%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{+54}:\\ \;\;\;\;\left|\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right|\\ \mathbf{elif}\;y.re \leq 48000000000:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \sin t\_2\\ \mathbf{if}\;y.re \leq -1.52 \cdot 10^{+22}:\\ \;\;\;\;\left|t\_3 \cdot {t\_0}^{\left(\frac{y.re}{2}\right)}\right|\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-57}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_2\right)\right) \cdot \frac{1}{e^{t\_1}}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+28}:\\ \;\;\;\;t\_3 \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+56}:\\ \;\;\;\;t\_3 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{t\_0}\right) - t\_1} \cdot \left(y.im \cdot \left(0.5 \cdot \log t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (sin t_2)))
   (if (<= y.re -1.52e+22)
     (fabs (* t_3 (pow t_0 (/ y.re 2.0))))
     (if (<= y.re 1.7e-57)
       (* (sin (fma (log (hypot x.re x.im)) y.im t_2)) (/ 1.0 (exp t_1)))
       (if (<= y.re 2.8e+28)
         (*
          t_3
          (/ (pow (hypot x.re x.im) y.re) (pow (exp y.im) (atan2 x.im x.re))))
         (if (<= y.re 3.8e+56)
           (* t_3 (pow x.re y.re))
           (*
            (exp (- (* y.re (log (sqrt t_0))) t_1))
            (* y.im (* 0.5 (log t_0))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = sin(t_2);
	double tmp;
	if (y_46_re <= -1.52e+22) {
		tmp = fabs((t_3 * pow(t_0, (y_46_re / 2.0))));
	} else if (y_46_re <= 1.7e-57) {
		tmp = sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_2)) * (1.0 / exp(t_1));
	} else if (y_46_re <= 2.8e+28) {
		tmp = t_3 * (pow(hypot(x_46_re, x_46_im), y_46_re) / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 3.8e+56) {
		tmp = t_3 * pow(x_46_re, y_46_re);
	} else {
		tmp = exp(((y_46_re * log(sqrt(t_0))) - t_1)) * (y_46_im * (0.5 * log(t_0)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = sin(t_2)
	tmp = 0.0
	if (y_46_re <= -1.52e+22)
		tmp = abs(Float64(t_3 * (t_0 ^ Float64(y_46_re / 2.0))));
	elseif (y_46_re <= 1.7e-57)
		tmp = Float64(sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_2)) * Float64(1.0 / exp(t_1)));
	elseif (y_46_re <= 2.8e+28)
		tmp = Float64(t_3 * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / (exp(y_46_im) ^ atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 3.8e+56)
		tmp = Float64(t_3 * (x_46_re ^ y_46_re));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(t_0))) - t_1)) * Float64(y_46_im * Float64(0.5 * log(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[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, If[LessEqual[y$46$re, -1.52e+22], N[Abs[N[(t$95$3 * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.7e-57], N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.8e+28], N[(t$95$3 * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.8e+56], N[(t$95$3 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[(y$46$im * N[(0.5 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.52e22

   1. Initial program 47.1%

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

      1. exp-diff 37.3%

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

      2. exp-to-pow 37.3%

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

      3. hypot-define 37.3%

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

      4. *-commutative 37.3%

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

      5. exp-prod 31.4%

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

      6. fma-define 31.4%

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

      7. hypot-define 68.6%

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

      8. *-commutative 68.6%

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

   3. Simplified 68.6%

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

   5. Taylor expanded in y.im around 0 82.5%

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

      1. unpow2 82.5%

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

      2. unpow2 82.5%

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

      3. hypot-undefine 82.5%

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

   7. Simplified 82.5%

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

   8. Taylor expanded in y.im around 0 82.5%

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

      1. add-sqr-sqrt 78.6%

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

      2. sqrt-unprod 84.5%

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

      3. pow2 84.5%

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

      4. *-commutative 84.5%

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

      5. *-commutative 84.5%

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

      6. hypot-undefine 84.5%

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

      7. +-commutative 84.5%

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

      8. sqrt-pow2 84.5%

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

   10. Applied egg-rr 84.5%

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

       1. unpow2 84.5%

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

       2. rem-sqrt-square 84.5%

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

       3. *-commutative 84.5%

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

   12. Simplified 84.5%

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

   if -1.52e22 < y.re < 1.70000000000000008e-57

   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. Step-by-step derivation

      1. exp-diff 37.9%

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

      2. exp-to-pow 37.9%

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

      3. hypot-define 37.9%

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

      4. *-commutative 37.9%

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

      5. exp-prod 37.7%

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

      6. fma-define 37.7%

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

      7. hypot-define 79.0%

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

      8. *-commutative 79.0%

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

   3. Simplified 79.0%

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

      1. frac-2neg 79.0%

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

      2. distribute-frac-neg 79.0%

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

      3. hypot-define 50.0%

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

      4. sqrt-pow2 50.0%

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

      5. div-inv 50.0%

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

      6. metadata-eval 50.0%

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

      7. neg-sub0 52.2%

         \[\leadsto \left(-\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{\color{blue}{0 - {\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}}\right) \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. pow-exp 52.4%

         \[\leadsto \left(-\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{0 - \color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\right) \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied egg-rr 52.4%

      \[\leadsto \color{blue}{\left(-\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{0 - e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Taylor expanded in y.re around 0 81.9%

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

   if 1.70000000000000008e-57 < y.re < 2.8000000000000001e28

   1. Initial program 42.9%

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

      1. exp-diff 35.7%

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

      2. exp-to-pow 35.7%

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

      3. hypot-define 35.7%

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

      4. *-commutative 35.7%

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

      5. exp-prod 35.7%

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

      6. fma-define 35.7%

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

      7. hypot-define 85.5%

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

      8. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   5. Taylor expanded in y.im around 0 85.5%

