powComplex, imaginary part

Percentage Accurate: 41.0% → 72.8%

Time: 46.0s

Alternatives: 16

Speedup: 2.6×

• 35.5% 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: 72.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{t_1 + 1}\\ t_3 := \frac{\sin t_0}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ t_4 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_1}\\ t_5 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_6 := t_4 \cdot \sin \left({\left(\sqrt[3]{t_5}\right)}^{3}\right)\\ \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+192}:\\ \;\;\;\;t_6\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{+71}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;t_4 \cdot \sin t_5\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-173}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 0.00016:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+189}:\\ \;\;\;\;t_6\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (hypot x.im x.re))))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2
         (*
          (sin (fma y.re (atan2 x.im x.re) (* (log (hypot x.re x.im)) y.im)))
          (/ (pow (hypot x.re x.im) y.re) (+ t_1 1.0))))
        (t_3 (/ (sin t_0) (pow (exp y.im) (atan2 x.im x.re))))
        (t_4
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_1)))
        (t_5 (* y.re (atan2 x.im x.re)))
        (t_6 (* t_4 (sin (pow (cbrt t_5) 3.0)))))
   (if (<= y.im -8.8e+192)
     t_6
     (if (<= y.im -1.05e+71)
       t_3
       (if (<= y.im -1.15e+17)
         (* t_4 (sin t_5))
         (if (<= y.im 4.2e-208)
           t_2
           (if (<= y.im 3.6e-173)
             (* t_0 (pow (hypot x.im x.re) y.re))
             (if (<= y.im 0.00016) t_2 (if (<= y.im 1.7e+189) t_6 t_3)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = sin(fma(y_46_re, atan2(x_46_im, x_46_re), (log(hypot(x_46_re, x_46_im)) * y_46_im))) * (pow(hypot(x_46_re, x_46_im), y_46_re) / (t_1 + 1.0));
	double t_3 = sin(t_0) / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	double t_4 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	double t_5 = y_46_re * atan2(x_46_im, x_46_re);
	double t_6 = t_4 * sin(pow(cbrt(t_5), 3.0));
	double tmp;
	if (y_46_im <= -8.8e+192) {
		tmp = t_6;
	} else if (y_46_im <= -1.05e+71) {
		tmp = t_3;
	} else if (y_46_im <= -1.15e+17) {
		tmp = t_4 * sin(t_5);
	} else if (y_46_im <= 4.2e-208) {
		tmp = t_2;
	} else if (y_46_im <= 3.6e-173) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 0.00016) {
		tmp = t_2;
	} else if (y_46_im <= 1.7e+189) {
		tmp = t_6;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))) * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(t_1 + 1.0)))
	t_3 = Float64(sin(t_0) / (exp(y_46_im) ^ atan(x_46_im, x_46_re)))
	t_4 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_1))
	t_5 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_6 = Float64(t_4 * sin((cbrt(t_5) ^ 3.0)))
	tmp = 0.0
	if (y_46_im <= -8.8e+192)
		tmp = t_6;
	elseif (y_46_im <= -1.05e+71)
		tmp = t_3;
	elseif (y_46_im <= -1.15e+17)
		tmp = Float64(t_4 * sin(t_5));
	elseif (y_46_im <= 4.2e-208)
		tmp = t_2;
	elseif (y_46_im <= 3.6e-173)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 0.00016)
		tmp = t_2;
	elseif (y_46_im <= 1.7e+189)
		tmp = t_6;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $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[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[t$95$0], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[Sin[N[Power[N[Power[t$95$5, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -8.8e+192], t$95$6, If[LessEqual[y$46$im, -1.05e+71], t$95$3, If[LessEqual[y$46$im, -1.15e+17], N[(t$95$4 * N[Sin[t$95$5], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e-208], t$95$2, If[LessEqual[y$46$im, 3.6e-173], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 0.00016], t$95$2, If[LessEqual[y$46$im, 1.7e+189], t$95$6, t$95$3]]]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.8000000000000003e192 or 1.60000000000000013e-4 < y.im < 1.69999999999999992e189

   1. Initial program 34.5%

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

   2. Taylor expanded in y.im around 0 65.9%

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

      1. *-commutative 65.9%

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

      2. add-sqr-sqrt 31.4%

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

      3. unpow2 31.4%

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

      4. add-cube-cbrt 38.3%

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

      5. pow3 38.3%

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

      6. unpow2 38.3%

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

      7. add-sqr-sqrt 74.5%

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

      8. *-commutative 74.5%

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

   4. Applied egg-rr 74.5%

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

   if -8.8000000000000003e192 < y.im < -1.04999999999999995e71 or 1.69999999999999992e189 < y.im

   1. Initial program 46.9%

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

      1. exp-diff 40.4%

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

      2. exp-to-pow 40.4%

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

      3. hypot-def 40.4%

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

      4. *-commutative 40.4%

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

      5. exp-prod 39.6%

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

      6. +-commutative 39.6%

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

      7. *-commutative 39.6%

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

      8. fma-def 39.6%

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

      9. +-commutative 39.6%

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

   3. Simplified 59.6%

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

      1. hypot-udef 39.6%

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

      2. add-cube-cbrt 39.3%

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

      3. pow3 39.2%

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

      4. *-commutative 39.2%

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

      5. hypot-udef 57.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in y.re around 0 39.6%

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

      1. pow-base-1 39.6%

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

      2. associate-*r* 39.6%

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

      3. unpow2 39.6%

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

      4. unpow2 39.6%

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

      5. hypot-def 59.6%

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

      6. *-lft-identity 59.6%

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

   8. Simplified 59.6%

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

   9. Taylor expanded in y.re around 0 44.8%

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

       1. unpow2 44.8%

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

       2. unpow2 44.8%

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

       3. hypot-def 73.5%

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

       4. *-commutative 73.5%

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

       5. exp-prod 72.8%

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

   11. Simplified 72.8%

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

   if -1.04999999999999995e71 < y.im < -1.15e17

   1. Initial program 29.9%

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

   2. Taylor expanded in y.im around 0 71.7%

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

   if -1.15e17 < y.im < 4.20000000000000024e-208 or 3.59999999999999972e-173 < y.im < 1.60000000000000013e-4

   1. Initial program 47.4%

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

      1. exp-diff 47.4%

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

      2. exp-to-pow 47.4%

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

      3. hypot-def 47.4%

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

      4. *-commutative 47.4%

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

      5. exp-prod 47.4%

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

      6. +-commutative 47.4%

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

      7. *-commutative 47.4%

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

      8. fma-def 47.4%

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

      9. +-commutative 47.4%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in y.im around 0 89.5%

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

   if 4.20000000000000024e-208 < y.im < 3.59999999999999972e-173

   1. Initial program 27.3%

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

      1. exp-diff 27.3%

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

      2. exp-to-pow 27.3%

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

      3. hypot-def 27.3%

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

      4. *-commutative 27.3%

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

      5. exp-prod 27.3%

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

      6. +-commutative 27.3%

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

      7. *-commutative 27.3%

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

      8. fma-def 27.3%

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

      9. +-commutative 27.3%

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

   3. Simplified 54.5%

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

      1. hypot-udef 27.3%

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

      2. add-cube-cbrt 27.3%

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

      3. pow3 27.3%

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

      4. *-commutative 27.3%

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

      5. hypot-udef 54.5%

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

   5. Applied egg-rr 54.5%

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

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

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

      1. pow-base-1 37.1%

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

      2. associate-*r* 37.1%

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

      3. unpow2 37.1%

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

      4. unpow2 37.1%

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

      5. hypot-def 91.6%

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

      6. *-lft-identity 91.6%

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

   8. Simplified 91.6%

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

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

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

       1. unpow2 82.5%

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

       2. unpow2 82.5%

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

       3. hypot-def 82.5%

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

       4. unpow2 82.5%

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

       5. unpow2 82.5%

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

       6. hypot-def 91.6%

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

   11. Simplified 91.6%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+192}:\\ \;\;\;\;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 \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{+71}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;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 \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-173}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 0.00016:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+189}:\\ \;\;\;\;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 \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 2: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_1 := e^{y.re \cdot t_0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := t_1 \cdot \sin \left(t_2 + y.im \cdot t_0\right)\\ \mathbf{if}\;t_3 \leq -20:\\ \;\;\;\;t_1 \cdot \sin \left({\left(\sqrt[3]{t_2}\right)}^{3}\right)\\ \mathbf{elif}\;t_3 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \end{array} \end{array} \]

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)))))
        (t_1 (exp (- (* y.re t_0) (* (atan2 x.im x.re) y.im))))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (* t_1 (sin (+ t_2 (* y.im t_0))))))
   (if (<= t_3 -20.0)
     (* t_1 (sin (pow (cbrt t_2) 3.0)))
     (if (<= t_3 INFINITY)
       t_3
       (*
        (pow (hypot x.re x.im) y.re)
        (sin
         (fma y.re (atan2 x.im x.re) (* (log (hypot x.re x.im)) y.im))))))))

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))));
	double t_1 = exp(((y_46_re * t_0) - (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 = t_1 * sin((t_2 + (y_46_im * t_0)));
	double tmp;
	if (t_3 <= -20.0) {
		tmp = t_1 * sin(pow(cbrt(t_2), 3.0));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = pow(hypot(x_46_re, x_46_im), y_46_re) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (log(hypot(x_46_re, x_46_im)) * y_46_im)));
	}
	return tmp;
}

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))))
	t_1 = exp(Float64(Float64(y_46_re * t_0) - 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 = Float64(t_1 * sin(Float64(t_2 + Float64(y_46_im * t_0))))
	tmp = 0.0
	if (t_3 <= -20.0)
		tmp = Float64(t_1 * sin((cbrt(t_2) ^ 3.0)));
	elseif (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64((hypot(x_46_re, x_46_im) ^ y_46_re) * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Sin[N[(t$95$2 + N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -20.0], N[(t$95$1 * N[Sin[N[Power[N[Power[t$95$2, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$3, N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -20

   1. Initial program 36.3%

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

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

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

      1. *-commutative 54.4%

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

      2. add-sqr-sqrt 27.3%

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

      3. unpow2 27.3%

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

      4. add-cube-cbrt 27.3%

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

      5. pow3 27.3%

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

      6. unpow2 27.3%

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

      7. add-sqr-sqrt 72.6%

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

      8. *-commutative 72.6%

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

   4. Applied egg-rr 72.6%

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

   if -20 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 82.8%

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

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 0.0%

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

      1. exp-diff 0.0%

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

      2. exp-to-pow 0.0%

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

      3. hypot-def 0.0%

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

      4. *-commutative 0.0%

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

      5. exp-prod 0.0%

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

      6. +-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. fma-def 0.0%

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

      9. +-commutative 0.0%

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

   3. Simplified 69.7%

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

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

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right) \leq -20:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \mathbf{elif}\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right) \leq \infty:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \end{array} \]

Alternative 3: 79.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq 3.6 \cdot 10^{+118}:\\ \;\;\;\;e^{y.re \cdot t_0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(t_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= y.re 3.6e+118)
     (*
      (exp (- (* y.re t_0) (* (atan2 x.im x.re) y.im)))
      (sin (fma t_0 y.im (* y.re (atan2 x.im x.re)))))
     (* (pow (hypot x.re x.im) y.re) (sin (* y.im (log (hypot x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= 3.6e+118) {
		tmp = exp(((y_46_re * t_0) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	} else {
		tmp = pow(hypot(x_46_re, x_46_im), y_46_re) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= 3.6e+118)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_0) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	else
		tmp = Float64((hypot(x_46_re, x_46_im) ^ y_46_re) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, 3.6e+118], N[(N[Exp[N[(N[(y$46$re * t$95$0), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < 3.6e118

