powComplex, imaginary part

Percentage Accurate: 39.2% → 79.8%

Time: 39.8s

Alternatives: 20

Speedup: 2.7×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 39.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.8% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	return exp(((t_0 * y_46_re) - (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))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(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)))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - N[(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]]

Derivation

1. Initial program 40.7%

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

3. Simplified 82.9%

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

4. Final simplification 82.9%

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

Alternative 2: 71.5% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (* (exp (- (* y.re t_2) t_0)) (sin (+ t_1 (* y.im t_2))))))
   (if (<= t_3 INFINITY)
     t_3
     (* (exp (- (* (log (hypot x.re x.im)) y.re) t_0)) (sin t_1)))))

C

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

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = Math.exp(((y_46_re * t_2) - t_0)) * Math.sin((t_1 + (y_46_im * t_2)));
	double tmp;
	if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = Math.exp(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * Math.sin(t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	t_3 = math.exp(((y_46_re * t_2) - t_0)) * math.sin((t_1 + (y_46_im * t_2)))
	tmp = 0
	if t_3 <= math.inf:
		tmp = t_3
	else:
		tmp = math.exp(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * math.sin(t_1)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = Float64(exp(Float64(Float64(y_46_re * t_2) - t_0)) * sin(Float64(t_1 + Float64(y_46_im * t_2))))
	tmp = 0.0
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(exp(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * sin(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	t_3 = exp(((y_46_re * t_2) - t_0)) * sin((t_1 + (y_46_im * t_2)));
	tmp = 0.0;
	if (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = exp(((log(hypot(x_46_re, x_46_im)) * y_46_re) - t_0)) * sin(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 83.5%

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

   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

   3. Simplified 82.3%

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

   4. Taylor expanded in y.im around 0 65.5%

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

4. Final simplification 74.3%

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

Alternative 3: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq -2600000 \lor \neg \left(y.re \leq 0.122\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)}{e^{t_0}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (or (<= y.re -2600000.0) (not (<= y.re 0.122)))
     (*
      (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
      (sin (* y.im (log (hypot x.im x.re)))))
     (/
      (*
       (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))))
      (exp t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if ((y_46_re <= -2600000.0) || !(y_46_re <= 0.122)) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} 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)))) / exp(t_0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2600000.0) || !(y_46_re <= 0.122))
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = Float64(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)))) / exp(t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -2600000.0], N[Not[LessEqual[y$46$re, 0.122]], $MachinePrecision]], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(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] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.6e6 or 0.122 < y.re

   1. Initial program 40.7%

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

   2. Taylor expanded in y.re around 0 44.3%

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

      1. unpow2 44.3%

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

      2. unpow2 44.3%

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

      3. hypot-def 81.6%

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

   4. Simplified 81.6%

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

   if -2.6e6 < y.re < 0.122

   1. Initial program 40.8%

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

      1. exp-diff 40.8%

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

      2. +-rgt-identity 40.8%

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

      3. +-rgt-identity 40.8%

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

      4. exp-to-pow 40.8%

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

      5. hypot-def 40.8%

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

      6. *-commutative 40.8%

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

      7. exp-prod 40.5%

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

      8. +-commutative 40.5%

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

      9. *-commutative 40.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in y.re around inf 82.0%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2600000 \lor \neg \left(y.re \leq 0.122\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array} \]

Alternative 4: 70.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := e^{t_2 \cdot y.re - t_0}\\ \mathbf{if}\;x.re \leq -1.06 \cdot 10^{-211}:\\ \;\;\;\;t_3 \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(-x.re\right)\right)\right)\\ \mathbf{elif}\;x.re \leq -8.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sin \left(t_2 \cdot y.im\right)}{e^{t_0}}\\ \mathbf{elif}\;x.re \leq 1.6 \cdot 10^{-288}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 7 \cdot 10^{-27}:\\ \;\;\;\;t_3 \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(t_1 + y.im \cdot \log x.re\right)}{e^{t_0 - y.re \cdot \log x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (exp (- (* t_2 y.re) t_0))))
   (if (<= x.re -1.06e-211)
     (* t_3 (sin (fma y.re (atan2 x.im x.re) (* y.im (log (- x.re))))))
     (if (<= x.re -8.6e-289)
       (/ (sin (* t_2 y.im)) (exp t_0))
       (if (<= x.re 1.6e-288)
         (* (sin (pow (cbrt t_1) 3.0)) (pow (hypot x.im x.re) y.re))
         (if (<= x.re 7e-27)
           (* t_3 (sin t_1))
           (/
            (sin (+ t_1 (* y.im (log x.re))))
            (exp (- t_0 (* y.re (log x.re)))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double tmp;
	if (x_46_re <= -1.06e-211) {
		tmp = t_3 * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * log(-x_46_re))));
	} else if (x_46_re <= -8.6e-289) {
		tmp = sin((t_2 * y_46_im)) / exp(t_0);
	} else if (x_46_re <= 1.6e-288) {
		tmp = sin(pow(cbrt(t_1), 3.0)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (x_46_re <= 7e-27) {
		tmp = t_3 * sin(t_1);
	} else {
		tmp = sin((t_1 + (y_46_im * log(x_46_re)))) / exp((t_0 - (y_46_re * log(x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	tmp = 0.0
	if (x_46_re <= -1.06e-211)
		tmp = Float64(t_3 * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * log(Float64(-x_46_re))))));
	elseif (x_46_re <= -8.6e-289)
		tmp = Float64(sin(Float64(t_2 * y_46_im)) / exp(t_0));
	elseif (x_46_re <= 1.6e-288)
		tmp = Float64(sin((cbrt(t_1) ^ 3.0)) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (x_46_re <= 7e-27)
		tmp = Float64(t_3 * sin(t_1));
	else
		tmp = Float64(sin(Float64(t_1 + Float64(y_46_im * log(x_46_re)))) / exp(Float64(t_0 - Float64(y_46_re * log(x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.06e-211], N[(t$95$3 * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -8.6e-289], N[(N[Sin[N[(t$95$2 * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.6e-288], N[(N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7e-27], N[(t$95$3 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$0 - N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.re < -1.0600000000000001e-211

   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

   3. Simplified 83.8%

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

   4. Taylor expanded in x.re around -inf 82.2%

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

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. *-commutative 82.2%

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

      4. unsub-neg 82.2%

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

      5. *-commutative 82.2%

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

   6. Simplified 82.2%

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

      1. expm1-log1p-u 82.2%

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

      2. expm1-udef 61.2%

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

      3. fma-neg 61.2%

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

      4. distribute-rgt-neg-in 61.2%

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

      5. neg-log 61.2%

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

      6. clear-num 61.2%

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

      7. div-inv 61.2%

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

      8. metadata-eval 61.2%

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

   8. Applied egg-rr 61.2%

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

      1. expm1-def 82.3%

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

      2. expm1-log1p 82.3%

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

      3. *-commutative 82.3%

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

      4. mul-1-neg 82.3%

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

   10. Simplified 82.3%

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

   if -1.0600000000000001e-211 < x.re < -8.5999999999999995e-289

   1. Initial program 62.3%

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

      1. exp-diff 62.3%

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

      2. +-rgt-identity 62.3%

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

      3. +-rgt-identity 62.3%

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

      4. exp-to-pow 62.3%

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

      5. hypot-def 62.3%

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

      6. *-commutative 62.3%

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

      7. exp-prod 62.3%

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

      8. +-commutative 62.3%

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

      9. *-commutative 62.3%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in y.re around 0 51.3%