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

   if 2.8000000000000001e28 < y.re < 3.79999999999999996e56

   1. Initial program 16.7%

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

      1. exp-diff 16.7%

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

      2. exp-to-pow 16.7%

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

      3. hypot-define 16.7%

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

      4. *-commutative 16.7%

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

      5. exp-prod 16.7%

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

      6. fma-define 16.7%

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

      7. hypot-define 50.0%

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

      8. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in y.im around 0 50.8%

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

      1. unpow2 50.8%

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

      2. unpow2 50.8%

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

      3. hypot-undefine 50.8%

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

   7. Simplified 50.8%

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

   8. Taylor expanded in y.im around 0 50.8%

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

   9. Taylor expanded in x.im around 0 67.2%

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

   if 3.79999999999999996e56 < y.re

   1. Initial program 30.8%

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

   3. Taylor expanded in y.re around 0 34.6%

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

      1. +-commutative 34.6%

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

      2. unpow2 34.6%

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

      3. unpow2 34.6%

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

      4. unpow1/2 34.6%

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

      5. exp-to-pow 34.6%

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

      6. *-commutative 34.6%

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

      7. rem-log-exp 34.6%

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

      8. associate-*r* 34.6%

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

   5. Simplified 34.6%

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

   6. Taylor expanded in y.im around 0 71.2%

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

      1. *-commutative 71.2%

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

      2. +-commutative 71.2%

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

      3. unpow2 71.2%

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

      4. unpow2 71.2%

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

      5. associate-*l* 71.2%

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

      6. *-commutative 71.2%

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

   8. Simplified 71.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.im \cdot \left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.52 \cdot 10^{+22}:\\ \;\;\;\;\left|\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right|\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-57}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+28}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+56}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-27}:\\ \;\;\;\;t\_2 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 180000000000:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_0\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1
         (*
          (sin t_0)
          (exp
           (-
            (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
            (* (atan2 x.im x.re) y.im)))))
        (t_2 (pow (hypot x.im x.re) y.re)))
   (if (<= y.im -1.7e+16)
     t_1
     (if (<= y.im -4e-27)
       (* t_2 (sin (* y.im (log (hypot x.im x.re)))))
       (if (<= y.im 180000000000.0)
         (* (sin (fma (log (hypot x.re x.im)) y.im t_0)) t_2)
         t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0) * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_2 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_im <= -1.7e+16) {
		tmp = t_1;
	} else if (y_46_im <= -4e-27) {
		tmp = t_2 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else if (y_46_im <= 180000000000.0) {
		tmp = sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(sin(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(atan(x_46_im, x_46_re) * y_46_im))))
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_im <= -1.7e+16)
		tmp = t_1;
	elseif (y_46_im <= -4e-27)
		tmp = Float64(t_2 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	elseif (y_46_im <= 180000000000.0)
		tmp = Float64(sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * t_2);
	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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[t$95$0], $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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+16], t$95$1, If[LessEqual[y$46$im, -4e-27], N[(t$95$2 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 180000000000.0], N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.7e16 or 1.8e11 < y.im

   1. Initial program 33.1%

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

   3. Taylor expanded in y.im around 0 56.4%

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

   if -1.7e16 < y.im < -4.0000000000000002e-27

   1. Initial program 30.0%

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

      1. exp-diff 30.0%

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

      2. exp-to-pow 30.0%

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

      3. hypot-define 30.0%

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

      4. *-commutative 30.0%

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

      5. exp-prod 30.0%

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

      6. fma-define 30.0%

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

      7. hypot-define 50.0%

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

      8. *-commutative 50.0%

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

   3. Simplified 50.0%

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

   5. Taylor expanded in y.im around 0 39.7%

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

      1. unpow2 39.7%

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

      2. unpow2 39.7%

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

      3. hypot-undefine 49.3%

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

   7. Simplified 49.3%

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

   8. Taylor expanded in y.re around 0 59.3%

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

      1. unpow2 59.3%

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

      2. unpow2 59.3%

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

      3. hypot-define 99.3%

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

   10. Simplified 99.3%

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

   if -4.0000000000000002e-27 < y.im < 1.8e11

   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. Step-by-step derivation

      1. exp-diff 43.0%

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

      2. exp-to-pow 43.0%

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

      3. hypot-define 43.0%

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

      4. *-commutative 43.0%

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

      5. exp-prod 43.0%

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

      6. fma-define 43.0%

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

      7. hypot-define 91.1%

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

      8. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in y.im around 0 63.7%

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

      1. unpow2 63.7%

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

      2. unpow2 63.7%

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

      3. hypot-undefine 91.6%

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

   7. Simplified 91.6%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-27}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 180000000000:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -0.88:\\ \;\;\;\;\left|\sin t\_2 \cdot {t\_0}^{\left(\frac{y.re}{2}\right)}\right|\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+31}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_2\right)\right) \cdot \frac{1}{e^{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{t\_0}\right) - t\_1} \cdot \left(y.im \cdot \left(0.5 \cdot \log t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -0.88)
     (fabs (* (sin t_2) (pow t_0 (/ y.re 2.0))))
     (if (<= y.re 1.2e+31)
       (* (sin (fma (log (hypot x.re x.im)) y.im t_2)) (/ 1.0 (exp t_1)))
       (*
        (exp (- (* y.re (log (sqrt t_0))) t_1))
        (* y.im (* 0.5 (log t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -0.88) {
		tmp = fabs((sin(t_2) * pow(t_0, (y_46_re / 2.0))));
	} else if (y_46_re <= 1.2e+31) {
		tmp = sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_2)) * (1.0 / exp(t_1));
	} else {
		tmp = exp(((y_46_re * log(sqrt(t_0))) - t_1)) * (y_46_im * (0.5 * log(t_0)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -0.88)
		tmp = abs(Float64(sin(t_2) * (t_0 ^ Float64(y_46_re / 2.0))));
	elseif (y_46_re <= 1.2e+31)
		tmp = Float64(sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_2)) * Float64(1.0 / exp(t_1)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(t_0))) - t_1)) * Float64(y_46_im * Float64(0.5 * log(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[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -0.88], N[Abs[N[(N[Sin[t$95$2], $MachinePrecision] * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.2e+31], N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[(y$46$im * N[(0.5 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.880000000000000004

   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. exp-diff 35.7%

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

      2. exp-to-pow 35.7%

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

      3. hypot-define 35.7%

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

      4. *-commutative 35.7%

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

      5. exp-prod 30.4%

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

      6. fma-define 30.4%

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

      7. hypot-define 67.9%

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

      8. *-commutative 67.9%

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

   3. Simplified 67.9%

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

   5. Taylor expanded in y.im around 0 80.6%

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

      1. unpow2 80.6%

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

      2. unpow2 80.6%

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

      3. hypot-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in y.im around 0 80.6%