   1. Initial program 44.4%

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

      1. hypot-def 44.4%

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

      2. fma-def 44.4%

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

      3. hypot-def 81.9%

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

      4. *-commutative 81.9%

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

   3. Simplified 81.9%

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

   if 3.6e118 < y.re

   1. Initial program 30.3%

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

      1. exp-diff 21.2%

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

      2. exp-to-pow 21.2%

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

      3. hypot-def 21.2%

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

      4. *-commutative 21.2%

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

      5. exp-prod 21.2%

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

      6. +-commutative 21.2%

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

      7. *-commutative 21.2%

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

      8. fma-def 21.2%

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

      9. +-commutative 21.2%

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

   3. Simplified 42.4%

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

      1. hypot-udef 21.2%

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

      2. add-cube-cbrt 21.2%

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

      3. pow3 21.2%

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

      4. *-commutative 21.2%

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

      5. hypot-udef 45.5%

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

   5. Applied egg-rr 45.5%

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

   6. Taylor expanded in y.re around 0 30.3%

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

      1. pow-base-1 30.3%

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

      2. associate-*r* 30.3%

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

      3. unpow2 30.3%

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

      4. unpow2 30.3%

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

      5. hypot-def 63.6%

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

      6. *-lft-identity 63.6%

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

   8. Simplified 63.6%

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

   9. Taylor expanded in y.im around 0 78.8%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 3.6 \cdot 10^{+118}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 4: 73.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ t_4 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_1}\\ t_5 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_6 := \sin t_5\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+17}:\\ \;\;\;\;t_4 \cdot \sin t_2\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;t_3 \cdot \frac{t_0}{t_1 + 1}\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{-173}:\\ \;\;\;\;t_5 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;t_0 \cdot t_3\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{+224}:\\ \;\;\;\;t_6 \cdot t_4\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+273}:\\ \;\;\;\;t_2 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_6}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3
         (sin (fma y.re (atan2 x.im x.re) (* (log (hypot x.re x.im)) y.im))))
        (t_4
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_1)))
        (t_5 (* y.im (log (hypot x.im x.re))))
        (t_6 (sin t_5)))
   (if (<= y.im -1.2e+17)
     (* t_4 (sin t_2))
     (if (<= y.im 4.2e-208)
       (* t_3 (/ t_0 (+ t_1 1.0)))
       (if (<= y.im 2.45e-173)
         (* t_5 (pow (hypot x.im x.re) y.re))
         (if (<= y.im 3.5e-15)
           (* t_0 t_3)
           (if (<= y.im 1.5e+224)
             (* t_6 t_4)
             (if (<= y.im 3.6e+273)
               (* t_2 (exp (* (atan2 x.im x.re) (- y.im))))
               (/ t_6 (pow (exp y.im) (atan2 x.im x.re)))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_re, x_46_im), y_46_re);
	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(fma(y_46_re, atan2(x_46_im, x_46_re), (log(hypot(x_46_re, x_46_im)) * y_46_im)));
	double t_4 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	double t_5 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_6 = sin(t_5);
	double tmp;
	if (y_46_im <= -1.2e+17) {
		tmp = t_4 * sin(t_2);
	} else if (y_46_im <= 4.2e-208) {
		tmp = t_3 * (t_0 / (t_1 + 1.0));
	} else if (y_46_im <= 2.45e-173) {
		tmp = t_5 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 3.5e-15) {
		tmp = t_0 * t_3;
	} else if (y_46_im <= 1.5e+224) {
		tmp = t_6 * t_4;
	} else if (y_46_im <= 3.6e+273) {
		tmp = t_2 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_6 / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	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(fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(hypot(x_46_re, x_46_im)) * y_46_im)))
	t_4 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_1))
	t_5 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_6 = sin(t_5)
	tmp = 0.0
	if (y_46_im <= -1.2e+17)
		tmp = Float64(t_4 * sin(t_2));
	elseif (y_46_im <= 4.2e-208)
		tmp = Float64(t_3 * Float64(t_0 / Float64(t_1 + 1.0)));
	elseif (y_46_im <= 2.45e-173)
		tmp = Float64(t_5 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 3.5e-15)
		tmp = Float64(t_0 * t_3);
	elseif (y_46_im <= 1.5e+224)
		tmp = Float64(t_6 * t_4);
	elseif (y_46_im <= 3.6e+273)
		tmp = Float64(t_2 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64(t_6 / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $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[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sin[t$95$5], $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e+17], N[(t$95$4 * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e-208], N[(t$95$3 * N[(t$95$0 / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.45e-173], N[(t$95$5 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-15], N[(t$95$0 * t$95$3), $MachinePrecision], If[LessEqual[y$46$im, 1.5e+224], N[(t$95$6 * t$95$4), $MachinePrecision], If[LessEqual[y$46$im, 3.6e+273], N[(t$95$2 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$6 / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if y.im < -1.2e17

   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. Taylor expanded in y.im around 0 63.7%

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

   if -1.2e17 < y.im < 4.20000000000000024e-208

   1. Initial program 47.6%

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

      1. exp-diff 47.6%

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

      2. exp-to-pow 47.6%

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

      3. hypot-def 47.6%

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

      4. *-commutative 47.6%

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

      5. exp-prod 47.6%

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

      6. +-commutative 47.6%

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

      7. *-commutative 47.6%

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

      8. fma-def 47.7%

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

      9. +-commutative 47.7%

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

   3. Simplified 88.6%

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

   4. Taylor expanded in y.im around 0 88.6%

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

   if 4.20000000000000024e-208 < y.im < 2.44999999999999996e-173

   1. Initial program 27.3%

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

      1. exp-diff 27.3%

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

      2. exp-to-pow 27.3%

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

      3. hypot-def 27.3%

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

      4. *-commutative 27.3%

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

      5. exp-prod 27.3%

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

      6. +-commutative 27.3%

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

      7. *-commutative 27.3%

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

      8. fma-def 27.3%

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

      9. +-commutative 27.3%

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

   3. Simplified 54.5%

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

      1. hypot-udef 27.3%

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

      2. add-cube-cbrt 27.3%

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

      3. pow3 27.3%

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

      4. *-commutative 27.3%

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

      5. hypot-udef 54.5%

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

   5. Applied egg-rr 54.5%

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

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

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

      1. pow-base-1 37.1%

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

      2. associate-*r* 37.1%

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

      3. unpow2 37.1%

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

      4. unpow2 37.1%

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

      5. hypot-def 91.6%

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

      6. *-lft-identity 91.6%

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

   8. Simplified 91.6%

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

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

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

       1. unpow2 82.5%

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

       2. unpow2 82.5%

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

       3. hypot-def 82.5%

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

       4. unpow2 82.5%

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

       5. unpow2 82.5%

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

       6. hypot-def 91.6%

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

   11. Simplified 91.6%

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

   if 2.44999999999999996e-173 < y.im < 3.5000000000000001e-15

   1. Initial program 46.9%

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

      1. exp-diff 46.9%

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

      2. exp-to-pow 46.9%

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

      3. hypot-def 46.9%

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

      4. *-commutative 46.9%

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

      5. exp-prod 46.9%

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

      6. +-commutative 46.9%

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

      7. *-commutative 46.9%

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

      8. fma-def 46.9%

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

      9. +-commutative 46.9%

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

   3. Simplified 93.9%

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

   4. Taylor expanded in y.im around 0 93.9%

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

   if 3.5000000000000001e-15 < y.im < 1.5000000000000001e224

   1. Initial program 29.4%

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

      1. add-sqr-sqrt 16.4%

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

      2. pow2 16.4%

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

   3. Applied egg-rr 16.4%

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

   4. Taylor expanded in y.re around 0 29.4%

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

      1. +-commutative 29.4%

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

      2. unpow2 29.4%

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

      3. unpow2 29.4%

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

      4. hypot-def 65.8%

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

      5. hypot-def 29.4%

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

      6. unpow2 29.4%

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

      7. unpow2 29.4%

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

      8. +-commutative 29.4%

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

      9. unpow2 29.4%

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

      10. unpow2 29.4%

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

      11. hypot-def 65.8%

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

   6. Simplified 65.8%

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

   if 1.5000000000000001e224 < y.im < 3.60000000000000024e273

   1. Initial program 26.5%

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

   2. Taylor expanded in y.im around 0 59.0%

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

   3. Taylor expanded in y.re around 0 42.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 42.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 42.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 42.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 42.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 42.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 75.3%

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

      1. *-commutative 75.3%

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

   8. Simplified 75.3%

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

   if 3.60000000000000024e273 < y.im

   1. Initial program 83.3%

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

      1. exp-diff 50.0%

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

      2. exp-to-pow 50.0%

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

      3. hypot-def 50.0%

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

      4. *-commutative 50.0%

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

      5. exp-prod 50.0%

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

      6. +-commutative 50.0%

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

      7. *-commutative 50.0%

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

      8. fma-def 50.0%

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

      9. +-commutative 50.0%

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

   3. Simplified 66.7%

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

      1. hypot-udef 50.0%

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

      2. add-cube-cbrt 50.0%

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

      3. pow3 50.0%

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

      4. *-commutative 50.0%

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

      5. hypot-udef 66.7%

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

   5. Applied egg-rr 66.7%

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

   6. Taylor expanded in y.re around 0 50.0%

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

      1. pow-base-1 50.0%

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

      2. associate-*r* 50.0%

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

      3. unpow2 50.0%

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

      4. unpow2 50.0%

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

      5. hypot-def 66.7%

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

      6. *-lft-identity 66.7%

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

   8. Simplified 66.7%

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

   9. Taylor expanded in y.re around 0 83.3%

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

       1. unpow2 83.3%

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

       2. unpow2 83.3%

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

       3. hypot-def 100.0%

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

       4. *-commutative 100.0%

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

       5. exp-prod 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \]
3. Recombined 7 regimes into one program.