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

      1. +-commutative 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

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

   6. Simplified 51.3%

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

      1. hypot-udef 82.4%

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

      2. expm1-log1p-u 82.4%

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

      3. expm1-udef 44.7%

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

   8. Applied egg-rr 44.7%

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

      1. expm1-def 82.4%

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

      2. expm1-log1p 82.4%

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

   10. Simplified 82.4%

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

   if -8.5999999999999995e-289 < x.re < 1.6e-288

   1. Initial program 42.9%

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

      1. exp-diff 28.6%

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

      2. +-rgt-identity 28.6%

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

      3. +-rgt-identity 28.6%

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

      4. exp-to-pow 28.6%

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

      5. hypot-def 28.6%

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

      6. *-commutative 28.6%

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

      7. exp-prod 28.6%

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

      8. +-commutative 28.6%

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

      9. *-commutative 28.6%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in y.im around 0 43.8%

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

      1. unpow2 43.8%

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

      2. unpow2 43.8%

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

      3. hypot-def 57.4%

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

   6. Simplified 57.4%

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

      1. *-commutative 43.3%

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

      2. add-cube-cbrt 71.9%

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

      3. pow3 71.9%

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

   8. Applied egg-rr 85.5%

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

   if 1.6e-288 < x.re < 7.0000000000000003e-27

   1. Initial program 56.8%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in y.im around 0 74.0%

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

   if 7.0000000000000003e-27 < x.re

   1. Initial program 24.5%

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

      1. exp-diff 22.9%

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

      2. +-rgt-identity 22.9%

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

      3. +-rgt-identity 22.9%

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

      4. exp-to-pow 22.9%

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

      5. hypot-def 22.9%

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

      6. *-commutative 22.9%

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

      7. exp-prod 22.4%

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

      8. +-commutative 22.4%

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

      9. *-commutative 22.4%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in x.re around inf 68.8%

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

      1. associate-/l* 68.8%

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

      2. +-commutative 68.8%

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

      3. mul-1-neg 68.8%

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

      4. unsub-neg 68.8%

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

      5. *-commutative 68.8%

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

      6. log-rec 68.8%

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

      7. *-commutative 68.8%

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

      8. *-commutative 68.8%

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

      9. div-exp 73.6%

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

      10. *-commutative 73.6%

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

   6. Simplified 73.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.06 \cdot 10^{-211}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(-x.re\right)\right)\right)\\ \mathbf{elif}\;x.re \leq -8.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;x.re \leq 1.6 \cdot 10^{-288}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 7 \cdot 10^{-27}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log x.re}}\\ \end{array} \]

Alternative 5: 70.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-106}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \left(t_1 \cdot y.im\right)}{e^{t_0}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)) (t_1 (log (hypot x.re x.im))))
   (if (<= y.re -2.1e-106)
     (* (exp (- (* t_1 y.re) t_0)) (sin (* y.re (atan2 x.im x.re))))
     (if (<= y.re 7.2e-6)
       (/ (sin (* t_1 y.im)) (exp t_0))
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
        (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 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -2.1e-106) {
		tmp = exp(((t_1 * y_46_re) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 7.2e-6) {
		tmp = sin((t_1 * y_46_im)) / exp(t_0);
	} else {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.log(Math.hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -2.1e-106) {
		tmp = Math.exp(((t_1 * y_46_re) - t_0)) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 7.2e-6) {
		tmp = Math.sin((t_1 * y_46_im)) / Math.exp(t_0);
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.log(math.hypot(x_46_re, x_46_im))
	tmp = 0
	if y_46_re <= -2.1e-106:
		tmp = math.exp(((t_1 * y_46_re) - t_0)) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 7.2e-6:
		tmp = math.sin((t_1 * y_46_im)) / math.exp(t_0)
	else:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * math.sin((y_46_im * math.log(math.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 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= -2.1e-106)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_0)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 7.2e-6)
		tmp = Float64(sin(Float64(t_1 * y_46_im)) / exp(t_0));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = log(hypot(x_46_re, x_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.1e-106)
		tmp = exp(((t_1 * y_46_re) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 7.2e-6)
		tmp = sin((t_1 * y_46_im)) / exp(t_0);
	else
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e-106], N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e-6], N[(N[Sin[N[(t$95$1 * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[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 3 regimes

2. if y.re < -2.10000000000000003e-106

   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

   3. Simplified 90.2%

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

   4. Taylor expanded in y.im around 0 81.9%

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

   if -2.10000000000000003e-106 < y.re < 7.19999999999999967e-6

   1. Initial program 37.4%

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

      1. exp-diff 37.4%

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

      2. +-rgt-identity 37.4%

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

      3. +-rgt-identity 37.4%

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

      4. exp-to-pow 37.4%

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

      5. hypot-def 37.4%

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

      6. *-commutative 37.4%

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

      7. exp-prod 37.0%

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

      8. +-commutative 37.0%

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

      9. *-commutative 37.0%

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

   3. Simplified 76.9%

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

   4. Taylor expanded in y.re around 0 33.1%

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

      1. +-commutative 33.1%

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

      2. unpow2 33.1%

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

      3. unpow2 33.1%

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

   6. Simplified 33.1%

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

      1. hypot-udef 63.4%

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

      2. expm1-log1p-u 63.4%

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

      3. expm1-udef 47.2%

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

   8. Applied egg-rr 47.2%

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

      1. expm1-def 63.4%

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

      2. expm1-log1p 63.4%

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

   10. Simplified 63.4%

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

   if 7.19999999999999967e-6 < y.re

   1. Initial program 43.2%

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

   2. Taylor expanded in y.re around 0 45.8%

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

      1. unpow2 45.8%

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

      2. unpow2 45.8%

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

      3. hypot-def 74.3%

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

   4. Simplified 74.3%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-106}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 6: 70.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := e^{t_2 \cdot y.re - t_0}\\ \mathbf{if}\;x.re \leq -1.06 \cdot 10^{-211}:\\ \;\;\;\;t_3 \cdot \sin \left(t_1 - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{elif}\;x.re \leq -9.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sin \left(t_2 \cdot y.im\right)}{e^{t_0}}\\ \mathbf{elif}\;x.re \leq 1.62 \cdot 10^{-288}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;t_3 \cdot \sin t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(t_1 + y.im \cdot \log x.re\right)}{e^{t_0 - y.re \cdot \log x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (exp (- (* t_2 y.re) t_0))))
   (if (<= x.re -1.06e-211)
     (* t_3 (sin (- t_1 (* y.im (log (/ -1.0 x.re))))))
     (if (<= x.re -9.6e-289)
       (/ (sin (* t_2 y.im)) (exp t_0))
       (if (<= x.re 1.62e-288)
         (* (sin (pow (cbrt t_1) 3.0)) (pow (hypot x.im x.re) y.re))
         (if (<= x.re 1.6e-26)
           (* t_3 (sin t_1))
           (/
            (sin (+ t_1 (* y.im (log x.re))))
            (exp (- t_0 (* y.re (log x.re)))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double tmp;
	if (x_46_re <= -1.06e-211) {
		tmp = t_3 * sin((t_1 - (y_46_im * log((-1.0 / x_46_re)))));
	} else if (x_46_re <= -9.6e-289) {
		tmp = sin((t_2 * y_46_im)) / exp(t_0);
	} else if (x_46_re <= 1.62e-288) {
		tmp = sin(pow(cbrt(t_1), 3.0)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (x_46_re <= 1.6e-26) {
		tmp = t_3 * sin(t_1);
	} else {
		tmp = sin((t_1 + (y_46_im * log(x_46_re)))) / exp((t_0 - (y_46_re * log(x_46_re))));
	}
	return tmp;
}