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

      1. add-sqr-sqrt 77.0%

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

      2. sqrt-unprod 82.4%

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

      3. pow2 82.4%

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

      4. *-commutative 82.4%

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

      5. *-commutative 82.4%

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

      6. hypot-undefine 82.4%

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

      7. +-commutative 82.4%

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

      8. sqrt-pow2 82.4%

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

   10. Applied egg-rr 82.4%

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

       1. unpow2 82.4%

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

       2. rem-sqrt-square 82.4%

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

       3. *-commutative 82.4%

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

   12. Simplified 82.4%

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

   if -0.880000000000000004 < y.re < 1.19999999999999991e31

   1. Initial program 38.7%

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

      1. exp-diff 38.0%

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

      2. exp-to-pow 38.0%

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

      3. hypot-define 38.0%

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

      4. *-commutative 38.0%

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

      5. exp-prod 37.9%

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

      6. fma-define 37.9%

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

      7. hypot-define 79.7%

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

      8. *-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in y.re around 0 79.6%

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

   if 1.19999999999999991e31 < y.re

   1. Initial program 29.8%

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

   3. Taylor expanded in y.re around 0 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%

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

      2. unpow2 33.3%

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

      3. unpow2 33.3%

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

      4. unpow1/2 33.3%

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

      5. exp-to-pow 33.3%

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

      6. *-commutative 33.3%

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

      7. rem-log-exp 33.3%

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

      8. associate-*r* 33.3%

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

   5. Simplified 33.3%

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

   6. Taylor expanded in y.im around 0 70.2%

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

      1. *-commutative 70.2%

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

      2. +-commutative 70.2%

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

      3. unpow2 70.2%

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

      4. unpow2 70.2%

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

      5. associate-*l* 70.2%

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

      6. *-commutative 70.2%

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

   8. Simplified 70.2%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.88:\\ \;\;\;\;\left|\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right|\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{+31}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := t\_2 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -5.2 \cdot 10^{-204}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-297}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{elif}\;y.im \leq 27500000000000:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1
         (*
          t_0
          (exp
           (-
            (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
            (* (atan2 x.im x.re) y.im)))))
        (t_2 (pow (hypot x.im x.re) y.re))
        (t_3 (* t_2 (sin (* y.im (log (hypot x.im x.re)))))))
   (if (<= y.im -1.7e+16)
     t_1
     (if (<= y.im -5.2e-204)
       t_3
       (if (<= y.im 3.6e-297)
         (* t_0 t_2)
         (if (<= y.im 27500000000000.0) t_3 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_2 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_2 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e+16) {
		tmp = t_1;
	} else if (y_46_im <= -5.2e-204) {
		tmp = t_3;
	} else if (y_46_im <= 3.6e-297) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 27500000000000.0) {
		tmp = t_3;
	} 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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double t_2 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_2 * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e+16) {
		tmp = t_1;
	} else if (y_46_im <= -5.2e-204) {
		tmp = t_3;
	} else if (y_46_im <= 3.6e-297) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 27500000000000.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = t_0 * math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	t_2 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_3 = t_2 * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if y_46_im <= -1.7e+16:
		tmp = t_1
	elif y_46_im <= -5.2e-204:
		tmp = t_3
	elif y_46_im <= 3.6e-297:
		tmp = t_0 * t_2
	elif y_46_im <= 27500000000000.0:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_0 * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im))))
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_3 = Float64(t_2 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -1.7e+16)
		tmp = t_1;
	elseif (y_46_im <= -5.2e-204)
		tmp = t_3;
	elseif (y_46_im <= 3.6e-297)
		tmp = Float64(t_0 * t_2);
	elseif (y_46_im <= 27500000000000.0)
		tmp = t_3;
	else
		tmp = 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = t_0 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_3 = t_2 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_im <= -1.7e+16)
		tmp = t_1;
	elseif (y_46_im <= -5.2e-204)
		tmp = t_3;
	elseif (y_46_im <= 3.6e-297)
		tmp = t_0 * t_2;
	elseif (y_46_im <= 27500000000000.0)
		tmp = t_3;
	else
		tmp = 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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+16], t$95$1, If[LessEqual[y$46$im, -5.2e-204], t$95$3, If[LessEqual[y$46$im, 3.6e-297], N[(t$95$0 * t$95$2), $MachinePrecision], If[LessEqual[y$46$im, 27500000000000.0], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.7e16 or 2.75e13 < y.im

   1. Initial program 33.1%

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

   3. Taylor expanded in y.im around 0 56.4%

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

   if -1.7e16 < y.im < -5.19999999999999965e-204 or 3.59999999999999994e-297 < y.im < 2.75e13

   1. Initial program 40.3%

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

      1. exp-diff 40.3%

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

      2. exp-to-pow 40.3%

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

      3. hypot-define 40.3%

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

      4. *-commutative 40.3%

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

      5. exp-prod 40.3%

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

      6. fma-define 40.3%

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

      7. hypot-define 86.3%

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

      8. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y.im around 0 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. hypot-undefine 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in y.re around 0 37.1%

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

      1. unpow2 37.1%

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

      2. unpow2 37.1%

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

      3. hypot-define 80.6%

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

   10. Simplified 80.6%

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

   if -5.19999999999999965e-204 < y.im < 3.59999999999999994e-297

   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. exp-diff 49.9%

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

      2. exp-to-pow 49.9%

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

      3. hypot-define 49.9%

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

      4. *-commutative 49.9%

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

      5. exp-prod 49.9%

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

      6. fma-define 50.0%

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

      7. hypot-define 96.0%

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

      8. *-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in y.im around 0 77.7%

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

      1. unpow2 77.7%

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

      2. unpow2 77.7%

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

      3. hypot-undefine 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y.im around 0 86.5%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.im \leq -5.2 \cdot 10^{-204}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-297}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 27500000000000:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := t\_2 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;\left(t\_1 \cdot {t\_0}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{elif}\;y.im \leq -3.7 \cdot 10^{-204}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{-297}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{elif}\;y.im \leq 10^{+17}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \frac{{t\_0}^{\left(y.re \cdot 0.5\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (sin (* y.re (atan2 x.im x.re))))
        (t_2 (pow (hypot x.im x.re) y.re))
        (t_3 (* t_2 (sin (* y.im (log (hypot x.im x.re)))))))
   (if (<= y.im -1.7e+16)
     (+ (+ (* t_1 (pow t_0 (/ y.re 2.0))) 1.0) -1.0)
     (if (<= y.im -3.7e-204)
       t_3
       (if (<= y.im 9.5e-297)
         (* t_1 t_2)
         (if (<= y.im 1e+17)
           t_3
           (*
            t_1
            (/ (pow t_0 (* y.re 0.5)) (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 t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_2 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e+16) {
		tmp = ((t_1 * pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0;
	} else if (y_46_im <= -3.7e-204) {
		tmp = t_3;
	} else if (y_46_im <= 9.5e-297) {
		tmp = t_1 * t_2;
	} else if (y_46_im <= 1e+17) {
		tmp = t_3;
	} else {
		tmp = t_1 * (pow(t_0, (y_46_re * 0.5)) / 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 t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_2 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_2 * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e+16) {
		tmp = ((t_1 * Math.pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0;
	} else if (y_46_im <= -3.7e-204) {
		tmp = t_3;
	} else if (y_46_im <= 9.5e-297) {
		tmp = t_1 * t_2;
	} else if (y_46_im <= 1e+17) {
		tmp = t_3;
	} else {
		tmp = t_1 * (Math.pow(t_0, (y_46_re * 0.5)) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_2 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_3 = t_2 * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if y_46_im <= -1.7e+16:
		tmp = ((t_1 * math.pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0
	elif y_46_im <= -3.7e-204:
		tmp = t_3
	elif y_46_im <= 9.5e-297:
		tmp = t_1 * t_2
	elif y_46_im <= 1e+17:
		tmp = t_3
	else:
		tmp = t_1 * (math.pow(t_0, (y_46_re * 0.5)) / 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)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_3 = Float64(t_2 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -1.7e+16)
		tmp = Float64(Float64(Float64(t_1 * (t_0 ^ Float64(y_46_re / 2.0))) + 1.0) + -1.0);
	elseif (y_46_im <= -3.7e-204)
		tmp = t_3;
	elseif (y_46_im <= 9.5e-297)
		tmp = Float64(t_1 * t_2);
	elseif (y_46_im <= 1e+17)
		tmp = t_3;
	else
		tmp = Float64(t_1 * Float64((t_0 ^ Float64(y_46_re * 0.5)) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_3 = t_2 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_im <= -1.7e+16)
		tmp = ((t_1 * (t_0 ^ (y_46_re / 2.0))) + 1.0) + -1.0;
	elseif (y_46_im <= -3.7e-204)
		tmp = t_3;
	elseif (y_46_im <= 9.5e-297)
		tmp = t_1 * t_2;
	elseif (y_46_im <= 1e+17)
		tmp = t_3;
	else
		tmp = t_1 * ((t_0 ^ (y_46_re * 0.5)) / exp((atan2(x_46_im, x_46_re) * y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+16], N[(N[(N[(t$95$1 * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], If[LessEqual[y$46$im, -3.7e-204], t$95$3, If[LessEqual[y$46$im, 9.5e-297], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[y$46$im, 1e+17], t$95$3, N[(t$95$1 * N[(N[Power[t$95$0, N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.7e16