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+17}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{-173}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-15}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{+224}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+273}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 5: 68.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := \sin t_1\\ t_3 := {\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1080000000:\\ \;\;\;\;t_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{t_0}{t_3}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{t_2}{t_3}\\ \mathbf{elif}\;y.re \leq 520000000:\\ \;\;\;\;t_0 \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re))
        (t_1 (* y.im (log (hypot x.im x.re))))
        (t_2 (sin t_1))
        (t_3 (pow (exp y.im) (atan2 x.im x.re))))
   (if (<= y.re -1080000000.0)
     (* t_1 (pow (hypot x.im x.re) y.re))
     (if (<= y.re -3.1e-145)
       (* (sin (* y.re (atan2 x.im x.re))) (/ t_0 t_3))
       (if (<= y.re 1.8e-141)
         (/ t_2 t_3)
         (if (<= y.re 520000000.0)
           (*
            t_0
            (sin
             (fma y.re (atan2 x.im x.re) (* (log (hypot x.re x.im)) y.im))))
           (*
            t_2
            (exp
             (-
              (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
              (* (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 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_2 = sin(t_1);
	double t_3 = pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -1080000000.0) {
		tmp = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -3.1e-145) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (t_0 / t_3);
	} else if (y_46_re <= 1.8e-141) {
		tmp = t_2 / t_3;
	} else if (y_46_re <= 520000000.0) {
		tmp = t_0 * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (log(hypot(x_46_re, x_46_im)) * y_46_im)));
	} else {
		tmp = t_2 * 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)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	t_1 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_2 = sin(t_1)
	t_3 = exp(y_46_im) ^ atan(x_46_im, x_46_re)
	tmp = 0.0
	if (y_46_re <= -1080000000.0)
		tmp = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= -3.1e-145)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * Float64(t_0 / t_3));
	elseif (y_46_re <= 1.8e-141)
		tmp = Float64(t_2 / t_3);
	elseif (y_46_re <= 520000000.0)
		tmp = Float64(t_0 * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))));
	else
		tmp = Float64(t_2 * 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))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1080000000.0], N[(t$95$1 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3.1e-145], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.8e-141], N[(t$95$2 / t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 520000000.0], N[(t$95$0 * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * 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]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.08e9

   1. Initial program 43.4%

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

      1. exp-diff 38.2%

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

      2. exp-to-pow 38.2%

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

      3. hypot-def 38.2%

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

      4. *-commutative 38.2%

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

      5. exp-prod 36.8%

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

      6. +-commutative 36.8%

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

      7. *-commutative 36.8%

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

      8. fma-def 36.8%

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

      9. +-commutative 36.8%

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

   3. Simplified 67.1%

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

      1. hypot-udef 36.8%

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

      2. add-cube-cbrt 36.8%

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

      3. pow3 36.8%

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

      4. *-commutative 36.8%

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

      5. hypot-udef 65.8%

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

   5. Applied egg-rr 65.8%

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

   6. Taylor expanded in y.re around 0 38.2%

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

      1. pow-base-1 38.2%

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

      2. associate-*r* 38.2%

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

      3. unpow2 38.2%

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

      4. unpow2 38.2%

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

      5. hypot-def 64.5%

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

      6. *-lft-identity 64.5%

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

   8. Simplified 64.5%

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

   9. Taylor expanded in y.im around 0 50.1%

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

       1. unpow2 50.1%

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

       2. unpow2 50.1%

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

       3. hypot-def 50.1%

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

       4. unpow2 50.1%

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

       5. unpow2 50.1%

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

       6. hypot-def 83.0%

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

   11. Simplified 83.0%

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

   if -1.08e9 < y.re < -3.1e-145

   1. Initial program 38.2%

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

      1. exp-diff 38.2%

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

      2. exp-to-pow 38.3%

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

      3. hypot-def 38.3%

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

      4. *-commutative 38.3%

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

      5. exp-prod 37.8%

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

      6. +-commutative 37.8%

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

      7. *-commutative 37.8%

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

      8. fma-def 37.8%

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

      9. +-commutative 37.8%

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

   3. Simplified 79.8%

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

      1. hypot-udef 37.8%

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

      2. add-cube-cbrt 41.1%

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

      3. pow3 41.1%

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

      4. *-commutative 41.1%

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

      5. hypot-udef 83.1%

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

   5. Applied egg-rr 83.1%

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

      1. add-cube-cbrt 83.0%

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

      2. rem-cbrt-cube 79.7%

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

      3. rem-cbrt-cube 79.7%

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

      4. pow2 79.7%

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

      5. rem-cbrt-cube 83.0%

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

      6. unpow3 83.0%

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

      7. add-cube-cbrt 79.7%

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

      8. hypot-udef 37.8%

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

      9. +-commutative 37.8%

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

      10. hypot-def 79.7%

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

   7. Applied egg-rr 83.1%

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

   8. Taylor expanded in y.im around 0 76.6%

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

   if -3.1e-145 < y.re < 1.80000000000000007e-141

   1. Initial program 52.8%

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

      1. exp-diff 52.8%

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

      2. exp-to-pow 52.8%

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

      3. hypot-def 52.8%

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

      4. *-commutative 52.8%

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

      5. exp-prod 52.6%

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

      6. +-commutative 52.6%

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

      7. *-commutative 52.6%

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

      8. fma-def 52.6%

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

      9. +-commutative 52.6%

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

   3. Simplified 83.5%

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

      1. hypot-udef 52.6%

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

      2. add-cube-cbrt 52.2%

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

      3. pow3 52.2%

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

      4. *-commutative 52.2%

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

      5. hypot-udef 83.0%

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

   5. Applied egg-rr 83.0%

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

   6. Taylor expanded in y.re around 0 45.9%

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

      1. pow-base-1 45.9%

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

      2. associate-*r* 45.9%

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

      3. unpow2 45.9%

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

      4. unpow2 45.9%

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

      5. hypot-def 72.9%

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

      6. *-lft-identity 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in y.re around 0 46.1%

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

       1. unpow2 46.1%

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

       2. unpow2 46.1%

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

       3. hypot-def 73.1%

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

       4. *-commutative 73.1%

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

       5. exp-prod 72.9%

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

   11. Simplified 72.9%

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

   if 1.80000000000000007e-141 < y.re < 5.2e8

   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 36.2%

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

      2. exp-to-pow 36.2%

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

      3. hypot-def 36.2%

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

      4. *-commutative 36.2%

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

      5. exp-prod 35.6%

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

      6. +-commutative 35.6%

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

      7. *-commutative 35.6%

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

      8. fma-def 35.7%

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

      9. +-commutative 35.7%

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

   3. Simplified 82.1%

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

   4. Taylor expanded in y.im around 0 75.9%

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

   if 5.2e8 < y.re

   1. Initial program 35.1%

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

      1. add-sqr-sqrt 15.8%

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

      2. pow2 15.8%

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

   3. Applied egg-rr 15.8%

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

   4. Taylor expanded in y.re around 0 36.8%

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

      1. +-commutative 36.8%

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

      2. unpow2 36.8%

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

      3. unpow2 36.8%

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

      4. hypot-def 70.3%

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

      5. hypot-def 36.8%

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

      6. unpow2 36.8%

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

      7. unpow2 36.8%

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

      8. +-commutative 36.8%

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

      9. unpow2 36.8%

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

      10. unpow2 36.8%

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

      11. hypot-def 70.3%

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

   6. Simplified 70.3%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1080000000:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-145}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 520000000:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 6: 68.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := \sin t_1\\ t_3 := {\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ t_4 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{t_0}{t_3}\\ \mathbf{if}\;y.re \leq -130000000:\\ \;\;\;\;t_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-146}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{t_2}{t_3}\\ \mathbf{elif}\;y.re \leq 2800000000:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re))
        (t_1 (* y.im (log (hypot x.im x.re))))
        (t_2 (sin t_1))
        (t_3 (pow (exp y.im) (atan2 x.im x.re)))
        (t_4 (* (sin (* y.re (atan2 x.im x.re))) (/ t_0 t_3))))
   (if (<= y.re -130000000.0)
     (* t_1 (pow (hypot x.im x.re) y.re))
     (if (<= y.re -9e-146)
       t_4
       (if (<= y.re 1.4e-147)
         (/ t_2 t_3)
         (if (<= y.re 2800000000.0) t_4 (* t_0 t_2)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_2 = sin(t_1);
	double t_3 = pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	double t_4 = sin((y_46_re * atan2(x_46_im, x_46_re))) * (t_0 / t_3);
	double tmp;
	if (y_46_re <= -130000000.0) {
		tmp = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -9e-146) {
		tmp = t_4;
	} else if (y_46_re <= 1.4e-147) {
		tmp = t_2 / t_3;
	} else if (y_46_re <= 2800000000.0) {
		tmp = t_4;
	} else {
		tmp = t_0 * t_2;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double t_2 = Math.sin(t_1);
	double t_3 = Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re));
	double t_4 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * (t_0 / t_3);
	double tmp;
	if (y_46_re <= -130000000.0) {
		tmp = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -9e-146) {
		tmp = t_4;
	} else if (y_46_re <= 1.4e-147) {
		tmp = t_2 / t_3;
	} else if (y_46_re <= 2800000000.0) {
		tmp = t_4;
	} else {
		tmp = t_0 * t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	t_1 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	t_2 = math.sin(t_1)
	t_3 = math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re))
	t_4 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * (t_0 / t_3)
	tmp = 0
	if y_46_re <= -130000000.0:
		tmp = t_1 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= -9e-146:
		tmp = t_4
	elif y_46_re <= 1.4e-147:
		tmp = t_2 / t_3
	elif y_46_re <= 2800000000.0:
		tmp = t_4
	else:
		tmp = t_0 * t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	t_1 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_2 = sin(t_1)
	t_3 = exp(y_46_im) ^ atan(x_46_im, x_46_re)
	t_4 = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * Float64(t_0 / t_3))
	tmp = 0.0
	if (y_46_re <= -130000000.0)
		tmp = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= -9e-146)
		tmp = t_4;
	elseif (y_46_re <= 1.4e-147)
		tmp = Float64(t_2 / t_3);
	elseif (y_46_re <= 2800000000.0)
		tmp = t_4;
	else
		tmp = Float64(t_0 * 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 = hypot(x_46_re, x_46_im) ^ y_46_re;
	t_1 = y_46_im * log(hypot(x_46_im, x_46_re));
	t_2 = sin(t_1);
	t_3 = exp(y_46_im) ^ atan2(x_46_im, x_46_re);
	t_4 = sin((y_46_re * atan2(x_46_im, x_46_re))) * (t_0 / t_3);
	tmp = 0.0;
	if (y_46_re <= -130000000.0)
		tmp = t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= -9e-146)
		tmp = t_4;
	elseif (y_46_re <= 1.4e-147)
		tmp = t_2 / t_3;
	elseif (y_46_re <= 2800000000.0)
		tmp = t_4;
	else
		tmp = t_0 * 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[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / t$95$3), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -130000000.0], N[(t$95$1 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9e-146], t$95$4, If[LessEqual[y$46$re, 1.4e-147], N[(t$95$2 / t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 2800000000.0], t$95$4, N[(t$95$0 * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.3e8