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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log(Math.hypot(x_46_re, x_46_im));
	double t_3 = Math.exp(((t_2 * y_46_re) - t_0));
	double tmp;
	if (x_46_re <= -1.06e-211) {
		tmp = t_3 * Math.sin((t_1 - (y_46_im * Math.log((-1.0 / x_46_re)))));
	} else if (x_46_re <= -9.6e-289) {
		tmp = Math.sin((t_2 * y_46_im)) / Math.exp(t_0);
	} else if (x_46_re <= 1.62e-288) {
		tmp = Math.sin(Math.pow(Math.cbrt(t_1), 3.0)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (x_46_re <= 1.6e-26) {
		tmp = t_3 * Math.sin(t_1);
	} else {
		tmp = Math.sin((t_1 + (y_46_im * Math.log(x_46_re)))) / Math.exp((t_0 - (y_46_re * Math.log(x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	tmp = 0.0
	if (x_46_re <= -1.06e-211)
		tmp = Float64(t_3 * sin(Float64(t_1 - Float64(y_46_im * log(Float64(-1.0 / x_46_re))))));
	elseif (x_46_re <= -9.6e-289)
		tmp = Float64(sin(Float64(t_2 * y_46_im)) / exp(t_0));
	elseif (x_46_re <= 1.62e-288)
		tmp = Float64(sin((cbrt(t_1) ^ 3.0)) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (x_46_re <= 1.6e-26)
		tmp = Float64(t_3 * sin(t_1));
	else
		tmp = Float64(sin(Float64(t_1 + Float64(y_46_im * log(x_46_re)))) / exp(Float64(t_0 - Float64(y_46_re * log(x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.06e-211], N[(t$95$3 * N[Sin[N[(t$95$1 - N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -9.6e-289], N[(N[Sin[N[(t$95$2 * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.62e-288], N[(N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.6e-26], N[(t$95$3 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$0 - N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.re < -1.0600000000000001e-211

   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

   3. Simplified 83.8%

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

   4. Taylor expanded in x.re around -inf 82.2%

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

      1. +-commutative 82.2%

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

      2. mul-1-neg 82.2%

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

      3. *-commutative 82.2%

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

      4. unsub-neg 82.2%

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

      5. *-commutative 82.2%

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

   6. Simplified 82.2%

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

   if -1.0600000000000001e-211 < x.re < -9.59999999999999975e-289

   1. Initial program 62.3%

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

      1. exp-diff 62.3%

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

      2. +-rgt-identity 62.3%

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

      3. +-rgt-identity 62.3%

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

      4. exp-to-pow 62.3%

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

      5. hypot-def 62.3%

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

      6. *-commutative 62.3%

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

      7. exp-prod 62.3%

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

      8. +-commutative 62.3%

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

      9. *-commutative 62.3%

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

   3. Simplified 80.9%

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

   4. Taylor expanded in y.re around 0 51.3%

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

      1. +-commutative 51.3%

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

      2. unpow2 51.3%

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

      3. unpow2 51.3%

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

   6. Simplified 51.3%

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

      1. hypot-udef 82.4%

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

      2. expm1-log1p-u 82.4%

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

      3. expm1-udef 44.7%

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

   8. Applied egg-rr 44.7%

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

      1. expm1-def 82.4%

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

      2. expm1-log1p 82.4%

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

   10. Simplified 82.4%

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

   if -9.59999999999999975e-289 < x.re < 1.6200000000000001e-288

   1. Initial program 42.9%

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

      1. exp-diff 28.6%

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

      2. +-rgt-identity 28.6%

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

      3. +-rgt-identity 28.6%

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

      4. exp-to-pow 28.6%

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

      5. hypot-def 28.6%

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

      6. *-commutative 28.6%

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

      7. exp-prod 28.6%

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

      8. +-commutative 28.6%

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

      9. *-commutative 28.6%

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

   3. Simplified 56.9%

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

   4. Taylor expanded in y.im around 0 43.8%

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

      1. unpow2 43.8%

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

      2. unpow2 43.8%

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

      3. hypot-def 57.4%

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

   6. Simplified 57.4%

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

      1. *-commutative 43.3%

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

      2. add-cube-cbrt 71.9%

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

      3. pow3 71.9%

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

   8. Applied egg-rr 85.5%

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

   if 1.6200000000000001e-288 < x.re < 1.6000000000000001e-26

   1. Initial program 56.8%

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

   3. Simplified 86.1%

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

   4. Taylor expanded in y.im around 0 74.0%

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

   if 1.6000000000000001e-26 < x.re

   1. Initial program 24.5%

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

      1. exp-diff 22.9%

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

      2. +-rgt-identity 22.9%

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

      3. +-rgt-identity 22.9%

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

      4. exp-to-pow 22.9%

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

      5. hypot-def 22.9%

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

      6. *-commutative 22.9%

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

      7. exp-prod 22.4%

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

      8. +-commutative 22.4%

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

      9. *-commutative 22.4%

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

   3. Simplified 71.0%

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

   4. Taylor expanded in x.re around inf 68.8%

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

      1. associate-/l* 68.8%

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

      2. +-commutative 68.8%

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

      3. mul-1-neg 68.8%

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

      4. unsub-neg 68.8%

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

      5. *-commutative 68.8%

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

      6. log-rec 68.8%

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

      7. *-commutative 68.8%

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

      8. *-commutative 68.8%

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

      9. div-exp 73.6%

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

      10. *-commutative 73.6%

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

   6. Simplified 73.6%

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

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.06 \cdot 10^{-211}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{elif}\;x.re \leq -9.6 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;x.re \leq 1.62 \cdot 10^{-288}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 1.6 \cdot 10^{-26}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - y.re \cdot \log x.re}}\\ \end{array} \]

Alternative 7: 70.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := e^{t_2 \cdot y.re - t_0} \cdot \sin t_1\\ \mathbf{if}\;y.re \leq -7.4 \cdot 10^{-106}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sin \left(t_2 \cdot y.im\right)}{e^{t_0}}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{+196} \lor \neg \left(y.re \leq 7.8 \cdot 10^{+279}\right):\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|t_1\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (* (exp (- (* t_2 y.re) t_0)) (sin t_1))))
   (if (<= y.re -7.4e-106)
     t_3
     (if (<= y.re 5.2e-24)
       (/ (sin (* t_2 y.im)) (exp t_0))
       (if (or (<= y.re 4.9e+196) (not (<= y.re 7.8e+279)))
         t_3
         (* (sin (fabs t_1)) (pow x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = exp(((t_2 * y_46_re) - t_0)) * sin(t_1);
	double tmp;
	if (y_46_re <= -7.4e-106) {
		tmp = t_3;
	} else if (y_46_re <= 5.2e-24) {
		tmp = sin((t_2 * y_46_im)) / exp(t_0);
	} else if ((y_46_re <= 4.9e+196) || !(y_46_re <= 7.8e+279)) {
		tmp = t_3;
	} else {
		tmp = sin(fabs(t_1)) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log(Math.hypot(x_46_re, x_46_im));
	double t_3 = Math.exp(((t_2 * y_46_re) - t_0)) * Math.sin(t_1);
	double tmp;
	if (y_46_re <= -7.4e-106) {
		tmp = t_3;
	} else if (y_46_re <= 5.2e-24) {
		tmp = Math.sin((t_2 * y_46_im)) / Math.exp(t_0);
	} else if ((y_46_re <= 4.9e+196) || !(y_46_re <= 7.8e+279)) {
		tmp = t_3;
	} else {
		tmp = Math.sin(Math.abs(t_1)) * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.log(math.hypot(x_46_re, x_46_im))
	t_3 = math.exp(((t_2 * y_46_re) - t_0)) * math.sin(t_1)
	tmp = 0
	if y_46_re <= -7.4e-106:
		tmp = t_3
	elif y_46_re <= 5.2e-24:
		tmp = math.sin((t_2 * y_46_im)) / math.exp(t_0)
	elif (y_46_re <= 4.9e+196) or not (y_46_re <= 7.8e+279):
		tmp = t_3
	else:
		tmp = math.sin(math.fabs(t_1)) * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(t_1))
	tmp = 0.0
	if (y_46_re <= -7.4e-106)
		tmp = t_3;
	elseif (y_46_re <= 5.2e-24)
		tmp = Float64(sin(Float64(t_2 * y_46_im)) / exp(t_0));
	elseif ((y_46_re <= 4.9e+196) || !(y_46_re <= 7.8e+279))
		tmp = t_3;
	else
		tmp = Float64(sin(abs(t_1)) * (x_46_im ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log(hypot(x_46_re, x_46_im));
	t_3 = exp(((t_2 * y_46_re) - t_0)) * sin(t_1);
	tmp = 0.0;
	if (y_46_re <= -7.4e-106)
		tmp = t_3;
	elseif (y_46_re <= 5.2e-24)
		tmp = sin((t_2 * y_46_im)) / exp(t_0);
	elseif ((y_46_re <= 4.9e+196) || ~((y_46_re <= 7.8e+279)))
		tmp = t_3;
	else
		tmp = sin(abs(t_1)) * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.4e-106], t$95$3, If[LessEqual[y$46$re, 5.2e-24], N[(N[Sin[N[(t$95$2 * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 4.9e+196], N[Not[LessEqual[y$46$re, 7.8e+279]], $MachinePrecision]], t$95$3, N[(N[Sin[N[Abs[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.39999999999999959e-106 or 5.2e-24 < y.re < 4.8999999999999999e196 or 7.80000000000000038e279 < y.re