   1. Initial program 39.0%

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

      1. exp-diff 32.2%

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

      2. exp-to-pow 32.2%

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

      3. hypot-define 32.2%

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

      4. *-commutative 32.2%

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

      5. exp-prod 27.1%

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

      6. fma-define 27.1%

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

      7. hypot-define 52.6%

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

      8. *-commutative 52.6%

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

   3. Simplified 52.6%

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

   5. Taylor expanded in y.im around 0 43.3%

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

      1. unpow2 43.3%

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

      2. unpow2 43.3%

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

      3. hypot-undefine 33.5%

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

   7. Simplified 33.5%

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

   8. Taylor expanded in y.im around 0 37.1%

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

      1. expm1-log1p-u 32.0%

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

      2. expm1-undefine 46.5%

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

      3. *-commutative 46.5%

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

      4. *-commutative 46.5%

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

      5. hypot-undefine 49.7%

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

      6. +-commutative 49.7%

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

      7. sqrt-pow2 49.7%

         \[\leadsto e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}}\right)} - 1 \]

   10. Applied egg-rr 49.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 49.7%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} + \left(-1\right)} \]

       2. log1p-undefine 49.7%

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

       3. rem-exp-log 54.8%

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

       4. *-commutative 54.8%

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

       5. metadata-eval 54.8%

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

   12. Simplified 54.8%

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

   if -1.7e16 < y.im < -3.6999999999999997e-204 or 9.5000000000000005e-297 < y.im < 1e17

   1. Initial program 40.3%

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

      1. exp-diff 40.3%

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

      2. exp-to-pow 40.3%

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

      3. hypot-define 40.3%

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

      4. *-commutative 40.3%

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

      5. exp-prod 40.3%

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

      6. fma-define 40.3%

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

      7. hypot-define 86.3%

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

      8. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y.im around 0 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. hypot-undefine 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in y.re around 0 37.1%

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

      1. unpow2 37.1%

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

      2. unpow2 37.1%

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

      3. hypot-define 80.6%

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

   10. Simplified 80.6%

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

   if -3.6999999999999997e-204 < y.im < 9.5000000000000005e-297

   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. exp-diff 49.9%

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

      2. exp-to-pow 49.9%

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

      3. hypot-define 49.9%

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

      4. *-commutative 49.9%

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

      5. exp-prod 49.9%

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

      6. fma-define 50.0%

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

      7. hypot-define 96.0%

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

      8. *-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in y.im around 0 77.7%

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

      1. unpow2 77.7%

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

      2. unpow2 77.7%

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

      3. hypot-undefine 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y.im around 0 86.5%

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

   if 1e17 < y.im

   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. Step-by-step derivation

      1. exp-diff 21.8%

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

      2. exp-to-pow 21.8%

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

      3. hypot-define 21.8%

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

      4. *-commutative 21.8%

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

      5. exp-prod 21.4%

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

      6. fma-define 21.4%

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

      7. hypot-define 48.5%

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

      8. *-commutative 48.5%

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

   3. Simplified 48.5%

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

      1. frac-2neg 48.5%

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

      2. distribute-frac-neg 48.5%

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

      3. hypot-define 44.5%

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

      4. sqrt-pow2 44.5%

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

      5. div-inv 44.5%

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

      6. metadata-eval 44.5%

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

      7. neg-sub0 44.5%

         \[\leadsto \left(-\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{\color{blue}{0 - {\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}}\right) \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. pow-exp 46.5%

         \[\leadsto \left(-\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{0 - \color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\right) \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied egg-rr 46.5%

      \[\leadsto \color{blue}{\left(-\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{0 - e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. Taylor expanded in y.im around 0 44.2%

      \[\leadsto \left(-\frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{0 - e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{elif}\;y.im \leq -3.7 \cdot 10^{-204}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{-297}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+17}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := \left(t\_0 \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := t\_2 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{-203}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-297}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{elif}\;y.im \leq 1750000000000:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1
         (+
          (+ (* t_0 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))) 1.0)
          -1.0))
        (t_2 (pow (hypot x.im x.re) y.re))
        (t_3 (* t_2 (sin (* y.im (log (hypot x.im x.re)))))))
   (if (<= y.im -1.7e+16)
     t_1
     (if (<= y.im -1.3e-203)
       t_3
       (if (<= y.im 9e-297)
         (* t_0 t_2)
         (if (<= y.im 1750000000000.0) t_3 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = ((t_0 * pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))) + 1.0) + -1.0;
	double t_2 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_2 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e+16) {
		tmp = t_1;
	} else if (y_46_im <= -1.3e-203) {
		tmp = t_3;
	} else if (y_46_im <= 9e-297) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 1750000000000.0) {
		tmp = t_3;
	} 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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = ((t_0 * Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))) + 1.0) + -1.0;
	double t_2 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_2 * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e+16) {
		tmp = t_1;
	} else if (y_46_im <= -1.3e-203) {
		tmp = t_3;
	} else if (y_46_im <= 9e-297) {
		tmp = t_0 * t_2;
	} else if (y_46_im <= 1750000000000.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = ((t_0 * math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))) + 1.0) + -1.0
	t_2 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_3 = t_2 * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if y_46_im <= -1.7e+16:
		tmp = t_1
	elif y_46_im <= -1.3e-203:
		tmp = t_3
	elif y_46_im <= 9e-297:
		tmp = t_0 * t_2
	elif y_46_im <= 1750000000000.0:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(Float64(Float64(t_0 * (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0))) + 1.0) + -1.0)
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_3 = Float64(t_2 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -1.7e+16)
		tmp = t_1;
	elseif (y_46_im <= -1.3e-203)
		tmp = t_3;
	elseif (y_46_im <= 9e-297)
		tmp = Float64(t_0 * t_2);
	elseif (y_46_im <= 1750000000000.0)
		tmp = t_3;
	else
		tmp = 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = ((t_0 * (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0))) + 1.0) + -1.0;
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_3 = t_2 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_im <= -1.7e+16)
		tmp = t_1;
	elseif (y_46_im <= -1.3e-203)
		tmp = t_3;
	elseif (y_46_im <= 9e-297)
		tmp = t_0 * t_2;
	elseif (y_46_im <= 1750000000000.0)
		tmp = t_3;
	else
		tmp = 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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(t$95$0 * N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e+16], t$95$1, If[LessEqual[y$46$im, -1.3e-203], t$95$3, If[LessEqual[y$46$im, 9e-297], N[(t$95$0 * t$95$2), $MachinePrecision], If[LessEqual[y$46$im, 1750000000000.0], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.7e16 or 1.75e12 < y.im