   1. Initial program 43.4%

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

      1. exp-diff 38.2%

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

      2. exp-to-pow 38.2%

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

      3. hypot-def 38.2%

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

      4. *-commutative 38.2%

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

      5. exp-prod 36.8%

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

      6. +-commutative 36.8%

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

      7. *-commutative 36.8%

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

      8. fma-def 36.8%

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

      9. +-commutative 36.8%

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

   3. Simplified 67.1%

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

      1. hypot-udef 36.8%

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

      2. add-cube-cbrt 36.8%

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

      3. pow3 36.8%

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

      4. *-commutative 36.8%

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

      5. hypot-udef 65.8%

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

   5. Applied egg-rr 65.8%

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

   6. Taylor expanded in y.re around 0 38.2%

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

      1. pow-base-1 38.2%

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

      2. associate-*r* 38.2%

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

      3. unpow2 38.2%

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

      4. unpow2 38.2%

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

      5. hypot-def 64.5%

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

      6. *-lft-identity 64.5%

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

   8. Simplified 64.5%

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

   9. Taylor expanded in y.im around 0 50.1%

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

       1. unpow2 50.1%

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

       2. unpow2 50.1%

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

       3. hypot-def 50.1%

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

       4. unpow2 50.1%

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

       5. unpow2 50.1%

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

       6. hypot-def 83.0%

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

   11. Simplified 83.0%

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

   if -1.3e8 < y.re < -9.0000000000000001e-146 or 1.4e-147 < y.re < 2.8e9

   1. Initial program 39.1%

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

      1. exp-diff 39.1%

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

      2. exp-to-pow 39.1%

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

      3. hypot-def 39.1%

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

      4. *-commutative 39.1%

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

      5. exp-prod 38.6%

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

      6. +-commutative 38.6%

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

      7. *-commutative 38.6%

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

      8. fma-def 38.6%

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

      9. +-commutative 38.6%

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

   3. Simplified 80.8%

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

      1. hypot-udef 38.6%

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

      2. add-cube-cbrt 41.7%

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

      3. pow3 40.0%

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

      4. *-commutative 40.0%

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

      5. hypot-udef 82.1%

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

   5. Applied egg-rr 82.1%

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

      1. add-cube-cbrt 83.6%

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

      2. rem-cbrt-cube 82.0%

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

      3. rem-cbrt-cube 80.5%

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

      4. pow2 80.5%

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

      5. rem-cbrt-cube 83.6%

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

      6. unpow3 83.6%

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

      7. add-cube-cbrt 80.5%

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

      8. hypot-udef 38.5%

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

      9. +-commutative 38.5%

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

      10. hypot-def 80.5%

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

   7. Applied egg-rr 83.7%

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

   8. Taylor expanded in y.im around 0 72.4%

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

   if -9.0000000000000001e-146 < y.re < 1.4e-147

   1. Initial program 52.2%

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

      1. exp-diff 52.2%

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

      2. exp-to-pow 52.2%

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

      3. hypot-def 52.2%

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

      4. *-commutative 52.2%

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

      5. exp-prod 52.0%

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

      6. +-commutative 52.0%

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

      7. *-commutative 52.0%

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

      8. fma-def 52.0%

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

      9. +-commutative 52.0%

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

   3. Simplified 83.8%

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

      1. hypot-udef 52.0%

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

      2. add-cube-cbrt 49.8%

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

      3. pow3 51.5%

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

      4. *-commutative 51.5%

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

      5. hypot-udef 83.3%

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

   5. Applied egg-rr 83.3%

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

   6. Taylor expanded in y.re around 0 46.2%

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

      1. pow-base-1 46.2%

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

      2. associate-*r* 46.2%

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

      3. unpow2 46.2%

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

      4. unpow2 46.2%

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

      5. hypot-def 73.9%

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

      6. *-lft-identity 73.9%

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

   8. Simplified 73.9%

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

   9. Taylor expanded in y.re around 0 46.5%

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

       1. unpow2 46.5%

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

       2. unpow2 46.5%

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

       3. hypot-def 74.2%

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

       4. *-commutative 74.2%

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

       5. exp-prod 74.0%

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

   11. Simplified 74.0%

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

   if 2.8e9 < y.re

   1. Initial program 35.1%

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

      1. exp-diff 26.3%

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

      2. exp-to-pow 26.3%

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

      3. hypot-def 26.3%

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

      4. *-commutative 26.3%

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

      5. exp-prod 24.6%

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

      6. +-commutative 24.6%

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

      7. *-commutative 24.6%

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

      8. fma-def 24.6%

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

      9. +-commutative 24.6%

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

   3. Simplified 42.1%

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

      1. hypot-udef 24.6%

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

      2. add-cube-cbrt 28.1%

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

      3. pow3 28.1%

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

      4. *-commutative 28.1%

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

      5. hypot-udef 45.6%

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

   5. Applied egg-rr 45.6%

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

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

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

      1. pow-base-1 28.1%

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

      2. associate-*r* 28.1%

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

      3. unpow2 28.1%

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

      4. unpow2 28.1%

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

      5. hypot-def 56.1%

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

      6. *-lft-identity 56.1%

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

   8. Simplified 56.1%

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

   9. Taylor expanded in y.im around 0 66.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -130000000:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-146}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-147}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 2800000000:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 7: 72.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_3 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;t_2 \cdot \sin t_1\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{-173}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 0.00016:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+148}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin t_0}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (hypot x.im x.re))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* (atan2 x.im x.re) y.im))))
        (t_3
         (*
          (pow (hypot x.re x.im) y.re)
          (sin
           (fma y.re (atan2 x.im x.re) (* (log (hypot x.re x.im)) y.im))))))
   (if (<= y.im -1.15e+17)
     (* t_2 (sin t_1))
     (if (<= y.im 4.2e-208)
       t_3
       (if (<= y.im 2.45e-173)
         (* t_0 (pow (hypot x.im x.re) y.re))
         (if (<= y.im 0.00016)
           t_3
           (if (<= y.im 4.6e+148)
             (* t_1 t_2)
             (/ (sin t_0) (pow (exp y.im) (atan2 x.im x.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = 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_3 = pow(hypot(x_46_re, x_46_im), y_46_re) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (log(hypot(x_46_re, x_46_im)) * y_46_im)));
	double tmp;
	if (y_46_im <= -1.15e+17) {
		tmp = t_2 * sin(t_1);
	} else if (y_46_im <= 4.2e-208) {
		tmp = t_3;
	} else if (y_46_im <= 2.45e-173) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 0.00016) {
		tmp = t_3;
	} else if (y_46_im <= 4.6e+148) {
		tmp = t_1 * t_2;
	} else {
		tmp = sin(t_0) / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = 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_3 = Float64((hypot(x_46_re, x_46_im) ^ y_46_re) * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))))
	tmp = 0.0
	if (y_46_im <= -1.15e+17)
		tmp = Float64(t_2 * sin(t_1));
	elseif (y_46_im <= 4.2e-208)
		tmp = t_3;
	elseif (y_46_im <= 2.45e-173)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 0.00016)
		tmp = t_3;
	elseif (y_46_im <= 4.6e+148)
		tmp = Float64(t_1 * t_2);
	else
		tmp = Float64(sin(t_0) / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.15e+17], N[(t$95$2 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.2e-208], t$95$3, If[LessEqual[y$46$im, 2.45e-173], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 0.00016], t$95$3, If[LessEqual[y$46$im, 4.6e+148], N[(t$95$1 * t$95$2), $MachinePrecision], N[(N[Sin[t$95$0], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.15e17

   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. Taylor expanded in y.im around 0 63.7%

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

   if -1.15e17 < y.im < 4.20000000000000024e-208 or 2.44999999999999996e-173 < y.im < 1.60000000000000013e-4

   1. Initial program 47.4%

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

      1. exp-diff 47.4%

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

      2. exp-to-pow 47.4%

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

      3. hypot-def 47.4%

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

      4. *-commutative 47.4%

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

      5. exp-prod 47.4%

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

      6. +-commutative 47.4%

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

      7. *-commutative 47.4%

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

      8. fma-def 47.4%

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

      9. +-commutative 47.4%

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

   3. Simplified 89.5%

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

   4. Taylor expanded in y.im around 0 87.9%

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

   if 4.20000000000000024e-208 < y.im < 2.44999999999999996e-173

   1. Initial program 27.3%

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

      1. exp-diff 27.3%

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

      2. exp-to-pow 27.3%

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

      3. hypot-def 27.3%

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

      4. *-commutative 27.3%

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

      5. exp-prod 27.3%

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

      6. +-commutative 27.3%

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

      7. *-commutative 27.3%

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

      8. fma-def 27.3%

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

      9. +-commutative 27.3%

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

   3. Simplified 54.5%

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

      1. hypot-udef 27.3%

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

      2. add-cube-cbrt 27.3%

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

      3. pow3 27.3%

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

      4. *-commutative 27.3%

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

      5. hypot-udef 54.5%

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

   5. Applied egg-rr 54.5%

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

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

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

      1. pow-base-1 37.1%

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

      2. associate-*r* 37.1%

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

      3. unpow2 37.1%

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

      4. unpow2 37.1%

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

      5. hypot-def 91.6%

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

      6. *-lft-identity 91.6%

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

   8. Simplified 91.6%

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

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

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

       1. unpow2 82.5%

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

       2. unpow2 82.5%

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

       3. hypot-def 82.5%

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

       4. unpow2 82.5%

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

       5. unpow2 82.5%

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

       6. hypot-def 91.6%

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

   11. Simplified 91.6%

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

   if 1.60000000000000013e-4 < y.im < 4.6000000000000001e148

   1. Initial program 29.1%

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

   2. Taylor expanded in y.im around 0 64.9%

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

   3. Taylor expanded in y.re around 0 68.1%

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

      1. *-commutative 68.1%

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

   5. Simplified 68.1%

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

   if 4.6000000000000001e148 < y.im

   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 31.2%

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

      2. exp-to-pow 31.2%

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

      3. hypot-def 31.2%

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

      4. *-commutative 31.2%

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

      5. exp-prod 30.7%

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

      6. +-commutative 30.7%

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

      7. *-commutative 30.7%

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

      8. fma-def 30.7%

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

      9. +-commutative 30.7%

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

   3. Simplified 50.7%

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

      1. hypot-udef 30.7%

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

      2. add-cube-cbrt 26.8%

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

      3. pow3 26.8%

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

      4. *-commutative 26.8%

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

      5. hypot-udef 46.8%

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

   5. Applied egg-rr 46.8%

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

   6. Taylor expanded in y.re around 0 30.7%

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

      1. pow-base-1 30.7%

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

      2. associate-*r* 30.7%

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

      3. unpow2 30.7%

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

      4. unpow2 30.7%

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

      5. hypot-def 50.7%

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

      6. *-lft-identity 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in y.re around 0 37.9%

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

       1. unpow2 37.9%

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

       2. unpow2 37.9%

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

       3. hypot-def 68.1%

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

       4. *-commutative 68.1%

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

       5. exp-prod 67.6%

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

   11. Simplified 67.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{-208}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \mathbf{elif}\;y.im \leq 2.45 \cdot 10^{-173}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 0.00016:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+148}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 8: 66.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ t_2 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_3 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_4 := \sin t_3\\ \mathbf{if}\;y.re \leq -470000:\\ \;\;\;\;t_3 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-145}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{t_4}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 6000000000:\\ \;\;\;\;t_2 \cdot \sin t_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+26}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* t_0 (exp (* (atan2 x.im x.re) (- y.im)))))
        (t_2 (pow (hypot x.re x.im) y.re))
        (t_3 (* y.im (log (hypot x.im x.re))))
        (t_4 (sin t_3)))
   (if (<= y.re -470000.0)
     (* t_3 (pow (hypot x.im x.re) y.re))
     (if (<= y.re -1.3e-145)
       t_1
       (if (<= y.re 1.3e-146)
         (/ t_4 (pow (exp y.im) (atan2 x.im x.re)))
         (if (<= y.re 3.9e-52)
           t_1
           (if (<= y.re 6000000000.0)
             (* t_2 (sin t_0))
             (if (<= y.re 9.5e+26) t_1 (* t_2 t_4)))))))))