   1. Initial program 41.3%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in y.im around 0 79.1%

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

   if -7.39999999999999959e-106 < y.re < 5.2e-24

   1. Initial program 39.2%

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

      1. exp-diff 39.2%

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

      2. +-rgt-identity 39.2%

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

      3. +-rgt-identity 39.2%

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

      4. exp-to-pow 39.2%

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

      5. hypot-def 39.2%

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

      6. *-commutative 39.2%

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

      7. exp-prod 38.8%

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

      8. +-commutative 38.8%

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

      9. *-commutative 38.8%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in y.re around 0 34.8%

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

      1. +-commutative 34.8%

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

      2. unpow2 34.8%

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

      3. unpow2 34.8%

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

   6. Simplified 34.8%

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

      1. hypot-udef 64.1%

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

      2. expm1-log1p-u 64.1%

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

      3. expm1-udef 47.0%

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

   8. Applied egg-rr 47.0%

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

      1. expm1-def 64.1%

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

      2. expm1-log1p 64.1%

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

   10. Simplified 64.1%

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

   if 4.8999999999999999e196 < y.re < 7.80000000000000038e279

   1. Initial program 42.9%

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

      1. exp-diff 42.9%

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

      2. +-rgt-identity 42.9%

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

      3. +-rgt-identity 42.9%

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

      4. exp-to-pow 42.9%

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

      5. hypot-def 42.9%

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

      6. *-commutative 42.9%

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

      7. exp-prod 42.9%

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

      8. +-commutative 42.9%

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

      9. *-commutative 42.9%

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

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

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

      1. unpow2 57.1%

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

      2. unpow2 57.1%

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

      3. hypot-def 57.1%

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

   6. Simplified 57.1%

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

   7. Taylor expanded in x.re around 0 52.6%

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

      1. *-commutative 52.6%

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

      2. add-sqr-sqrt 24.0%

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

      3. sqrt-unprod 19.2%

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

      4. pow2 19.2%

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

   9. Applied egg-rr 19.2%

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

       1. *-commutative 19.2%

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

       2. unpow2 19.2%

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

       3. rem-sqrt-square 76.4%

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

   11. Simplified 76.4%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.4 \cdot 10^{-106}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{+196} \lor \neg \left(y.re \leq 7.8 \cdot 10^{+279}\right):\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 8: 61.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -4.7 \cdot 10^{-39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t_2 \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot y.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot t_2\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+213}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|t_0\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* t_0 (pow (hypot x.im x.re) y.re)))
        (t_2 (log (hypot x.im x.re))))
   (if (<= y.re -4.7e-39)
     t_1
     (if (<= y.re -4.3e-82)
       (/
        (sin (* (log (hypot x.re x.im)) y.im))
        (exp (* (atan2 x.im x.re) y.im)))
       (if (<= y.re -7.8e-106)
         (fma
          y.re
          (atan2 x.im x.re)
          (* t_2 (* (atan2 x.im x.re) (* y.re y.re))))
         (if (<= y.re 1.1e+62)
           (/ (sin (* y.im t_2)) (pow (exp y.im) (atan2 x.im x.re)))
           (if (<= y.re 3.45e+133)
             t_1
             (if (<= y.re 2.4e+213)
               (* (sin (pow (cbrt t_0) 3.0)) (pow x.im y.re))
               (* (sin (fabs t_0)) (pow x.im y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -4.7e-39) {
		tmp = t_1;
	} else if (y_46_re <= -4.3e-82) {
		tmp = sin((log(hypot(x_46_re, x_46_im)) * y_46_im)) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else if (y_46_re <= -7.8e-106) {
		tmp = fma(y_46_re, atan2(x_46_im, x_46_re), (t_2 * (atan2(x_46_im, x_46_re) * (y_46_re * y_46_re))));
	} else if (y_46_re <= 1.1e+62) {
		tmp = sin((y_46_im * t_2)) / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 3.45e+133) {
		tmp = t_1;
	} else if (y_46_re <= 2.4e+213) {
		tmp = sin(pow(cbrt(t_0), 3.0)) * pow(x_46_im, y_46_re);
	} else {
		tmp = sin(fabs(t_0)) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	t_2 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -4.7e-39)
		tmp = t_1;
	elseif (y_46_re <= -4.3e-82)
		tmp = Float64(sin(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im)) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= -7.8e-106)
		tmp = fma(y_46_re, atan(x_46_im, x_46_re), Float64(t_2 * Float64(atan(x_46_im, x_46_re) * Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 1.1e+62)
		tmp = Float64(sin(Float64(y_46_im * t_2)) / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 3.45e+133)
		tmp = t_1;
	elseif (y_46_re <= 2.4e+213)
		tmp = Float64(sin((cbrt(t_0) ^ 3.0)) * (x_46_im ^ y_46_re));
	else
		tmp = Float64(sin(abs(t_0)) * (x_46_im ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -4.7e-39], t$95$1, If[LessEqual[y$46$re, -4.3e-82], N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -7.8e-106], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(t$95$2 * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.1e+62], N[(N[Sin[N[(y$46$im * t$95$2), $MachinePrecision]], $MachinePrecision] / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.45e+133], t$95$1, If[LessEqual[y$46$re, 2.4e+213], N[(N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.re < -4.7000000000000002e-39 or 1.10000000000000007e62 < y.re < 3.4500000000000001e133

   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 31.5%

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

      2. +-rgt-identity 31.5%

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

      3. +-rgt-identity 31.5%

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

      4. exp-to-pow 31.5%

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

      5. hypot-def 31.5%

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

      6. *-commutative 31.5%

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

      7. exp-prod 31.5%

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

      8. +-commutative 31.5%

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

      9. *-commutative 31.5%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in y.im around 0 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. hypot-def 75.4%