   1. Initial program 33.1%

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

      1. exp-diff 27.1%

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

      2. exp-to-pow 27.1%

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

      3. hypot-define 27.1%

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

      4. *-commutative 27.1%

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

      5. exp-prod 24.3%

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

      6. fma-define 24.3%

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

      7. hypot-define 50.5%

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

      8. *-commutative 50.5%

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

   3. Simplified 50.5%

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

   5. Taylor expanded in y.im around 0 37.6%

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

      1. unpow2 37.6%

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

      2. unpow2 37.6%

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

      3. hypot-undefine 27.0%

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

   7. Simplified 27.0%

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

   8. Taylor expanded in y.im around 0 27.6%

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

      1. expm1-log1p-u 25.0%

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

      2. expm1-undefine 41.4%

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

      3. *-commutative 41.4%

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

      4. *-commutative 41.4%

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

      5. hypot-undefine 42.1%

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

      6. +-commutative 42.1%

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

      7. sqrt-pow2 42.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}}\right)} - 1 \]

   10. Applied egg-rr 42.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 42.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} + \left(-1\right)} \]

       2. log1p-undefine 42.1%

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

       3. rem-exp-log 49.0%

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

       4. *-commutative 49.0%

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

       5. metadata-eval 49.0%

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

   12. Simplified 49.0%

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

   if -1.7e16 < y.im < -1.29999999999999988e-203 or 8.99999999999999951e-297 < y.im < 1.75e12

   1. Initial program 40.3%

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

      1. exp-diff 40.3%

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

      2. exp-to-pow 40.3%

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

      3. hypot-define 40.3%

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

      4. *-commutative 40.3%

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

      5. exp-prod 40.3%

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

      6. fma-define 40.3%

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

      7. hypot-define 86.3%

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

      8. *-commutative 86.3%

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

   3. Simplified 86.3%

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

   5. Taylor expanded in y.im around 0 58.5%

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

      1. unpow2 58.5%

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

      2. unpow2 58.5%

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

      3. hypot-undefine 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in y.re around 0 37.1%

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

      1. unpow2 37.1%

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

      2. unpow2 37.1%

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

      3. hypot-define 80.6%

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

   10. Simplified 80.6%

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

   if -1.29999999999999988e-203 < y.im < 8.99999999999999951e-297

   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. exp-diff 49.9%

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

      2. exp-to-pow 49.9%

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

      3. hypot-define 49.9%

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

      4. *-commutative 49.9%

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

      5. exp-prod 49.9%

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

      6. fma-define 50.0%

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

      7. hypot-define 96.0%

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

      8. *-commutative 96.0%

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

   3. Simplified 96.0%

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

   5. Taylor expanded in y.im around 0 77.7%

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

      1. unpow2 77.7%

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

      2. unpow2 77.7%

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

      3. hypot-undefine 96.0%

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y.im around 0 86.5%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16}:\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{-203}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-297}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1750000000000:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := \left(t\_1 \cdot {t\_0}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{if}\;y.im \leq -95:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-45}:\\ \;\;\;\;\left(y.im \cdot \left(0.5 \cdot \log t\_0\right)\right) \cdot e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;t\_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (sin (* y.re (atan2 x.im x.re))))
        (t_2 (+ (+ (* t_1 (pow t_0 (/ y.re 2.0))) 1.0) -1.0)))
   (if (<= y.im -95.0)
     t_2
     (if (<= y.im -1.6e-45)
       (*
        (* y.im (* 0.5 (log t_0)))
        (exp (- (* y.re (log (- x.re))) (* (atan2 x.im x.re) y.im))))
       (if (<= y.im 1.6e+38) (* t_1 (pow (hypot x.im x.re) y.re)) t_2)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = ((t_1 * pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0;
	double tmp;
	if (y_46_im <= -95.0) {
		tmp = t_2;
	} else if (y_46_im <= -1.6e-45) {
		tmp = (y_46_im * (0.5 * log(t_0))) * exp(((y_46_re * log(-x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_im <= 1.6e+38) {
		tmp = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_2 = ((t_1 * Math.pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0;
	double tmp;
	if (y_46_im <= -95.0) {
		tmp = t_2;
	} else if (y_46_im <= -1.6e-45) {
		tmp = (y_46_im * (0.5 * Math.log(t_0))) * Math.exp(((y_46_re * Math.log(-x_46_re)) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_im <= 1.6e+38) {
		tmp = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_2 = ((t_1 * math.pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0
	tmp = 0
	if y_46_im <= -95.0:
		tmp = t_2
	elif y_46_im <= -1.6e-45:
		tmp = (y_46_im * (0.5 * math.log(t_0))) * math.exp(((y_46_re * math.log(-x_46_re)) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_im <= 1.6e+38:
		tmp = t_1 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_2 = Float64(Float64(Float64(t_1 * (t_0 ^ Float64(y_46_re / 2.0))) + 1.0) + -1.0)
	tmp = 0.0
	if (y_46_im <= -95.0)
		tmp = t_2;
	elseif (y_46_im <= -1.6e-45)
		tmp = Float64(Float64(y_46_im * Float64(0.5 * log(t_0))) * exp(Float64(Float64(y_46_re * log(Float64(-x_46_re))) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	elseif (y_46_im <= 1.6e+38)
		tmp = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_2 = ((t_1 * (t_0 ^ (y_46_re / 2.0))) + 1.0) + -1.0;
	tmp = 0.0;
	if (y_46_im <= -95.0)
		tmp = t_2;
	elseif (y_46_im <= -1.6e-45)
		tmp = (y_46_im * (0.5 * log(t_0))) * exp(((y_46_re * log(-x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_im <= 1.6e+38)
		tmp = t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]}, If[LessEqual[y$46$im, -95.0], t$95$2, If[LessEqual[y$46$im, -1.6e-45], N[(N[(y$46$im * N[(0.5 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+38], N[(t$95$1 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -95 or 1.59999999999999993e38 < y.im

   1. Initial program 33.8%

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

      1. exp-diff 27.8%

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

      2. exp-to-pow 27.8%

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

      3. hypot-define 27.8%

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

      4. *-commutative 27.8%

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

      5. exp-prod 25.2%

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

      6. fma-define 25.2%

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

      7. hypot-define 51.4%

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

      8. *-commutative 51.4%

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

   3. Simplified 51.4%

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

   5. Taylor expanded in y.im around 0 38.3%

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

      1. unpow2 38.3%

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

      2. unpow2 38.3%

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

      3. hypot-undefine 27.7%

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

   7. Simplified 27.7%

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

   8. Taylor expanded in y.im around 0 27.6%

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

      1. expm1-log1p-u 25.0%

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

      2. expm1-undefine 41.4%

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

      3. *-commutative 41.4%

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

      4. *-commutative 41.4%

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

      5. hypot-undefine 42.1%

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

      6. +-commutative 42.1%

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

      7. sqrt-pow2 42.1%

         \[\leadsto e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}}\right)} - 1 \]

   10. Applied egg-rr 42.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 42.1%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} + \left(-1\right)} \]

       2. log1p-undefine 42.1%

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

       3. rem-exp-log 49.0%

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

       4. *-commutative 49.0%

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

       5. metadata-eval 49.0%

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

   12. Simplified 49.0%

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

   if -95 < y.im < -1.60000000000000004e-45

   1. Initial program 38.2%

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

   3. Taylor expanded in y.re around 0 61.3%

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

      1. +-commutative 61.3%

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

      2. unpow2 61.3%

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

      3. unpow2 61.3%

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

      4. unpow1/2 61.3%

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

      5. exp-to-pow 61.3%

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

      6. *-commutative 61.3%

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

      7. rem-log-exp 61.3%

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

      8. associate-*r* 61.3%

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

   5. Simplified 61.3%

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

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

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

      1. *-commutative 76.9%

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

      2. +-commutative 76.9%

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

      3. unpow2 76.9%

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

      4. unpow2 76.9%

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

      5. associate-*l* 76.9%

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

      6. *-commutative 76.9%

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

   8. Simplified 76.9%

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

   9. Taylor expanded in x.re around -inf 61.5%

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

       1. mul-1-neg 61.5%

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

       2. neg-sub0 61.5%

          \[\leadsto e^{\log \color{blue}{\left(0 - x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right) \]