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 = t_0 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double t_2 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_4 = sin(t_3);
	double tmp;
	if (y_46_re <= -470000.0) {
		tmp = t_3 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -1.3e-145) {
		tmp = t_1;
	} else if (y_46_re <= 1.3e-146) {
		tmp = t_4 / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 3.9e-52) {
		tmp = t_1;
	} else if (y_46_re <= 6000000000.0) {
		tmp = t_2 * sin(t_0);
	} else if (y_46_re <= 9.5e+26) {
		tmp = t_1;
	} else {
		tmp = t_2 * t_4;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	double t_2 = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double t_3 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double t_4 = Math.sin(t_3);
	double tmp;
	if (y_46_re <= -470000.0) {
		tmp = t_3 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -1.3e-145) {
		tmp = t_1;
	} else if (y_46_re <= 1.3e-146) {
		tmp = t_4 / Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 3.9e-52) {
		tmp = t_1;
	} else if (y_46_re <= 6000000000.0) {
		tmp = t_2 * Math.sin(t_0);
	} else if (y_46_re <= 9.5e+26) {
		tmp = t_1;
	} else {
		tmp = t_2 * t_4;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = t_0 * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	t_2 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	t_3 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	t_4 = math.sin(t_3)
	tmp = 0
	if y_46_re <= -470000.0:
		tmp = t_3 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= -1.3e-145:
		tmp = t_1
	elif y_46_re <= 1.3e-146:
		tmp = t_4 / math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re))
	elif y_46_re <= 3.9e-52:
		tmp = t_1
	elif y_46_re <= 6000000000.0:
		tmp = t_2 * math.sin(t_0)
	elif y_46_re <= 9.5e+26:
		tmp = t_1
	else:
		tmp = t_2 * t_4
	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(t_0 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))))
	t_2 = hypot(x_46_re, x_46_im) ^ y_46_re
	t_3 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_4 = sin(t_3)
	tmp = 0.0
	if (y_46_re <= -470000.0)
		tmp = Float64(t_3 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= -1.3e-145)
		tmp = t_1;
	elseif (y_46_re <= 1.3e-146)
		tmp = Float64(t_4 / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 3.9e-52)
		tmp = t_1;
	elseif (y_46_re <= 6000000000.0)
		tmp = Float64(t_2 * sin(t_0));
	elseif (y_46_re <= 9.5e+26)
		tmp = t_1;
	else
		tmp = Float64(t_2 * t_4);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = t_0 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	t_2 = hypot(x_46_re, x_46_im) ^ y_46_re;
	t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	t_4 = sin(t_3);
	tmp = 0.0;
	if (y_46_re <= -470000.0)
		tmp = t_3 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= -1.3e-145)
		tmp = t_1;
	elseif (y_46_re <= 1.3e-146)
		tmp = t_4 / (exp(y_46_im) ^ atan2(x_46_im, x_46_re));
	elseif (y_46_re <= 3.9e-52)
		tmp = t_1;
	elseif (y_46_re <= 6000000000.0)
		tmp = t_2 * sin(t_0);
	elseif (y_46_re <= 9.5e+26)
		tmp = t_1;
	else
		tmp = t_2 * t_4;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$3], $MachinePrecision]}, If[LessEqual[y$46$re, -470000.0], N[(t$95$3 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.3e-145], t$95$1, If[LessEqual[y$46$re, 1.3e-146], N[(t$95$4 / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.9e-52], t$95$1, If[LessEqual[y$46$re, 6000000000.0], N[(t$95$2 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e+26], t$95$1, N[(t$95$2 * t$95$4), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.7e5

   1. Initial program 44.2%

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

      1. exp-diff 39.0%

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

      2. exp-to-pow 39.0%

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

      3. hypot-def 39.0%

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

      4. *-commutative 39.0%

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

      5. exp-prod 37.7%

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

      6. +-commutative 37.7%

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

      7. *-commutative 37.7%

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

      8. fma-def 37.7%

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

      9. +-commutative 37.7%

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

   3. Simplified 67.5%

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

      1. hypot-udef 37.7%

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

      2. add-cube-cbrt 37.7%

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

      3. pow3 37.7%

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

      4. *-commutative 37.7%

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

      5. hypot-udef 66.2%

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

   5. Applied egg-rr 66.2%

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

   6. Taylor expanded in y.re around 0 39.0%

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

      1. pow-base-1 39.0%

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

      2. associate-*r* 39.0%

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

      3. unpow2 39.0%

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

      4. unpow2 39.0%

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

      5. hypot-def 64.9%

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

      6. *-lft-identity 64.9%

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

   8. Simplified 64.9%

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

   9. Taylor expanded in y.im around 0 50.7%

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

       1. unpow2 50.7%

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

       2. unpow2 50.7%

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

       3. hypot-def 50.7%

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

       4. unpow2 50.7%

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

       5. unpow2 50.7%

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

       6. hypot-def 83.2%

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

   11. Simplified 83.2%

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

   if -4.7e5 < y.re < -1.3e-145 or 1.29999999999999993e-146 < y.re < 3.90000000000000018e-52 or 6e9 < y.re < 9.50000000000000054e26

   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. Taylor expanded in y.im around 0 52.7%

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

   3. Taylor expanded in y.re around 0 51.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 51.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 51.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 51.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 51.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 51.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 72.4%

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

      1. *-commutative 72.4%

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

   8. Simplified 72.4%

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

   if -1.3e-145 < y.re < 1.29999999999999993e-146

   1. Initial program 52.2%

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

      1. exp-diff 52.2%

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

      2. exp-to-pow 52.2%

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

      3. hypot-def 52.2%

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

      4. *-commutative 52.2%

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

      5. exp-prod 52.0%

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

      6. +-commutative 52.0%

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

      7. *-commutative 52.0%

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

      8. fma-def 52.0%

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

      9. +-commutative 52.0%

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

   3. Simplified 83.8%

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

      1. hypot-udef 52.0%

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

      2. add-cube-cbrt 49.8%

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

      3. pow3 51.5%

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

      4. *-commutative 51.5%

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

      5. hypot-udef 83.3%

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

   5. Applied egg-rr 83.3%

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

   6. Taylor expanded in y.re around 0 46.2%

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

      1. pow-base-1 46.2%

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

      2. associate-*r* 46.2%

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

      3. unpow2 46.2%

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

      4. unpow2 46.2%

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

      5. hypot-def 73.9%

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

      6. *-lft-identity 73.9%

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

   8. Simplified 73.9%

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

   9. Taylor expanded in y.re around 0 46.5%

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

       1. unpow2 46.5%

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

       2. unpow2 46.5%

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

       3. hypot-def 74.2%

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

       4. *-commutative 74.2%

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

       5. exp-prod 74.0%

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

   11. Simplified 74.0%

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

   if 3.90000000000000018e-52 < y.re < 6e9

   1. Initial program 38.1%

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

      1. exp-diff 38.1%

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

      2. exp-to-pow 38.1%

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

      3. hypot-def 38.1%

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

      4. *-commutative 38.1%

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

      5. exp-prod 38.1%

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

      6. +-commutative 38.1%

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

      7. *-commutative 38.1%

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

      8. fma-def 38.1%

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

      9. +-commutative 38.1%

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

   3. Simplified 99.6%

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

      1. hypot-udef 38.1%

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

      2. add-cube-cbrt 37.8%

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

      3. pow3 37.7%

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

      4. *-commutative 37.7%

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

      5. hypot-udef 98.9%

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

   5. Applied egg-rr 98.9%

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

      1. add-cube-cbrt 98.8%

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

      2. rem-cbrt-cube 98.8%

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

      3. rem-cbrt-cube 98.9%

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

      4. pow2 98.9%

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

      5. rem-cbrt-cube 98.8%

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

      6. unpow3 98.8%

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

      7. add-cube-cbrt 98.9%

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

      8. hypot-udef 37.7%

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

      9. +-commutative 37.7%

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

      10. hypot-def 98.9%

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

   7. Applied egg-rr 98.9%

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

   8. Taylor expanded in y.im around 0 91.5%

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

   9. Taylor expanded in y.im around 0 63.2%

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

   if 9.50000000000000054e26 < y.re

   1. Initial program 34.0%

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

      1. exp-diff 28.3%

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

      2. exp-to-pow 28.3%

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

      3. hypot-def 28.3%

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

      4. *-commutative 28.3%

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

      5. exp-prod 26.4%

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

      6. +-commutative 26.4%

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

      7. *-commutative 26.4%

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

      8. fma-def 26.4%

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

      9. +-commutative 26.4%

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

   3. Simplified 45.3%

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

      1. hypot-udef 26.4%

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

      2. add-cube-cbrt 30.2%

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

      3. pow3 30.2%

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

      4. *-commutative 30.2%

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

      5. hypot-udef 49.1%

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

   5. Applied egg-rr 49.1%

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

   6. Taylor expanded in y.re around 0 30.2%

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

      1. pow-base-1 30.2%

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

      2. associate-*r* 30.2%

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

      3. unpow2 30.2%

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

      4. unpow2 30.2%

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

      5. hypot-def 60.4%

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

      6. *-lft-identity 60.4%

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

   8. Simplified 60.4%

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

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

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -470000:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-145}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-146}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{-52}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.re \leq 6000000000:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+26}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 9: 64.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_2 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_3 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_4 := t_2 \cdot \sin t_3\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.25 \cdot 10^{-212}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-208}:\\ \;\;\;\;t_2 \cdot \sin t_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-28}:\\ \;\;\;\;t_3 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+222}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 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.re x.im) y.re))
        (t_3 (* y.im (log (hypot x.im x.re))))
        (t_4 (* t_2 (sin t_3))))
   (if (<= y.im -1.15e+17)
     t_1
     (if (<= y.im -2.25e-212)
       t_4
       (if (<= y.im 2.1e-208)
         (* t_2 (sin t_0))
         (if (<= y.im 7e-28)
           (* t_3 (pow (hypot x.im x.re) y.re))
           (if (<= y.im 1.8e+135)
             t_1
             (if (<= y.im 6.6e+222)
               t_4
               (* t_0 (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 = 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_re, x_46_im), y_46_re);
	double t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_4 = t_2 * sin(t_3);
	double tmp;
	if (y_46_im <= -1.15e+17) {
		tmp = t_1;
	} else if (y_46_im <= -2.25e-212) {
		tmp = t_4;
	} else if (y_46_im <= 2.1e-208) {
		tmp = t_2 * sin(t_0);
	} else if (y_46_im <= 7e-28) {
		tmp = t_3 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.8e+135) {
		tmp = t_1;
	} else if (y_46_im <= 6.6e+222) {
		tmp = t_4;
	} else {
		tmp = t_0 * 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 = 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_re, x_46_im), y_46_re);
	double t_3 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double t_4 = t_2 * Math.sin(t_3);
	double tmp;
	if (y_46_im <= -1.15e+17) {
		tmp = t_1;
	} else if (y_46_im <= -2.25e-212) {
		tmp = t_4;
	} else if (y_46_im <= 2.1e-208) {
		tmp = t_2 * Math.sin(t_0);
	} else if (y_46_im <= 7e-28) {
		tmp = t_3 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.8e+135) {
		tmp = t_1;
	} else if (y_46_im <= 6.6e+222) {
		tmp = t_4;
	} else {
		tmp = t_0 * 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 = 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_re, x_46_im), y_46_re)
	t_3 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	t_4 = t_2 * math.sin(t_3)
	tmp = 0
	if y_46_im <= -1.15e+17:
		tmp = t_1
	elif y_46_im <= -2.25e-212:
		tmp = t_4
	elif y_46_im <= 2.1e-208:
		tmp = t_2 * math.sin(t_0)
	elif y_46_im <= 7e-28:
		tmp = t_3 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_im <= 1.8e+135:
		tmp = t_1
	elif y_46_im <= 6.6e+222:
		tmp = t_4
	else:
		tmp = t_0 * 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(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_re, x_46_im) ^ y_46_re
	t_3 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_4 = Float64(t_2 * sin(t_3))
	tmp = 0.0
	if (y_46_im <= -1.15e+17)
		tmp = t_1;
	elseif (y_46_im <= -2.25e-212)
		tmp = t_4;
	elseif (y_46_im <= 2.1e-208)
		tmp = Float64(t_2 * sin(t_0));
	elseif (y_46_im <= 7e-28)
		tmp = Float64(t_3 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 1.8e+135)
		tmp = t_1;
	elseif (y_46_im <= 6.6e+222)
		tmp = t_4;
	else
		tmp = Float64(t_0 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-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 = 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_re, x_46_im) ^ y_46_re;
	t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	t_4 = t_2 * sin(t_3);
	tmp = 0.0;
	if (y_46_im <= -1.15e+17)
		tmp = t_1;
	elseif (y_46_im <= -2.25e-212)
		tmp = t_4;
	elseif (y_46_im <= 2.1e-208)
		tmp = t_2 * sin(t_0);
	elseif (y_46_im <= 7e-28)
		tmp = t_3 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_im <= 1.8e+135)
		tmp = t_1;
	elseif (y_46_im <= 6.6e+222)
		tmp = t_4;
	else
		tmp = t_0 * 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $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$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.15e+17], t$95$1, If[LessEqual[y$46$im, -2.25e-212], t$95$4, If[LessEqual[y$46$im, 2.1e-208], N[(t$95$2 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7e-28], N[(t$95$3 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+135], t$95$1, If[LessEqual[y$46$im, 6.6e+222], t$95$4, N[(t$95$0 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.15e17 or 6.9999999999999999e-28 < y.im < 1.7999999999999999e135