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

   6. Simplified 75.4%

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

      1. *-commutative 75.4%

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

      2. add-exp-log 43.7%

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

   8. Applied egg-rr 43.7%

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

   9. Taylor expanded in y.re around 0 84.3%

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

   if -4.7000000000000002e-39 < y.re < -4.30000000000000019e-82

   1. Initial program 33.2%

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

      1. exp-diff 33.2%

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

      2. +-rgt-identity 33.2%

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

      3. +-rgt-identity 33.2%

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

      4. exp-to-pow 33.2%

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

      5. hypot-def 33.2%

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

      6. *-commutative 33.2%

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

      7. exp-prod 33.2%

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

      8. +-commutative 33.2%

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

      9. *-commutative 33.2%

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

   3. Simplified 89.1%

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

   4. Taylor expanded in y.re around 0 22.3%

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

      1. +-commutative 22.3%

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

      2. unpow2 22.3%

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

      3. unpow2 22.3%

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

   6. Simplified 22.3%

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

      1. hypot-udef 69.1%

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

      2. expm1-log1p-u 69.1%

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

      3. expm1-udef 58.1%

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

   8. Applied egg-rr 58.1%

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

      1. expm1-def 69.1%

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

      2. expm1-log1p 69.1%

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

   10. Simplified 69.1%

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

   if -4.30000000000000019e-82 < y.re < -7.80000000000000019e-106

   1. Initial program 63.4%

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

      1. exp-diff 63.4%

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

      2. +-rgt-identity 63.4%

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

      3. +-rgt-identity 63.4%

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

      4. exp-to-pow 63.4%

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

      5. hypot-def 63.4%

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

      6. *-commutative 63.4%

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

      7. exp-prod 63.4%

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

      8. +-commutative 63.4%

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

      9. *-commutative 63.4%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y.im around 0 51.8%

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

      1. unpow2 51.8%

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

      2. unpow2 51.8%

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

      3. hypot-def 70.3%

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

   6. Simplified 70.3%

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

      1. *-commutative 70.3%

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

      2. add-exp-log 27.0%

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

   8. Applied egg-rr 27.0%

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

   9. Taylor expanded in y.re around 0 50.8%

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

       1. fma-def 50.8%

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

       2. associate-*r* 50.8%

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

       3. unpow2 50.8%

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

       4. unpow2 50.8%

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

       5. unpow2 50.8%

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

       6. hypot-def 70.3%

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

   11. Simplified 70.3%

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

   if -7.80000000000000019e-106 < y.re < 1.10000000000000007e62

   1. Initial program 38.6%

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

      1. exp-diff 37.6%

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

      2. +-rgt-identity 37.6%

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

      3. +-rgt-identity 37.6%

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

      4. exp-to-pow 37.6%

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

      5. hypot-def 37.6%

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

      6. *-commutative 37.6%

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

      7. exp-prod 37.3%

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

      8. +-commutative 37.3%

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

      9. *-commutative 37.3%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in y.re around 0 31.4%

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

      1. unpow2 31.4%

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

      2. unpow2 31.4%

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

      3. hypot-def 60.5%

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

      4. *-commutative 60.5%

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

      5. exp-prod 60.9%

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

   6. Simplified 60.9%

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

   if 3.4500000000000001e133 < y.re < 2.4e213

   1. Initial program 47.8%

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

      1. exp-diff 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. +-rgt-identity 39.1%

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

      3. +-rgt-identity 39.1%

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

      4. exp-to-pow 39.1%

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

      5. hypot-def 39.1%

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

      6. *-commutative 39.1%

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

      7. exp-prod 39.1%

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

      8. +-commutative 39.1%

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

      9. *-commutative 39.1%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in y.im around 0 73.9%

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

      1. unpow2 73.9%

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

      2. unpow2 73.9%

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

      3. hypot-def 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in x.re around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. add-cube-cbrt 82.7%

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

      3. pow3 82.7%

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

   9. Applied egg-rr 82.7%

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

   if 2.4e213 < y.re

   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. Step-by-step derivation

      1. exp-diff 45.5%

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

      2. +-rgt-identity 45.5%

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

      3. +-rgt-identity 45.5%

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

      4. exp-to-pow 45.5%

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

      5. hypot-def 45.5%

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

      6. *-commutative 45.5%

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

      7. exp-prod 45.5%

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

      8. +-commutative 45.5%

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

      9. *-commutative 45.5%

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

   3. Simplified 72.7%

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

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

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

      1. unpow2 63.7%

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

      2. unpow2 63.7%

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

      3. hypot-def 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in x.re around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. add-sqr-sqrt 22.9%

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

      3. sqrt-unprod 9.2%

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

      4. pow2 9.2%

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

   9. Applied egg-rr 9.2%

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

       1. *-commutative 9.2%

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

       2. unpow2 9.2%

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

       3. rem-sqrt-square 73.0%

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

   11. Simplified 73.0%

       \[\leadsto \sin \color{blue}{\left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)} \cdot {x.im}^{y.re} \]
3. Recombined 6 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{-39}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -4.3 \cdot 10^{-82}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot y.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.45 \cdot 10^{+133}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+213}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 9: 62.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+211}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|t_1\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (sin (* (log (hypot x.re x.im)) y.im))
          (exp (* (atan2 x.im x.re) y.im))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* t_1 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -1.75e-39)
     t_2
     (if (<= y.re -2.2e-80)
       t_0
       (if (<= y.re -7.8e-106)
         t_1
         (if (<= y.re 4e+62)
           t_0
           (if (<= y.re 3.8e+134)
             t_2
             (if (<= y.re 6.2e+211)
               (* (sin (pow (cbrt t_1) 3.0)) (pow x.im y.re))
               (* (sin (fabs t_1)) (pow x.im y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((log(hypot(x_46_re, x_46_im)) * y_46_im)) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.75e-39) {
		tmp = t_2;
	} else if (y_46_re <= -2.2e-80) {
		tmp = t_0;
	} else if (y_46_re <= -7.8e-106) {
		tmp = t_1;
	} else if (y_46_re <= 4e+62) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e+134) {
		tmp = t_2;
	} else if (y_46_re <= 6.2e+211) {
		tmp = sin(pow(cbrt(t_1), 3.0)) * pow(x_46_im, y_46_re);
	} else {
		tmp = sin(fabs(t_1)) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im)) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.75e-39) {
		tmp = t_2;
	} else if (y_46_re <= -2.2e-80) {
		tmp = t_0;
	} else if (y_46_re <= -7.8e-106) {
		tmp = t_1;
	} else if (y_46_re <= 4e+62) {
		tmp = t_0;
	} else if (y_46_re <= 3.8e+134) {
		tmp = t_2;
	} else if (y_46_re <= 6.2e+211) {
		tmp = Math.sin(Math.pow(Math.cbrt(t_1), 3.0)) * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = Math.sin(Math.abs(t_1)) * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im)) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.75e-39)
		tmp = t_2;
	elseif (y_46_re <= -2.2e-80)
		tmp = t_0;
	elseif (y_46_re <= -7.8e-106)
		tmp = t_1;
	elseif (y_46_re <= 4e+62)
		tmp = t_0;
	elseif (y_46_re <= 3.8e+134)
		tmp = t_2;
	elseif (y_46_re <= 6.2e+211)
		tmp = Float64(sin((cbrt(t_1) ^ 3.0)) * (x_46_im ^ y_46_re));
	else
		tmp = Float64(sin(abs(t_1)) * (x_46_im ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.75e-39], t$95$2, If[LessEqual[y$46$re, -2.2e-80], t$95$0, If[LessEqual[y$46$re, -7.8e-106], t$95$1, If[LessEqual[y$46$re, 4e+62], t$95$0, If[LessEqual[y$46$re, 3.8e+134], t$95$2, If[LessEqual[y$46$re, 6.2e+211], N[(N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[Abs[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.75e-39 or 4.00000000000000014e62 < y.re < 3.79999999999999998e134

   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 31.5%

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

      2. +-rgt-identity 31.5%

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

      3. +-rgt-identity 31.5%

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

      4. exp-to-pow 31.5%

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

      5. hypot-def 31.5%

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

      6. *-commutative 31.5%

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

      7. exp-prod 31.5%

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

      8. +-commutative 31.5%

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

      9. *-commutative 31.5%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in y.im around 0 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. hypot-def 75.4%

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

   6. Simplified 75.4%

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

      1. *-commutative 75.4%

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

      2. add-exp-log 43.7%

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

   8. Applied egg-rr 43.7%

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

   9. Taylor expanded in y.re around 0 84.3%

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

   if -1.75e-39 < y.re < -2.2000000000000001e-80 or -7.80000000000000019e-106 < y.re < 4.00000000000000014e62