   11. Simplified 61.5%

       \[\leadsto e^{\log \color{blue}{\left(0 - x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.im \cdot \left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right) \]

   if -1.60000000000000004e-45 < y.im < 1.59999999999999993e38

   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. exp-diff 41.9%

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

      2. exp-to-pow 41.9%

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

      3. hypot-define 41.9%

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

      4. *-commutative 41.9%

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

      5. exp-prod 41.7%

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

      6. fma-define 41.7%

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

      7. hypot-define 89.3%

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

      8. *-commutative 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y.im around 0 63.0%

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

      1. unpow2 63.0%

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

      2. unpow2 63.0%

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

      3. hypot-undefine 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in y.im around 0 56.7%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -95:\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-45}:\\ \;\;\;\;\left(y.im \cdot \left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right) \cdot e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := \left(t\_1 \cdot {t\_0}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ t_3 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -850000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{-45}:\\ \;\;\;\;t\_3 \cdot \sin \left(\log t\_0 \cdot \left(y.im \cdot 0.5\right)\right)\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;t\_1 \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (sin (* y.re (atan2 x.im x.re))))
        (t_2 (+ (+ (* t_1 (pow t_0 (/ y.re 2.0))) 1.0) -1.0))
        (t_3 (pow (hypot x.im x.re) y.re)))
   (if (<= y.im -850000000000.0)
     t_2
     (if (<= y.im -1.32e-45)
       (* t_3 (sin (* (log t_0) (* y.im 0.5))))
       (if (<= y.im 1.6e+38) (* t_1 t_3) t_2)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = ((t_1 * pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0;
	double t_3 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_im <= -850000000000.0) {
		tmp = t_2;
	} else if (y_46_im <= -1.32e-45) {
		tmp = t_3 * sin((log(t_0) * (y_46_im * 0.5)));
	} else if (y_46_im <= 1.6e+38) {
		tmp = t_1 * t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_2 = ((t_1 * Math.pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0;
	double t_3 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_im <= -850000000000.0) {
		tmp = t_2;
	} else if (y_46_im <= -1.32e-45) {
		tmp = t_3 * Math.sin((Math.log(t_0) * (y_46_im * 0.5)));
	} else if (y_46_im <= 1.6e+38) {
		tmp = t_1 * t_3;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_2 = ((t_1 * math.pow(t_0, (y_46_re / 2.0))) + 1.0) + -1.0
	t_3 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_im <= -850000000000.0:
		tmp = t_2
	elif y_46_im <= -1.32e-45:
		tmp = t_3 * math.sin((math.log(t_0) * (y_46_im * 0.5)))
	elif y_46_im <= 1.6e+38:
		tmp = t_1 * t_3
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_2 = Float64(Float64(Float64(t_1 * (t_0 ^ Float64(y_46_re / 2.0))) + 1.0) + -1.0)
	t_3 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_im <= -850000000000.0)
		tmp = t_2;
	elseif (y_46_im <= -1.32e-45)
		tmp = Float64(t_3 * sin(Float64(log(t_0) * Float64(y_46_im * 0.5))));
	elseif (y_46_im <= 1.6e+38)
		tmp = Float64(t_1 * t_3);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_2 = ((t_1 * (t_0 ^ (y_46_re / 2.0))) + 1.0) + -1.0;
	t_3 = hypot(x_46_im, x_46_re) ^ y_46_re;
	tmp = 0.0;
	if (y_46_im <= -850000000000.0)
		tmp = t_2;
	elseif (y_46_im <= -1.32e-45)
		tmp = t_3 * sin((log(t_0) * (y_46_im * 0.5)));
	elseif (y_46_im <= 1.6e+38)
		tmp = t_1 * t_3;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(t$95$1 * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$im, -850000000000.0], t$95$2, If[LessEqual[y$46$im, -1.32e-45], N[(t$95$3 * N[Sin[N[(N[Log[t$95$0], $MachinePrecision] * N[(y$46$im * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+38], N[(t$95$1 * t$95$3), $MachinePrecision], t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.5e11 or 1.59999999999999993e38 < y.im

   1. Initial program 33.2%

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

      1. exp-diff 27.1%

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

      2. exp-to-pow 27.1%

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

      3. hypot-define 27.1%

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

      4. *-commutative 27.1%

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

      5. exp-prod 24.5%

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

      6. fma-define 24.5%

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

      7. hypot-define 51.0%

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

      8. *-commutative 51.0%

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

   3. Simplified 51.0%

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

   5. Taylor expanded in y.im around 0 37.7%

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

      1. unpow2 37.7%

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

      2. unpow2 37.7%

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

      3. hypot-undefine 27.1%

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

   7. Simplified 27.1%

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

   8. Taylor expanded in y.im around 0 26.9%

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

      1. expm1-log1p-u 24.3%

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

      2. expm1-undefine 40.8%

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

      3. *-commutative 40.8%

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

      4. *-commutative 40.8%

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

      5. hypot-undefine 41.6%

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

      6. +-commutative 41.6%

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

      7. sqrt-pow2 41.6%

         \[\leadsto e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}}\right)} - 1 \]

   10. Applied egg-rr 41.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 41.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} + \left(-1\right)} \]

       2. log1p-undefine 41.6%

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

       3. rem-exp-log 48.5%

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

       4. *-commutative 48.5%

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

       5. metadata-eval 48.5%

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

   12. Simplified 48.5%

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

   if -8.5e11 < y.im < -1.32000000000000005e-45

   1. Initial program 42.6%

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

      1. exp-diff 42.6%

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

      2. exp-to-pow 42.6%

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

      3. hypot-define 42.6%

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

      4. *-commutative 42.6%

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

      5. exp-prod 42.6%

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

      6. fma-define 42.6%

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

      7. hypot-define 71.2%

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

      8. *-commutative 71.2%

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

   3. Simplified 71.2%

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

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

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

      1. unpow2 50.0%

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

      2. unpow2 50.0%

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

      3. hypot-undefine 70.7%

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

   7. Simplified 70.7%

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

   8. Taylor expanded in y.im around inf 63.6%

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

      1. +-commutative 63.6%

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

      2. unpow2 63.6%

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

      3. unpow2 63.6%

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

      4. unpow1/2 63.6%

         \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \color{blue}{\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{0.5}\right)}\right) \]

      5. exp-to-pow 63.6%

         \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \color{blue}{\left(e^{\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot 0.5}\right)}\right) \]

      6. *-commutative 63.6%

         \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(e^{\color{blue}{0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)}}\right)\right) \]

      7. rem-log-exp 63.6%

         \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \color{blue}{\left(0.5 \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}\right) \]