   1. Initial program 39.3%

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

   2. Taylor expanded in y.im around 0 63.2%

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

   3. Taylor expanded in y.re around 0 63.6%

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

      1. *-commutative 63.6%

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

   5. Simplified 63.6%

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

   if -1.15e17 < y.im < -2.2499999999999999e-212 or 1.7999999999999999e135 < y.im < 6.59999999999999967e222

   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 38.6%

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

      2. exp-to-pow 38.6%

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

      3. hypot-def 38.6%

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

      4. *-commutative 38.6%

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

      5. exp-prod 38.4%

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

      6. +-commutative 38.4%

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

      7. *-commutative 38.4%

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

      8. fma-def 38.4%

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

      9. +-commutative 38.4%

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

   3. Simplified 78.2%

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

      1. hypot-udef 38.4%

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

      2. add-cube-cbrt 36.6%

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

      3. pow3 36.5%

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

      4. *-commutative 36.5%

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

      5. hypot-udef 74.6%

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

   5. Applied egg-rr 74.6%

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

   6. Taylor expanded in y.re around 0 38.6%

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

      1. pow-base-1 38.6%

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

      2. associate-*r* 38.6%

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

      3. unpow2 38.6%

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

      4. unpow2 38.6%

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

      5. hypot-def 74.6%

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

      6. *-lft-identity 74.6%

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

   8. Simplified 74.6%

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

   9. Taylor expanded in y.im around 0 76.2%

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

   if -2.2499999999999999e-212 < y.im < 2.10000000000000012e-208

   1. Initial program 54.8%

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

      1. exp-diff 54.8%

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

      2. exp-to-pow 54.8%

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

      3. hypot-def 54.8%

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

      4. *-commutative 54.8%

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

      5. exp-prod 54.8%

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

      6. +-commutative 54.8%

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

      7. *-commutative 54.8%

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

      8. fma-def 54.9%

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

      9. +-commutative 54.9%

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

   3. Simplified 92.3%

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

      1. hypot-udef 54.9%

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

      2. add-cube-cbrt 54.7%

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

      3. pow3 54.7%

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

      4. *-commutative 54.7%

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

      5. hypot-udef 92.0%

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

   5. Applied egg-rr 92.0%

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

      1. add-cube-cbrt 91.9%

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

      2. rem-cbrt-cube 91.9%

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

      3. rem-cbrt-cube 91.9%

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

      4. pow2 91.9%

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

      5. rem-cbrt-cube 91.9%

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

      6. unpow3 91.9%

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

      7. add-cube-cbrt 91.9%

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

      8. hypot-udef 54.6%

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

      9. +-commutative 54.6%

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

      10. hypot-def 91.9%

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

   7. Applied egg-rr 92.0%

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

   8. Taylor expanded in y.im around 0 92.0%

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

   9. Taylor expanded in y.im around 0 79.3%

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

   if 2.10000000000000012e-208 < y.im < 6.9999999999999999e-28

   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 40.4%

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

      3. hypot-def 40.4%

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

      4. *-commutative 40.4%

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

      5. exp-prod 40.4%

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

      6. +-commutative 40.4%

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

      7. *-commutative 40.4%

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

      8. fma-def 40.4%

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

      9. +-commutative 40.4%

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

   3. Simplified 83.2%

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

      1. hypot-udef 40.4%

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

      2. add-cube-cbrt 40.1%

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

      3. pow3 40.1%

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

      4. *-commutative 40.1%

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

      5. hypot-udef 82.6%

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

   5. Applied egg-rr 82.6%

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

   6. Taylor expanded in y.re around 0 38.6%

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

      1. pow-base-1 38.6%

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

      2. associate-*r* 38.6%

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

      3. unpow2 38.6%

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

      4. unpow2 38.6%

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

      5. hypot-def 76.6%

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

      6. *-lft-identity 76.6%

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

   8. Simplified 76.6%

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

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

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

       1. unpow2 50.8%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 50.8%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 51.0%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 51.0%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 51.0%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 76.6%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 76.6%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   if 6.59999999999999967e222 < y.im

   1. Initial program 45.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 67.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 61.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 61.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 61.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 61.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 61.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 61.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 78.0%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 78.0%

         \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   8. Simplified 78.0%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+17}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.25 \cdot 10^{-212}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-208}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-28}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+135}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+222}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 10: 62.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ t_3 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-212}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 10^{-208}:\\ \;\;\;\;t_3 \cdot \sin t_1\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+222}:\\ \;\;\;\;t_3 \cdot \sin t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (hypot x.im x.re))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* t_1 (exp (* (atan2 x.im x.re) (- y.im)))))
        (t_3 (pow (hypot x.re x.im) y.re)))
   (if (<= y.im -1.05e+80)
     t_2
     (if (<= y.im -2e-212)
       (* t_0 (pow (hypot x.im x.re) y.re))
       (if (<= y.im 1e-208)
         (* t_3 (sin t_1))
         (if (<= y.im 6.6e+222) (* t_3 (sin t_0)) t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double t_3 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_im <= -1.05e+80) {
		tmp = t_2;
	} else if (y_46_im <= -2e-212) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1e-208) {
		tmp = t_3 * sin(t_1);
	} else if (y_46_im <= 6.6e+222) {
		tmp = t_3 * sin(t_0);
	} 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 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = t_1 * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	double t_3 = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_im <= -1.05e+80) {
		tmp = t_2;
	} else if (y_46_im <= -2e-212) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1e-208) {
		tmp = t_3 * Math.sin(t_1);
	} else if (y_46_im <= 6.6e+222) {
		tmp = t_3 * Math.sin(t_0);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = t_1 * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	t_3 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_im <= -1.05e+80:
		tmp = t_2
	elif y_46_im <= -2e-212:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_im <= 1e-208:
		tmp = t_3 * math.sin(t_1)
	elif y_46_im <= 6.6e+222:
		tmp = t_3 * math.sin(t_0)
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(t_1 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))))
	t_3 = hypot(x_46_re, x_46_im) ^ y_46_re
	tmp = 0.0
	if (y_46_im <= -1.05e+80)
		tmp = t_2;
	elseif (y_46_im <= -2e-212)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 1e-208)
		tmp = Float64(t_3 * sin(t_1));
	elseif (y_46_im <= 6.6e+222)
		tmp = Float64(t_3 * sin(t_0));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	t_3 = hypot(x_46_re, x_46_im) ^ y_46_re;
	tmp = 0.0;
	if (y_46_im <= -1.05e+80)
		tmp = t_2;
	elseif (y_46_im <= -2e-212)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_im <= 1e-208)
		tmp = t_3 * sin(t_1);
	elseif (y_46_im <= 6.6e+222)
		tmp = t_3 * sin(t_0);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e+80], t$95$2, If[LessEqual[y$46$im, -2e-212], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1e-208], N[(t$95$3 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.6e+222], N[(t$95$3 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.05000000000000001e80 or 6.59999999999999967e222 < y.im

   1. Initial program 45.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 62.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 52.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 52.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 54.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 69.5%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 69.5%

         \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   8. Simplified 69.5%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if -1.05000000000000001e80 < y.im < -1.99999999999999991e-212

   1. Initial program 42.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 37.5%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 37.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 37.5%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 37.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 38.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 38.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 38.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 76.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 76.8%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 35.9%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 35.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 35.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 35.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 35.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 69.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 69.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 69.4%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 45.2%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 45.2%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 45.2%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 45.5%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 45.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 45.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 70.7%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 70.7%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   if -1.99999999999999991e-212 < y.im < 1.0000000000000001e-208

   1. Initial program 54.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 54.8%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 54.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 54.8%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 54.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 54.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 92.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 54.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 54.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 54.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 54.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 92.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 92.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}}\right)\right) \]

      2. rem-cbrt-cube 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \left(\color{blue}{\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      3. rem-cbrt-cube 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \color{blue}{\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      4. pow2 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      5. rem-cbrt-cube 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)}}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      6. unpow3 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      7. add-cube-cbrt 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      8. hypot-udef 54.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      9. +-commutative 54.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right)}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      10. hypot-def 91.9%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   7. Applied egg-rr 92.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2} \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}\right)\right) \]

   8. Taylor expanded in y.im around 0 92.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{1}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2} \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)\right) \]

   9. Taylor expanded in y.im around 0 79.3%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   if 1.0000000000000001e-208 < y.im < 6.59999999999999967e222

   1. Initial program 35.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 33.5%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 33.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 33.5%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 33.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 63.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 34.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 33.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 33.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 63.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 63.4%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 29.4%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 29.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 29.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 29.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 29.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 59.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 59.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 59.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 64.6%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{1}} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{+80}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-212}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 10^{-208}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+222}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 11: 62.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-212}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-209}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin t_1\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+222}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (* y.im (log (hypot x.im x.re))) (pow (hypot x.im x.re) y.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* t_1 (exp (* (atan2 x.im x.re) (- y.im))))))
   (if (<= y.im -2.8e+80)
     t_2
     (if (<= y.im -2.2e-212)
       t_0
       (if (<= y.im 3.5e-209)
         (* (pow (hypot x.re x.im) y.re) (sin t_1))
         (if (<= y.im 6.6e+222) t_0 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * log(hypot(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double tmp;
	if (y_46_im <= -2.8e+80) {
		tmp = t_2;
	} else if (y_46_im <= -2.2e-212) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-209) {
		tmp = pow(hypot(x_46_re, x_46_im), y_46_re) * sin(t_1);
	} else if (y_46_im <= 6.6e+222) {
		tmp = t_0;
	} 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 = (y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = t_1 * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	double tmp;
	if (y_46_im <= -2.8e+80) {
		tmp = t_2;
	} else if (y_46_im <= -2.2e-212) {
		tmp = t_0;
	} else if (y_46_im <= 3.5e-209) {
		tmp = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) * Math.sin(t_1);
	} else if (y_46_im <= 6.6e+222) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_im * math.log(math.hypot(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = t_1 * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	tmp = 0
	if y_46_im <= -2.8e+80:
		tmp = t_2
	elif y_46_im <= -2.2e-212:
		tmp = t_0
	elif y_46_im <= 3.5e-209:
		tmp = math.pow(math.hypot(x_46_re, x_46_im), y_46_re) * math.sin(t_1)
	elif y_46_im <= 6.6e+222:
		tmp = t_0
	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(y_46_im * log(hypot(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(t_1 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))))
	tmp = 0.0
	if (y_46_im <= -2.8e+80)
		tmp = t_2;
	elseif (y_46_im <= -2.2e-212)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-209)
		tmp = Float64((hypot(x_46_re, x_46_im) ^ y_46_re) * sin(t_1));
	elseif (y_46_im <= 6.6e+222)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_im * log(hypot(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	tmp = 0.0;
	if (y_46_im <= -2.8e+80)
		tmp = t_2;
	elseif (y_46_im <= -2.2e-212)
		tmp = t_0;
	elseif (y_46_im <= 3.5e-209)
		tmp = (hypot(x_46_re, x_46_im) ^ y_46_re) * sin(t_1);
	elseif (y_46_im <= 6.6e+222)
		tmp = t_0;
	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[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.8e+80], t$95$2, If[LessEqual[y$46$im, -2.2e-212], t$95$0, If[LessEqual[y$46$im, 3.5e-209], N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.6e+222], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.79999999999999984e80 or 6.59999999999999967e222 < y.im