   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 37.2%

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

      2. +-rgt-identity 37.2%

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

      3. +-rgt-identity 37.2%

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

      4. exp-to-pow 37.2%

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

      5. hypot-def 37.2%

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

      6. *-commutative 37.2%

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

      7. exp-prod 37.0%

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

      8. +-commutative 37.0%

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

      9. *-commutative 37.0%

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

   3. Simplified 77.1%

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

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

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

      1. +-commutative 30.7%

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

      2. unpow2 30.7%

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

      3. unpow2 30.7%

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

   6. Simplified 30.7%

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

      1. hypot-udef 61.2%

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

      2. expm1-log1p-u 61.2%

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

      3. expm1-udef 46.1%

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

   8. Applied egg-rr 46.1%

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

      1. expm1-def 61.2%

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

      2. expm1-log1p 61.2%

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

   10. Simplified 61.2%

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

   if -2.2000000000000001e-80 < y.re < -7.80000000000000019e-106

   1. Initial program 63.4%

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

      1. exp-diff 63.4%

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

      2. +-rgt-identity 63.4%

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

      3. +-rgt-identity 63.4%

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

      4. exp-to-pow 63.4%

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

      5. hypot-def 63.4%

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

      6. *-commutative 63.4%

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

      7. exp-prod 63.4%

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

      8. +-commutative 63.4%

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

      9. *-commutative 63.4%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y.im around 0 51.8%

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

      1. unpow2 51.8%

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

      2. unpow2 51.8%

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

      3. hypot-def 70.3%

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

   6. Simplified 70.3%

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

   7. Taylor expanded in x.im around 0 27.3%

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

   8. Taylor expanded in y.re around 0 70.3%

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

   if 3.79999999999999998e134 < y.re < 6.2000000000000003e211

   1. Initial program 47.8%

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

      1. exp-diff 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. +-rgt-identity 39.1%

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

      3. +-rgt-identity 39.1%

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

      4. exp-to-pow 39.1%

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

      5. hypot-def 39.1%

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

      6. *-commutative 39.1%

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

      7. exp-prod 39.1%

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

      8. +-commutative 39.1%

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

      9. *-commutative 39.1%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in y.im around 0 73.9%

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

      1. unpow2 73.9%

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

      2. unpow2 73.9%

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

      3. hypot-def 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in x.re around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. add-cube-cbrt 82.7%

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

      3. pow3 82.7%

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

   9. Applied egg-rr 82.7%

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

   if 6.2000000000000003e211 < y.re

   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. Step-by-step derivation

      1. exp-diff 45.5%

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

      2. +-rgt-identity 45.5%

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

      3. +-rgt-identity 45.5%

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

      4. exp-to-pow 45.5%

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

      5. hypot-def 45.5%

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

      6. *-commutative 45.5%

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

      7. exp-prod 45.5%

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

      8. +-commutative 45.5%

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

      9. *-commutative 45.5%

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

   3. Simplified 72.7%

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

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

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

      1. unpow2 63.7%

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

      2. unpow2 63.7%

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

      3. hypot-def 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in x.re around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. add-sqr-sqrt 22.9%

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

      3. sqrt-unprod 9.2%

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

      4. pow2 9.2%

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

   9. Applied egg-rr 9.2%

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

       1. *-commutative 9.2%

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

       2. unpow2 9.2%

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

       3. rem-sqrt-square 73.0%

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

   11. Simplified 73.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.75 \cdot 10^{-39}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+62}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+134}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+211}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 10: 62.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.32 \cdot 10^{-38}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot y.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+134}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+214}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|t_1\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (sin (* (log (hypot x.re x.im)) y.im))
          (exp (* (atan2 x.im x.re) y.im))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* t_1 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -2.32e-38)
     t_2
     (if (<= y.re -3.1e-81)
       t_0
       (if (<= y.re -7.8e-106)
         (fma
          y.re
          (atan2 x.im x.re)
          (* (log (hypot x.im x.re)) (* (atan2 x.im x.re) (* y.re y.re))))
         (if (<= y.re 5.2e+57)
           t_0
           (if (<= y.re 3.5e+134)
             t_2
             (if (<= y.re 4e+214)
               (* (sin (pow (cbrt t_1) 3.0)) (pow x.im y.re))
               (* (sin (fabs t_1)) (pow x.im y.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((log(hypot(x_46_re, x_46_im)) * y_46_im)) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.32e-38) {
		tmp = t_2;
	} else if (y_46_re <= -3.1e-81) {
		tmp = t_0;
	} else if (y_46_re <= -7.8e-106) {
		tmp = fma(y_46_re, atan2(x_46_im, x_46_re), (log(hypot(x_46_im, x_46_re)) * (atan2(x_46_im, x_46_re) * (y_46_re * y_46_re))));
	} else if (y_46_re <= 5.2e+57) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e+134) {
		tmp = t_2;
	} else if (y_46_re <= 4e+214) {
		tmp = sin(pow(cbrt(t_1), 3.0)) * pow(x_46_im, y_46_re);
	} else {
		tmp = sin(fabs(t_1)) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im)) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.32e-38)
		tmp = t_2;
	elseif (y_46_re <= -3.1e-81)
		tmp = t_0;
	elseif (y_46_re <= -7.8e-106)
		tmp = fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(hypot(x_46_im, x_46_re)) * Float64(atan(x_46_im, x_46_re) * Float64(y_46_re * y_46_re))));
	elseif (y_46_re <= 5.2e+57)
		tmp = t_0;
	elseif (y_46_re <= 3.5e+134)
		tmp = t_2;
	elseif (y_46_re <= 4e+214)
		tmp = Float64(sin((cbrt(t_1) ^ 3.0)) * (x_46_im ^ y_46_re));
	else
		tmp = Float64(sin(abs(t_1)) * (x_46_im ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.32e-38], t$95$2, If[LessEqual[y$46$re, -3.1e-81], t$95$0, If[LessEqual[y$46$re, -7.8e-106], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.2e+57], t$95$0, If[LessEqual[y$46$re, 3.5e+134], t$95$2, If[LessEqual[y$46$re, 4e+214], N[(N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[Abs[t$95$1], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.3199999999999999e-38 or 5.2e57 < y.re < 3.50000000000000003e134

   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 31.5%

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

      2. +-rgt-identity 31.5%

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

      3. +-rgt-identity 31.5%

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

      4. exp-to-pow 31.5%

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

      5. hypot-def 31.5%

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

      6. *-commutative 31.5%

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

      7. exp-prod 31.5%

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

      8. +-commutative 31.5%

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

      9. *-commutative 31.5%

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

   3. Simplified 67.4%

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

   4. Taylor expanded in y.im around 0 71.1%

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

      1. unpow2 71.1%

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

      2. unpow2 71.1%

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

      3. hypot-def 75.4%

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

   6. Simplified 75.4%

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

      1. *-commutative 75.4%

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

      2. add-exp-log 43.7%

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

   8. Applied egg-rr 43.7%

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

   9. Taylor expanded in y.re around 0 84.3%

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

   if -2.3199999999999999e-38 < y.re < -3.09999999999999988e-81 or -7.80000000000000019e-106 < y.re < 5.2e57