      8. associate-*r* 63.6%

         \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \color{blue}{\left(\left(y.im \cdot 0.5\right) \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)} \]

   10. Simplified 63.6%

       \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \color{blue}{\left(\left(y.im \cdot 0.5\right) \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)} \]

   if -1.32000000000000005e-45 < y.im < 1.59999999999999993e38

   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. exp-diff 41.9%

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

      2. exp-to-pow 41.9%

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

      3. hypot-define 41.9%

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

      4. *-commutative 41.9%

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

      5. exp-prod 41.7%

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

      6. fma-define 41.7%

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

      7. hypot-define 89.3%

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

      8. *-commutative 89.3%

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

   3. Simplified 89.3%

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

   5. Taylor expanded in y.im around 0 63.0%

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

      1. unpow2 63.0%

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

      2. unpow2 63.0%

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

      3. hypot-undefine 90.0%

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

   7. Simplified 90.0%

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

   8. Taylor expanded in y.im around 0 56.7%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -850000000000:\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{elif}\;y.im \leq -1.32 \cdot 10^{-45}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \sin \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(y.im \cdot 0.5\right)\right)\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+38}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16} \lor \neg \left(y.im \leq 3.75 \cdot 10^{-159}\right):\\ \;\;\;\;\left(\sin t\_0 \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.im -1.7e+16) (not (<= y.im 3.75e-159)))
     (+
      (+ (* (sin t_0) (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))) 1.0)
      -1.0)
     (* 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 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_im <= -1.7e+16) || !(y_46_im <= 3.75e-159)) {
		tmp = ((sin(t_0) * pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))) + 1.0) + -1.0;
	} else {
		tmp = t_0 * 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 t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_im <= -1.7e+16) || !(y_46_im <= 3.75e-159)) {
		tmp = ((Math.sin(t_0) * Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))) + 1.0) + -1.0;
	} else {
		tmp = t_0 * 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):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if (y_46_im <= -1.7e+16) or not (y_46_im <= 3.75e-159):
		tmp = ((math.sin(t_0) * math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))) + 1.0) + -1.0
	else:
		tmp = t_0 * 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)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_im <= -1.7e+16) || !(y_46_im <= 3.75e-159))
		tmp = Float64(Float64(Float64(sin(t_0) * (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0))) + 1.0) + -1.0);
	else
		tmp = Float64(t_0 * (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)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if ((y_46_im <= -1.7e+16) || ~((y_46_im <= 3.75e-159)))
		tmp = ((sin(t_0) * (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0))) + 1.0) + -1.0;
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -1.7e+16], N[Not[LessEqual[y$46$im, 3.75e-159]], $MachinePrecision]], N[(N[(N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision], N[(t$95$0 * 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 < -1.7e16 or 3.75e-159 < y.im

   1. Initial program 32.3%

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

      1. exp-diff 27.6%

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

      2. exp-to-pow 27.6%

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

      3. hypot-define 27.6%

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

      4. *-commutative 27.6%

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

      5. exp-prod 25.5%

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

      6. fma-define 25.5%

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

      7. hypot-define 58.8%

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

      8. *-commutative 58.8%

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

   3. Simplified 58.8%

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

   5. Taylor expanded in y.im around 0 41.8%

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

      1. unpow2 41.8%

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

      2. unpow2 41.8%

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

      3. hypot-undefine 41.0%

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

   7. Simplified 41.0%

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

   8. Taylor expanded in y.im around 0 31.8%

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

      1. expm1-log1p-u 27.7%

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

      2. expm1-undefine 39.7%

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

      3. *-commutative 39.7%

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

      4. *-commutative 39.7%

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

      5. hypot-undefine 40.2%

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

      6. +-commutative 40.2%

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

      7. sqrt-pow2 40.2%

         \[\leadsto e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \color{blue}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}}\right)} - 1 \]

   10. Applied egg-rr 40.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} - 1} \]
   11. Step-by-step derivation

       1. sub-neg 40.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)} + \left(-1\right)} \]

       2. log1p-undefine 40.2%

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

       3. rem-exp-log 47.5%

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

       4. *-commutative 47.5%

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

       5. metadata-eval 47.5%

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

   12. Simplified 47.5%

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

   if -1.7e16 < y.im < 3.75e-159

   1. Initial program 46.2%

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

      1. exp-diff 46.2%

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

      2. exp-to-pow 46.2%

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

      3. hypot-define 46.2%

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

      4. *-commutative 46.2%

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

      5. exp-prod 46.2%

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

      6. fma-define 46.2%

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

      7. hypot-define 88.5%

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

      8. *-commutative 88.5%

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

   3. Simplified 88.5%

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

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

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

      1. unpow2 63.9%

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

      2. unpow2 63.9%

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

      3. hypot-undefine 88.5%

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

   7. Simplified 88.5%

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

   8. Taylor expanded in y.im around 0 54.4%

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

   9. Taylor expanded in y.re around 0 56.3%

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

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+16} \lor \neg \left(y.im \leq 3.75 \cdot 10^{-159}\right):\\ \;\;\;\;\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} + 1\right) + -1\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -5400000000000 \lor \neg \left(y.im \leq 5.6 \cdot 10^{-186}\right):\\ \;\;\;\;\sin t\_0 \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.im -5400000000000.0) (not (<= y.im 5.6e-186)))
     (* (sin t_0) (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0)))
     (* 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 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_im <= -5400000000000.0) || !(y_46_im <= 5.6e-186)) {
		tmp = sin(t_0) * pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else {
		tmp = t_0 * 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 t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_im <= -5400000000000.0) || !(y_46_im <= 5.6e-186)) {
		tmp = Math.sin(t_0) * Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	} else {
		tmp = t_0 * 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):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if (y_46_im <= -5400000000000.0) or not (y_46_im <= 5.6e-186):
		tmp = math.sin(t_0) * math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	else:
		tmp = t_0 * 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)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_im <= -5400000000000.0) || !(y_46_im <= 5.6e-186))
		tmp = Float64(sin(t_0) * (Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0)));
	else
		tmp = Float64(t_0 * (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)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if ((y_46_im <= -5400000000000.0) || ~((y_46_im <= 5.6e-186)))
		tmp = sin(t_0) * (((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0));
	else
		tmp = t_0 * (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_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -5400000000000.0], N[Not[LessEqual[y$46$im, 5.6e-186]], $MachinePrecision]], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 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 < -5.4e12 or 5.59999999999999966e-186 < y.im

   1. Initial program 33.0%

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

      1. exp-diff 28.4%

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

      2. exp-to-pow 28.4%

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

      3. hypot-define 28.4%

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

      4. *-commutative 28.4%

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

      5. exp-prod 26.3%

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

      6. fma-define 26.3%

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

      7. hypot-define 58.9%

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

      8. *-commutative 58.9%

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

   3. Simplified 58.9%

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

   5. Taylor expanded in y.im around 0 42.3%

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

      1. unpow2 42.3%

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

      2. unpow2 42.3%

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

      3. hypot-undefine 41.6%

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

   7. Simplified 41.6%

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

   8. Taylor expanded in y.im around 0 31.8%

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

      1. hypot-undefine 40.5%

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

      2. +-commutative 40.5%

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

      3. sqrt-pow2 40.5%

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

      4. *-commutative 40.5%

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

   10. Applied egg-rr 40.5%

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

   if -5.4e12 < y.im < 5.59999999999999966e-186

   1. Initial program 45.6%

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

      1. exp-diff 45.6%

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

      2. exp-to-pow 45.6%

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

      3. hypot-define 45.6%

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

      4. *-commutative 45.6%

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

      5. exp-prod 45.6%

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

      6. fma-define 45.6%

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

      7. hypot-define 89.2%

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

      8. *-commutative 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in y.im around 0 63.8%