   1. Initial program 45.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 62.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 52.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 52.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 54.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 69.5%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 69.5%

         \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   8. Simplified 69.5%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if -2.79999999999999984e80 < y.im < -2.20000000000000003e-212 or 3.50000000000000002e-209 < y.im < 6.59999999999999967e222

   1. Initial program 38.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 35.2%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 35.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 35.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 35.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 34.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 34.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 34.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 34.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 34.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 68.9%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 34.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 36.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 35.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 35.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 69.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 69.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 32.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 63.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 63.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 63.8%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 41.7%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 41.7%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 41.7%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 42.5%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 42.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 42.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 65.9%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 65.9%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   if -2.20000000000000003e-212 < y.im < 3.50000000000000002e-209

   1. Initial program 54.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 54.8%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 54.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 54.8%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 54.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 54.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 54.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 92.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 54.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 54.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 54.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 54.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 92.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 92.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]
   6. Step-by-step derivation

      1. add-cube-cbrt 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}}\right)\right) \]

      2. rem-cbrt-cube 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \left(\color{blue}{\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      3. rem-cbrt-cube 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \color{blue}{\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right) \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      4. pow2 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{2}} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      5. rem-cbrt-cube 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\color{blue}{\left(\sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)}}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      6. unpow3 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)} \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      7. add-cube-cbrt 91.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      8. hypot-udef 54.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      9. +-commutative 54.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right)}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

      10. hypot-def 91.9%

          \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}\right)}^{2} \cdot \sqrt[3]{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   7. Applied egg-rr 92.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2} \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}\right)\right) \]

   8. Taylor expanded in y.im around 0 92.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{1}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2} \cdot \sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)\right) \]

   9. Taylor expanded in y.im around 0 79.3%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{1} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.8 \cdot 10^{+80}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-212}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-209}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+222}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 12: 59.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+79} \lor \neg \left(y.im \leq 6.6 \cdot 10^{+222}\right):\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9.5e+79) (not (<= y.im 6.6e+222)))
   (* (* y.re (atan2 x.im x.re)) (exp (* (atan2 x.im x.re) (- y.im))))
   (* (* y.im (log (hypot x.im x.re))) (pow (hypot x.im x.re) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.5e+79) || !(y_46_im <= 6.6e+222)) {
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = (y_46_im * log(hypot(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9.5e+79) || !(y_46_im <= 6.6e+222)) {
		tmp = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = (y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -9.5e+79) or not (y_46_im <= 6.6e+222):
		tmp = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = (y_46_im * math.log(math.hypot(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9.5e+79) || !(y_46_im <= 6.6e+222))
		tmp = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64(Float64(y_46_im * log(hypot(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -9.5e+79) || ~((y_46_im <= 6.6e+222)))
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = (y_46_im * log(hypot(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9.5e+79], N[Not[LessEqual[y$46$im, 6.6e+222]], $MachinePrecision]], N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -9.49999999999999994e79 or 6.59999999999999967e222 < y.im

   1. Initial program 45.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 62.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 52.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 52.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 54.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 54.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 69.5%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 69.5%

         \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   8. Simplified 69.5%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if -9.49999999999999994e79 < y.im < 6.59999999999999967e222

   1. Initial program 41.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 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-def 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 38.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 38.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 38.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 38.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 38.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 38.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 39.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 39.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 39.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 73.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 73.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 32.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 32.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 60.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 60.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 60.6%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 40.3%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 40.3%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 40.3%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 41.0%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 41.0%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 41.0%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 62.3%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 62.3%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+79} \lor \neg \left(y.im \leq 6.6 \cdot 10^{+222}\right):\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 13: 44.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(-y.im\right)\right)\\ \mathbf{if}\;x.re \leq -6.2 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -2.8 \cdot 10^{-176}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.re \leq 6.8 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (* (* y.re (atan2 x.im x.re)) (exp (* (atan2 x.im x.re) (- y.im)))))
        (t_1
         (* (pow (hypot x.im x.re) y.re) (* (log (/ -1.0 x.re)) (- y.im)))))
   (if (<= x.re -6.2e+74)
     t_1
     (if (<= x.re -2.6e+20)
       t_0
       (if (<= x.re -2.8e-176)
         t_1
         (if (<= x.re 6.8e-87)
           t_0
           (* y.im (* (log x.re) (pow x.re y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double t_1 = pow(hypot(x_46_im, x_46_re), y_46_re) * (log((-1.0 / x_46_re)) * -y_46_im);
	double tmp;
	if (x_46_re <= -6.2e+74) {
		tmp = t_1;
	} else if (x_46_re <= -2.6e+20) {
		tmp = t_0;
	} else if (x_46_re <= -2.8e-176) {
		tmp = t_1;
	} else if (x_46_re <= 6.8e-87) {
		tmp = t_0;
	} else {
		tmp = y_46_im * (log(x_46_re) * pow(x_46_re, y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re) * (Math.log((-1.0 / x_46_re)) * -y_46_im);
	double tmp;
	if (x_46_re <= -6.2e+74) {
		tmp = t_1;
	} else if (x_46_re <= -2.6e+20) {
		tmp = t_0;
	} else if (x_46_re <= -2.8e-176) {
		tmp = t_1;
	} else if (x_46_re <= 6.8e-87) {
		tmp = t_0;
	} else {
		tmp = y_46_im * (Math.log(x_46_re) * Math.pow(x_46_re, y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re) * (math.log((-1.0 / x_46_re)) * -y_46_im)
	tmp = 0
	if x_46_re <= -6.2e+74:
		tmp = t_1
	elif x_46_re <= -2.6e+20:
		tmp = t_0
	elif x_46_re <= -2.8e-176:
		tmp = t_1
	elif x_46_re <= 6.8e-87:
		tmp = t_0
	else:
		tmp = y_46_im * (math.log(x_46_re) * math.pow(x_46_re, y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))))
	t_1 = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) * Float64(log(Float64(-1.0 / x_46_re)) * Float64(-y_46_im)))
	tmp = 0.0
	if (x_46_re <= -6.2e+74)
		tmp = t_1;
	elseif (x_46_re <= -2.6e+20)
		tmp = t_0;
	elseif (x_46_re <= -2.8e-176)
		tmp = t_1;
	elseif (x_46_re <= 6.8e-87)
		tmp = t_0;
	else
		tmp = Float64(y_46_im * Float64(log(x_46_re) * (x_46_re ^ y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	t_1 = (hypot(x_46_im, x_46_re) ^ y_46_re) * (log((-1.0 / x_46_re)) * -y_46_im);
	tmp = 0.0;
	if (x_46_re <= -6.2e+74)
		tmp = t_1;
	elseif (x_46_re <= -2.6e+20)
		tmp = t_0;
	elseif (x_46_re <= -2.8e-176)
		tmp = t_1;
	elseif (x_46_re <= 6.8e-87)
		tmp = t_0;
	else
		tmp = y_46_im * (log(x_46_re) * (x_46_re ^ y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] * N[(N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -6.2e+74], t$95$1, If[LessEqual[x$46$re, -2.6e+20], t$95$0, If[LessEqual[x$46$re, -2.8e-176], t$95$1, If[LessEqual[x$46$re, 6.8e-87], t$95$0, N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -6.20000000000000043e74 or -2.6e20 < x.re < -2.8000000000000001e-176

   1. Initial program 44.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.0%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 42.0%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 42.0%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 44.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 76.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 76.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 35.5%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 35.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 35.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 35.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 35.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 60.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 60.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 60.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 39.8%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 39.8%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 39.8%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 39.9%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 39.9%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 39.9%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 58.9%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 58.9%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   12. Taylor expanded in x.re around -inf 55.3%

       \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \color{blue}{\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\right)} \]

   if -6.20000000000000043e74 < x.re < -2.6e20 or -2.8000000000000001e-176 < x.re < 6.7999999999999997e-87

   1. Initial program 46.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 59.4%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 48.4%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 48.4%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 49.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 49.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 49.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 49.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 47.9%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 47.9%

         \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   8. Simplified 47.9%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if 6.7999999999999997e-87 < x.re

   1. Initial program 36.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 35.4%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 35.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 35.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 35.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 30.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 30.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 30.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 56.6%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 28.5%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 58.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 58.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 58.2%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 39.2%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 39.2%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 39.2%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 40.8%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 40.8%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 40.8%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 60.9%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 60.9%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   12. Taylor expanded in x.im around 0 53.5%

       \[\leadsto \color{blue}{{x.re}^{y.re} \cdot \left(\log x.re \cdot y.im\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 53.5%

          \[\leadsto \color{blue}{\left({x.re}^{y.re} \cdot \log x.re\right) \cdot y.im} \]

       2. *-commutative 53.5%

          \[\leadsto \color{blue}{\left(\log x.re \cdot {x.re}^{y.re}\right)} \cdot y.im \]

       3. *-commutative 53.5%

          \[\leadsto \color{blue}{y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)} \]

   14. Simplified 53.5%

       \[\leadsto \color{blue}{y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -6.2 \cdot 10^{+74}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(-y.im\right)\right)\\ \mathbf{elif}\;x.re \leq -2.6 \cdot 10^{+20}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.re \leq -2.8 \cdot 10^{-176}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(-y.im\right)\right)\\ \mathbf{elif}\;x.re \leq 6.8 \cdot 10^{-87}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)\\ \end{array} \]