   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 37.2%

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

      2. +-rgt-identity 37.2%

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

      3. +-rgt-identity 37.2%

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

      4. exp-to-pow 37.2%

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

      5. hypot-def 37.2%

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

      6. *-commutative 37.2%

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

      7. exp-prod 37.0%

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

      8. +-commutative 37.0%

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

      9. *-commutative 37.0%

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

   3. Simplified 77.1%

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

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

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

      1. +-commutative 30.7%

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

      2. unpow2 30.7%

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

      3. unpow2 30.7%

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

   6. Simplified 30.7%

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

      1. hypot-udef 61.2%

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

      2. expm1-log1p-u 61.2%

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

      3. expm1-udef 46.1%

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

   8. Applied egg-rr 46.1%

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

      1. expm1-def 61.2%

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

      2. expm1-log1p 61.2%

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

   10. Simplified 61.2%

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

   if -3.09999999999999988e-81 < y.re < -7.80000000000000019e-106

   1. Initial program 63.4%

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

      1. exp-diff 63.4%

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

      2. +-rgt-identity 63.4%

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

      3. +-rgt-identity 63.4%

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

      4. exp-to-pow 63.4%

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

      5. hypot-def 63.4%

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

      6. *-commutative 63.4%

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

      7. exp-prod 63.4%

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

      8. +-commutative 63.4%

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

      9. *-commutative 63.4%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in y.im around 0 51.8%

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

      1. unpow2 51.8%

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

      2. unpow2 51.8%

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

      3. hypot-def 70.3%

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

   6. Simplified 70.3%

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

      1. *-commutative 70.3%

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

      2. add-exp-log 27.0%

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

   8. Applied egg-rr 27.0%

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

   9. Taylor expanded in y.re around 0 50.8%

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

       1. fma-def 50.8%

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

       2. associate-*r* 50.8%

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

       3. unpow2 50.8%

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

       4. unpow2 50.8%

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

       5. unpow2 50.8%

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

       6. hypot-def 70.3%

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

   11. Simplified 70.3%

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

   if 3.50000000000000003e134 < y.re < 3.9999999999999998e214

   1. Initial program 47.8%

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

      1. exp-diff 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. +-rgt-identity 39.1%

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

      3. +-rgt-identity 39.1%

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

      4. exp-to-pow 39.1%

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

      5. hypot-def 39.1%

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

      6. *-commutative 39.1%

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

      7. exp-prod 39.1%

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

      8. +-commutative 39.1%

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

      9. *-commutative 39.1%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in y.im around 0 73.9%

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

      1. unpow2 73.9%

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

      2. unpow2 73.9%

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

      3. hypot-def 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in x.re around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. add-cube-cbrt 82.7%

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

      3. pow3 82.7%

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

   9. Applied egg-rr 82.7%

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

   if 3.9999999999999998e214 < y.re

   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. Step-by-step derivation

      1. exp-diff 45.5%

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

      2. +-rgt-identity 45.5%

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

      3. +-rgt-identity 45.5%

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

      4. exp-to-pow 45.5%

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

      5. hypot-def 45.5%

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

      6. *-commutative 45.5%

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

      7. exp-prod 45.5%

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

      8. +-commutative 45.5%

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

      9. *-commutative 45.5%

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

   3. Simplified 72.7%

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

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

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

      1. unpow2 63.7%

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

      2. unpow2 63.7%

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

      3. hypot-def 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in x.re around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. add-sqr-sqrt 22.9%

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

      3. sqrt-unprod 9.2%

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

      4. pow2 9.2%

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

   9. Applied egg-rr 9.2%

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

       1. *-commutative 9.2%

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

       2. unpow2 9.2%

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

       3. rem-sqrt-square 73.0%

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

   11. Simplified 73.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.32 \cdot 10^{-38}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(y.re \cdot y.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{+134}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+214}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 11: 51.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|t_0\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* t_0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -7.8e-106)
     t_1
     (if (<= y.re 7.5e+59)
       (/
        (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (exp (* (atan2 x.im x.re) y.im)))
       (if (<= y.re 3.7e+133)
         t_1
         (if (<= y.re 2e+214)
           (* (sin (pow (cbrt t_0) 3.0)) (pow x.im y.re))
           (* (sin (fabs t_0)) (pow x.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -7.8e-106) {
		tmp = t_1;
	} else if (y_46_re <= 7.5e+59) {
		tmp = (y_46_im * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else if (y_46_re <= 3.7e+133) {
		tmp = t_1;
	} else if (y_46_re <= 2e+214) {
		tmp = sin(pow(cbrt(t_0), 3.0)) * pow(x_46_im, y_46_re);
	} else {
		tmp = sin(fabs(t_0)) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -7.8e-106) {
		tmp = t_1;
	} else if (y_46_re <= 7.5e+59) {
		tmp = (y_46_im * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else if (y_46_re <= 3.7e+133) {
		tmp = t_1;
	} else if (y_46_re <= 2e+214) {
		tmp = Math.sin(Math.pow(Math.cbrt(t_0), 3.0)) * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = Math.sin(Math.abs(t_0)) * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -7.8e-106)
		tmp = t_1;
	elseif (y_46_re <= 7.5e+59)
		tmp = Float64(Float64(y_46_im * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 3.7e+133)
		tmp = t_1;
	elseif (y_46_re <= 2e+214)
		tmp = Float64(sin((cbrt(t_0) ^ 3.0)) * (x_46_im ^ y_46_re));
	else
		tmp = Float64(sin(abs(t_0)) * (x_46_im ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.8e-106], t$95$1, If[LessEqual[y$46$re, 7.5e+59], N[(N[(y$46$im * 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[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.7e+133], t$95$1, If[LessEqual[y$46$re, 2e+214], N[(N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.80000000000000019e-106 or 7.4999999999999996e59 < y.re < 3.70000000000000023e133

   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 34.8%

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

      2. +-rgt-identity 34.8%

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

      3. +-rgt-identity 34.8%

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

      4. exp-to-pow 34.8%

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

      5. hypot-def 34.8%

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

      6. *-commutative 34.8%

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

      7. exp-prod 34.8%

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

      8. +-commutative 34.8%

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

      9. *-commutative 34.8%

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

   3. Simplified 72.4%

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

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

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

      1. unpow2 66.9%

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

      2. unpow2 66.9%

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

      3. hypot-def 70.5%

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

   6. Simplified 70.5%

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

      1. *-commutative 70.5%

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

      2. add-exp-log 40.2%

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

   8. Applied egg-rr 40.2%

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

   9. Taylor expanded in y.re around 0 77.8%

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

   if -7.80000000000000019e-106 < y.re < 7.4999999999999996e59

   1. Initial program 38.6%

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

      1. exp-diff 37.6%

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

      2. +-rgt-identity 37.6%

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

      3. +-rgt-identity 37.6%

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

      4. exp-to-pow 37.6%

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

      5. hypot-def 37.6%

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

      6. *-commutative 37.6%

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

      7. exp-prod 37.3%

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

      8. +-commutative 37.3%

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

      9. *-commutative 37.3%

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

   3. Simplified 76.0%

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

   4. Taylor expanded in y.re around 0 31.4%

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

      1. +-commutative 31.4%

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

      2. unpow2 31.4%

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

      3. unpow2 31.4%

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

   6. Simplified 31.4%

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

   7. Taylor expanded in y.im around 0 46.3%

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

      1. unpow2 46.3%

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

      2. unpow2 46.3%

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

   9. Simplified 46.3%

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

   if 3.70000000000000023e133 < y.re < 1.9999999999999999e214

   1. Initial program 47.8%

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

      1. exp-diff 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. +-rgt-identity 39.1%