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

      1. unpow2 63.8%

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

      2. unpow2 63.8%

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

      3. hypot-undefine 89.1%

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

   7. Simplified 89.1%

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

   8. Taylor expanded in y.im around 0 54.9%

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

   9. Taylor expanded in y.re around 0 56.9%

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

4. Final simplification 47.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5400000000000 \lor \neg \left(y.im \leq 5.6 \cdot 10^{-186}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 37.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -37 \lor \neg \left(y.re \leq 0.0004\right):\\ \;\;\;\;\sin t\_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.re -37.0) (not (<= y.re 0.0004)))
     (* (sin t_0) (pow x.im y.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 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -37.0) || !(y_46_re <= 0.0004)) {
		tmp = sin(t_0) * pow(x_46_im, y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    if ((y_46re <= (-37.0d0)) .or. (.not. (y_46re <= 0.0004d0))) then
        tmp = sin(t_0) * (x_46im ** y_46re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -37.0) || !(y_46_re <= 0.0004)) {
		tmp = Math.sin(t_0) * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if (y_46_re <= -37.0) or not (y_46_re <= 0.0004):
		tmp = math.sin(t_0) * math.pow(x_46_im, y_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(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_re <= -37.0) || !(y_46_re <= 0.0004))
		tmp = Float64(sin(t_0) * (x_46_im ^ y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if ((y_46_re <= -37.0) || ~((y_46_re <= 0.0004)))
		tmp = sin(t_0) * (x_46_im ^ y_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -37.0], N[Not[LessEqual[y$46$re, 0.0004]], $MachinePrecision]], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if y.re < -37 or 4.00000000000000019e-4 < y.re

   1. Initial program 36.2%

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

      1. exp-diff 30.2%

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

      2. exp-to-pow 30.2%

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

      3. hypot-define 30.2%

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

      4. *-commutative 30.2%

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

      5. exp-prod 27.6%

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

      6. fma-define 27.6%

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

      7. hypot-define 58.6%

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

      8. *-commutative 58.6%

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

   3. Simplified 58.6%

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

   5. Taylor expanded in y.im around 0 66.6%

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

      1. unpow2 66.6%

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

      2. unpow2 66.6%

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

      3. hypot-undefine 66.6%

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

   7. Simplified 66.6%

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

   8. Taylor expanded in y.im around 0 64.9%

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

   9. Taylor expanded in x.re around 0 48.0%

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

   if -37 < y.re < 4.00000000000000019e-4

   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. exp-diff 39.5%

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

      2. exp-to-pow 39.5%

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

      3. hypot-define 39.5%

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

      4. *-commutative 39.5%

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

      5. exp-prod 39.4%

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

      6. fma-define 39.4%

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

      7. hypot-define 81.4%

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

      8. *-commutative 81.4%

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

   3. Simplified 81.4%

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

   5. Taylor expanded in y.im around 0 38.0%

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

      1. unpow2 38.0%

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

      2. unpow2 38.0%

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

      3. hypot-undefine 55.8%

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

   7. Simplified 55.8%

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

   8. Taylor expanded in y.im around 0 21.4%

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

   9. Taylor expanded in y.re around 0 19.8%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -37 \lor \neg \left(y.re \leq 0.0004\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 36.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq 1.55 \cdot 10^{-27}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

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 1.55e-27) (* t_0 (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 <= 1.55e-27) {
		tmp = t_0 * pow(x_46_re, y_46_re);
	} else {
		tmp = t_0 * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46im <= 1.55d-27) then
        tmp = t_0 * (x_46re ** y_46re)
    else
        tmp = t_0 * (x_46im ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= 1.55e-27) {
		tmp = t_0 * 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 <= 1.55e-27:
		tmp = t_0 * 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 <= 1.55e-27)
		tmp = Float64(t_0 * (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 <= 1.55e-27)
		tmp = t_0 * (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, 1.55e-27], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.5499999999999999e-27

   1. Initial program 40.2%

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

      1. exp-diff 37.6%

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

      2. exp-to-pow 37.6%

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

      3. hypot-define 37.6%

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

      4. *-commutative 37.6%

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

      5. exp-prod 36.0%

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

      6. fma-define 36.0%

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

      7. hypot-define 72.8%

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

      8. *-commutative 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in y.im around 0 54.3%

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

      1. unpow2 54.3%

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

      2. unpow2 54.3%

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

      3. hypot-undefine 62.1%

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

   7. Simplified 62.1%

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

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

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

   9. Taylor expanded in x.im around 0 32.6%

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

   if 1.5499999999999999e-27 < x.im

   1. Initial program 31.1%

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

      1. exp-diff 27.8%

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

      2. exp-to-pow 27.8%

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

      3. hypot-define 27.8%

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

      4. *-commutative 27.8%

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

      5. exp-prod 27.8%

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

      6. fma-define 27.8%

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

      7. hypot-define 65.5%

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

      8. *-commutative 65.5%

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

   3. Simplified 65.5%

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

   5. Taylor expanded in y.im around 0 40.4%

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

      1. unpow2 40.4%

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

      2. unpow2 40.4%

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

      3. hypot-undefine 56.2%

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

   7. Simplified 56.2%

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

   8. Taylor expanded in y.im around 0 42.6%

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

   9. Taylor expanded in x.re around 0 42.5%

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

Alternative 14: 44.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (* y.re (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * atan2(x_46_im, x_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (y_46_re * math.atan2(x_46_im, x_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (y_46_re * atan2(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.0%

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

   1. exp-diff 35.3%

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

   2. exp-to-pow 35.3%

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

   3. hypot-define 35.3%

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

   4. *-commutative 35.3%

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

   5. exp-prod 34.0%

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

   6. fma-define 34.0%

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

   7. hypot-define 71.1%

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

   8. *-commutative 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in y.im around 0 50.9%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   3. hypot-undefine 60.7%

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

7. Simplified 60.7%

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

8. Taylor expanded in y.im around 0 41.1%

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

9. Taylor expanded in y.re around 0 41.1%

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

10. Final simplification 41.1%

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

Alternative 15: 13.9% 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 38.0%

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

   1. exp-diff 35.3%

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

   2. exp-to-pow 35.3%

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

   3. hypot-define 35.3%

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

   4. *-commutative 35.3%

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

   5. exp-prod 34.0%

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

   6. fma-define 34.0%

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

   7. hypot-define 71.1%

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

   8. *-commutative 71.1%

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

3. Simplified 71.1%

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

5. Taylor expanded in y.im around 0 50.9%

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

   1. unpow2 50.9%

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

   2. unpow2 50.9%

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

   3. hypot-undefine 60.7%

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

7. Simplified 60.7%

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

8. Taylor expanded in y.im around 0 41.1%

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

9. Taylor expanded in y.re around 0 13.2%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024107 
(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)))))