Alternative 14: 45.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;x.re \leq -1.25 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.re \leq -3.7 \cdot 10^{-175}:\\ \;\;\;\;y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right)\\ \mathbf{elif}\;x.re \leq 8.6 \cdot 10^{-90}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (* (* y.re (atan2 x.im x.re)) (exp (* (atan2 x.im x.re) (- y.im))))))
   (if (<= x.re -1.25e-121)
     t_0
     (if (<= x.re -3.7e-175)
       (* y.im (* (log x.im) (pow x.im y.re)))
       (if (<= x.re 8.6e-90) t_0 (* y.im (* (log x.re) (pow x.re y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double tmp;
	if (x_46_re <= -1.25e-121) {
		tmp = t_0;
	} else if (x_46_re <= -3.7e-175) {
		tmp = y_46_im * (log(x_46_im) * pow(x_46_im, y_46_re));
	} else if (x_46_re <= 8.6e-90) {
		tmp = t_0;
	} else {
		tmp = y_46_im * (log(x_46_re) * pow(x_46_re, y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_46re * atan2(x_46im, x_46re)) * exp((atan2(x_46im, x_46re) * -y_46im))
    if (x_46re <= (-1.25d-121)) then
        tmp = t_0
    else if (x_46re <= (-3.7d-175)) then
        tmp = y_46im * (log(x_46im) * (x_46im ** y_46re))
    else if (x_46re <= 8.6d-90) then
        tmp = t_0
    else
        tmp = y_46im * (log(x_46re) * (x_46re ** y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	double tmp;
	if (x_46_re <= -1.25e-121) {
		tmp = t_0;
	} else if (x_46_re <= -3.7e-175) {
		tmp = y_46_im * (Math.log(x_46_im) * Math.pow(x_46_im, y_46_re));
	} else if (x_46_re <= 8.6e-90) {
		tmp = t_0;
	} else {
		tmp = y_46_im * (Math.log(x_46_re) * Math.pow(x_46_re, y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	tmp = 0
	if x_46_re <= -1.25e-121:
		tmp = t_0
	elif x_46_re <= -3.7e-175:
		tmp = y_46_im * (math.log(x_46_im) * math.pow(x_46_im, y_46_re))
	elif x_46_re <= 8.6e-90:
		tmp = t_0
	else:
		tmp = y_46_im * (math.log(x_46_re) * math.pow(x_46_re, y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))))
	tmp = 0.0
	if (x_46_re <= -1.25e-121)
		tmp = t_0;
	elseif (x_46_re <= -3.7e-175)
		tmp = Float64(y_46_im * Float64(log(x_46_im) * (x_46_im ^ y_46_re)));
	elseif (x_46_re <= 8.6e-90)
		tmp = t_0;
	else
		tmp = Float64(y_46_im * Float64(log(x_46_re) * (x_46_re ^ y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	tmp = 0.0;
	if (x_46_re <= -1.25e-121)
		tmp = t_0;
	elseif (x_46_re <= -3.7e-175)
		tmp = y_46_im * (log(x_46_im) * (x_46_im ^ y_46_re));
	elseif (x_46_re <= 8.6e-90)
		tmp = t_0;
	else
		tmp = y_46_im * (log(x_46_re) * (x_46_re ^ y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1.25e-121], t$95$0, If[LessEqual[x$46$re, -3.7e-175], N[(y$46$im * N[(N[Log[x$46$im], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 8.6e-90], t$95$0, N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.24999999999999997e-121 or -3.69999999999999998e-175 < x.re < 8.6000000000000004e-90

   1. Initial program 43.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.im around 0 59.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. Taylor expanded in y.re around 0 45.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left({y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   4. Step-by-step derivation

      1. fma-def 45.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {y.re}^{3} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. cube-prod 46.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, \color{blue}{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative 46.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}^{3}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-commutative 46.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right) \]

   5. Simplified 46.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\mathsf{fma}\left(-0.16666666666666666, {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0 46.8%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-commutative 46.8%

         \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   8. Simplified 46.8%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if -1.24999999999999997e-121 < x.re < -3.69999999999999998e-175

   1. Initial program 64.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 56.6%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 56.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 56.6%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 69.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 56.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 56.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 56.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 68.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 68.9%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 49.2%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 49.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 49.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 49.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 49.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 54.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 54.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 54.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 72.5%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 72.5%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 72.5%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 72.5%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 72.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 72.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 74.0%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 74.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   12. Taylor expanded in x.re around 0 54.2%

       \[\leadsto \color{blue}{y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right)} \]

   if 8.6000000000000004e-90 < x.re

   1. Initial program 36.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 35.4%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 35.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 35.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 35.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 59.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 30.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 30.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 30.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 56.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 56.6%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 28.5%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 28.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 58.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 58.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 58.2%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 39.2%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 39.2%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 39.2%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 40.8%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 40.8%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 40.8%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 60.9%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 60.9%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   12. Taylor expanded in x.im around 0 53.5%

       \[\leadsto \color{blue}{{x.re}^{y.re} \cdot \left(\log x.re \cdot y.im\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 53.5%

          \[\leadsto \color{blue}{\left({x.re}^{y.re} \cdot \log x.re\right) \cdot y.im} \]

       2. *-commutative 53.5%

          \[\leadsto \color{blue}{\left(\log x.re \cdot {x.re}^{y.re}\right)} \cdot y.im \]

       3. *-commutative 53.5%

          \[\leadsto \color{blue}{y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)} \]

   14. Simplified 53.5%

       \[\leadsto \color{blue}{y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.25 \cdot 10^{-121}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.re \leq -3.7 \cdot 10^{-175}:\\ \;\;\;\;y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right)\\ \mathbf{elif}\;x.re \leq 8.6 \cdot 10^{-90}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)\\ \end{array} \]

Alternative 15: 29.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 5.5 \cdot 10^{-278}:\\ \;\;\;\;y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 5.5e-278)
   (* y.im (* (log x.im) (pow x.im y.re)))
   (* y.im (* (log x.re) (pow x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 5.5e-278) {
		tmp = y_46_im * (log(x_46_im) * pow(x_46_im, y_46_re));
	} else {
		tmp = y_46_im * (log(x_46_re) * pow(x_46_re, y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 5.5d-278) then
        tmp = y_46im * (log(x_46im) * (x_46im ** y_46re))
    else
        tmp = y_46im * (log(x_46re) * (x_46re ** y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 5.5e-278) {
		tmp = y_46_im * (Math.log(x_46_im) * Math.pow(x_46_im, y_46_re));
	} else {
		tmp = y_46_im * (Math.log(x_46_re) * Math.pow(x_46_re, y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 5.5e-278:
		tmp = y_46_im * (math.log(x_46_im) * math.pow(x_46_im, y_46_re))
	else:
		tmp = y_46_im * (math.log(x_46_re) * math.pow(x_46_re, y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 5.5e-278)
		tmp = Float64(y_46_im * Float64(log(x_46_im) * (x_46_im ^ y_46_re)));
	else
		tmp = Float64(y_46_im * Float64(log(x_46_re) * (x_46_re ^ y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 5.5e-278)
		tmp = y_46_im * (log(x_46_im) * (x_46_im ^ y_46_re));
	else
		tmp = y_46_im * (log(x_46_re) * (x_46_re ^ y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 5.5e-278], N[(y$46$im * N[(N[Log[x$46$im], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 5.49999999999999989e-278

   1. Initial program 46.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 43.6%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 43.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 43.6%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 43.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 43.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 43.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 43.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 43.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 43.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 76.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 43.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 45.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 45.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 45.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 78.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 78.3%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 37.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 37.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 37.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 37.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 37.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 61.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 61.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 61.5%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 35.9%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 35.9%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 35.9%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 36.1%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 36.1%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 36.1%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 53.2%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 53.2%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   12. Taylor expanded in x.re around 0 18.5%

       \[\leadsto \color{blue}{y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right)} \]

   if 5.49999999999999989e-278 < x.re

   1. Initial program 38.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 33.9%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. exp-to-pow 33.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. hypot-def 33.9%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-commutative 33.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. exp-prod 31.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. +-commutative 31.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      7. *-commutative 31.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

      8. fma-def 31.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

      9. +-commutative 31.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

   3. Simplified 60.4%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
   4. Step-by-step derivation

      1. hypot-udef 31.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

      2. add-cube-cbrt 31.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

      3. pow3 31.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

      4. *-commutative 31.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

      5. hypot-udef 59.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

   5. Applied egg-rr 59.4%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

   6. Taylor expanded in y.re around 0 28.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
   7. Step-by-step derivation

      1. pow-base-1 28.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      2. associate-*r* 28.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. unpow2 28.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

      4. unpow2 28.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

      5. hypot-def 55.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

      6. *-lft-identity 55.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 55.7%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   9. Taylor expanded in y.im around 0 35.6%

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   10. Step-by-step derivation

       1. unpow2 35.6%

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       2. unpow2 35.6%

          \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       3. hypot-def 37.5%

          \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

       4. unpow2 37.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

       5. unpow2 37.5%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

       6. hypot-def 55.0%

          \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   11. Simplified 55.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

   12. Taylor expanded in x.im around 0 44.2%

       \[\leadsto \color{blue}{{x.re}^{y.re} \cdot \left(\log x.re \cdot y.im\right)} \]
   13. Step-by-step derivation

       1. associate-*r* 44.2%

          \[\leadsto \color{blue}{\left({x.re}^{y.re} \cdot \log x.re\right) \cdot y.im} \]

       2. *-commutative 44.2%

          \[\leadsto \color{blue}{\left(\log x.re \cdot {x.re}^{y.re}\right)} \cdot y.im \]

       3. *-commutative 44.2%

          \[\leadsto \color{blue}{y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)} \]

   14. Simplified 44.2%

       \[\leadsto \color{blue}{y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 5.5 \cdot 10^{-278}:\\ \;\;\;\;y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)\\ \end{array} \]

Alternative 16: 17.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* y.im (* (log x.im) (pow x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * (log(x_46_im) * pow(x_46_im, y_46_re));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46im * (log(x_46im) * (x_46im ** y_46re))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * (Math.log(x_46_im) * Math.pow(x_46_im, y_46_re));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_im * (math.log(x_46_im) * math.pow(x_46_im, y_46_re))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_im * Float64(log(x_46_im) * (x_46_im ^ y_46_re)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_im * (log(x_46_im) * (x_46_im ^ y_46_re));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$im * N[(N[Log[x$46$im], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

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 39.0%

      \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. exp-to-pow 39.0%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. hypot-def 39.0%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. *-commutative 39.0%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. exp-prod 38.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. +-commutative 38.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

   7. *-commutative 38.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   8. fma-def 38.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)\right)} \]

   9. +-commutative 38.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.im\right)\right) \]

3. Simplified 68.8%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]
4. Step-by-step derivation

   1. hypot-udef 38.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} \cdot y.im\right)\right) \]

   2. add-cube-cbrt 39.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im} \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right) \cdot \sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}}\right)\right) \]

   3. pow3 39.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im}\right)}^{3}}\right)\right) \]

   4. *-commutative 39.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}}\right)}^{3}\right)\right) \]

   5. hypot-udef 69.4%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, {\left(\sqrt[3]{y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right)}^{3}\right)\right) \]

5. Applied egg-rr 69.4%

   \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{{\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{3}}\right)\right) \]

6. Taylor expanded in y.re around 0 33.1%

   \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left({1}^{0.3333333333333333} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)} \]
7. Step-by-step derivation

   1. pow-base-1 33.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{1} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

   2. associate-*r* 33.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   3. unpow2 33.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

   4. unpow2 33.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

   5. hypot-def 58.8%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\left(1 \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

   6. *-lft-identity 58.8%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

8. Simplified 58.8%

   \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

9. Taylor expanded in y.im around 0 35.7%

   \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
10. Step-by-step derivation

    1. unpow2 35.7%

       \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

    2. unpow2 35.7%

       \[\leadsto {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

    3. hypot-def 36.7%

       \[\leadsto {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

    4. unpow2 36.7%

       \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)\right) \]

    5. unpow2 36.7%

       \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)\right) \]

    6. hypot-def 54.0%

       \[\leadsto {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right) \]

11. Simplified 54.0%

    \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)} \]

12. Taylor expanded in x.re around 0 18.3%

    \[\leadsto \color{blue}{y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right)} \]

13. Final simplification 18.3%

    \[\leadsto y.im \cdot \left(\log x.im \cdot {x.im}^{y.re}\right) \]

Reproduce

herbie shell --seed 2023279 
(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)))))