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

      3. +-rgt-identity 39.1%

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

      4. exp-to-pow 39.1%

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

      5. hypot-def 39.1%

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

      6. *-commutative 39.1%

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

      7. exp-prod 39.1%

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

      8. +-commutative 39.1%

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

      9. *-commutative 39.1%

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

   3. Simplified 65.2%

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

   4. Taylor expanded in y.im around 0 73.9%

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

      1. unpow2 73.9%

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

      2. unpow2 73.9%

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

      3. hypot-def 73.9%

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

   6. Simplified 73.9%

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

   7. Taylor expanded in x.re around 0 69.7%

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

      1. *-commutative 69.7%

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

      2. add-cube-cbrt 82.7%

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

      3. pow3 82.7%

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

   9. Applied egg-rr 82.7%

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

   if 1.9999999999999999e214 < y.re

   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. Step-by-step derivation

      1. exp-diff 45.5%

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

      2. +-rgt-identity 45.5%

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

      3. +-rgt-identity 45.5%

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

      4. exp-to-pow 45.5%

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

      5. hypot-def 45.5%

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

      6. *-commutative 45.5%

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

      7. exp-prod 45.5%

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

      8. +-commutative 45.5%

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

      9. *-commutative 45.5%

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

   3. Simplified 72.7%

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

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

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

      1. unpow2 63.7%

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

      2. unpow2 63.7%

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

      3. hypot-def 63.7%

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

   6. Simplified 63.7%

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

   7. Taylor expanded in x.re around 0 54.8%

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

      1. *-commutative 54.8%

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

      2. add-sqr-sqrt 22.9%

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

      3. sqrt-unprod 9.2%

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

      4. pow2 9.2%

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

   9. Applied egg-rr 9.2%

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

       1. *-commutative 9.2%

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

       2. unpow2 9.2%

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

       3. rem-sqrt-square 73.0%

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

   11. Simplified 73.0%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+59}:\\ \;\;\;\;\frac{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{+133}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+214}:\\ \;\;\;\;\sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 12: 45.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := t_0 \cdot t_1\\ t_3 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -9.8 \cdot 10^{-141}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{t_3}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{t_3}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+195} \lor \neg \left(y.re \leq 1.35 \cdot 10^{+279}\right):\\ \;\;\;\;\sin t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|t_0\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (pow (hypot x.im x.re) y.re))
        (t_2 (* t_0 t_1))
        (t_3 (exp (* (atan2 x.im x.re) y.im))))
   (if (<= y.re -9.8e-141)
     t_2
     (if (<= y.re 2.2e-163)
       (/ (sin (* y.im (log (- x.re)))) t_3)
       (if (<= y.re 2.9e+47)
         (/ (sin (* y.im (log (- x.im)))) t_3)
         (if (<= y.re 5e+131)
           t_2
           (if (or (<= y.re 8.2e+195) (not (<= y.re 1.35e+279)))
             (* (sin t_0) t_1)
             (* (sin (fabs t_0)) (pow x.im y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = t_0 * t_1;
	double t_3 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	double tmp;
	if (y_46_re <= -9.8e-141) {
		tmp = t_2;
	} else if (y_46_re <= 2.2e-163) {
		tmp = sin((y_46_im * log(-x_46_re))) / t_3;
	} else if (y_46_re <= 2.9e+47) {
		tmp = sin((y_46_im * log(-x_46_im))) / t_3;
	} else if (y_46_re <= 5e+131) {
		tmp = t_2;
	} else if ((y_46_re <= 8.2e+195) || !(y_46_re <= 1.35e+279)) {
		tmp = sin(t_0) * t_1;
	} else {
		tmp = sin(fabs(t_0)) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = t_0 * t_1;
	double t_3 = Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	double tmp;
	if (y_46_re <= -9.8e-141) {
		tmp = t_2;
	} else if (y_46_re <= 2.2e-163) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_re))) / t_3;
	} else if (y_46_re <= 2.9e+47) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_im))) / t_3;
	} else if (y_46_re <= 5e+131) {
		tmp = t_2;
	} else if ((y_46_re <= 8.2e+195) || !(y_46_re <= 1.35e+279)) {
		tmp = Math.sin(t_0) * t_1;
	} else {
		tmp = Math.sin(Math.abs(t_0)) * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_2 = t_0 * t_1
	t_3 = math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	tmp = 0
	if y_46_re <= -9.8e-141:
		tmp = t_2
	elif y_46_re <= 2.2e-163:
		tmp = math.sin((y_46_im * math.log(-x_46_re))) / t_3
	elif y_46_re <= 2.9e+47:
		tmp = math.sin((y_46_im * math.log(-x_46_im))) / t_3
	elif y_46_re <= 5e+131:
		tmp = t_2
	elif (y_46_re <= 8.2e+195) or not (y_46_re <= 1.35e+279):
		tmp = math.sin(t_0) * t_1
	else:
		tmp = math.sin(math.fabs(t_0)) * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_2 = Float64(t_0 * t_1)
	t_3 = exp(Float64(atan(x_46_im, x_46_re) * y_46_im))
	tmp = 0.0
	if (y_46_re <= -9.8e-141)
		tmp = t_2;
	elseif (y_46_re <= 2.2e-163)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_re)))) / t_3);
	elseif (y_46_re <= 2.9e+47)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_im)))) / t_3);
	elseif (y_46_re <= 5e+131)
		tmp = t_2;
	elseif ((y_46_re <= 8.2e+195) || !(y_46_re <= 1.35e+279))
		tmp = Float64(sin(t_0) * t_1);
	else
		tmp = Float64(sin(abs(t_0)) * (x_46_im ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_2 = t_0 * t_1;
	t_3 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -9.8e-141)
		tmp = t_2;
	elseif (y_46_re <= 2.2e-163)
		tmp = sin((y_46_im * log(-x_46_re))) / t_3;
	elseif (y_46_re <= 2.9e+47)
		tmp = sin((y_46_im * log(-x_46_im))) / t_3;
	elseif (y_46_re <= 5e+131)
		tmp = t_2;
	elseif ((y_46_re <= 8.2e+195) || ~((y_46_re <= 1.35e+279)))
		tmp = sin(t_0) * t_1;
	else
		tmp = sin(abs(t_0)) * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9.8e-141], t$95$2, If[LessEqual[y$46$re, 2.2e-163], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 2.9e+47], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 5e+131], t$95$2, If[Or[LessEqual[y$46$re, 8.2e+195], N[Not[LessEqual[y$46$re, 1.35e+279]], $MachinePrecision]], N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -9.80000000000000012e-141 or 2.8999999999999998e47 < y.re < 4.99999999999999995e131

   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 35.0%

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

      2. +-rgt-identity 35.0%

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

      3. +-rgt-identity 35.0%

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

      4. exp-to-pow 35.0%

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

      5. hypot-def 35.0%

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

      6. *-commutative 35.0%

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

      7. exp-prod 34.8%

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

      8. +-commutative 34.8%

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

      9. *-commutative 34.8%

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

   3. Simplified 72.3%

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

   4. Taylor expanded in y.im around 0 65.2%

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

      1. unpow2 65.2%

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

      2. unpow2 65.2%

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

      3. hypot-def 68.7%

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

   6. Simplified 68.7%

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

      1. *-commutative 68.7%

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

      2. add-exp-log 39.2%

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

   8. Applied egg-rr 39.2%

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

   9. Taylor expanded in y.re around 0 75.8%

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

   if -9.80000000000000012e-141 < y.re < 2.20000000000000011e-163

   1. Initial program 42.5%

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

      1. exp-diff 42.5%

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

      2. +-rgt-identity 42.5%

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

      3. +-rgt-identity 42.5%

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

      4. exp-to-pow 42.5%

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

      5. hypot-def 42.5%

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

      6. *-commutative 42.5%

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

      7. exp-prod 42.5%

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

      8. +-commutative 42.5%

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

      9. *-commutative 42.5%

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

   3. Simplified 78.4%

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

   4. Taylor expanded in y.re around 0 37.6%

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

      1. +-commutative 37.6%

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

      2. unpow2 37.6%

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

      3. unpow2 37.6%

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

   6. Simplified 37.6%

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

   7. Taylor expanded in x.re around -inf 38.0%

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

      1. mul-1-neg 38.0%

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

   9. Simplified 38.0%

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

   if 2.20000000000000011e-163 < y.re < 2.8999999999999998e47

   1. Initial program 33.8%

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

      1. exp-diff 31.5%

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

      2. +-rgt-identity 31.5%

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

      3. +-rgt-identity 31.5%

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

      4. exp-to-pow 31.5%

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

      5. hypot-def 31.5%

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

      6. *-commutative 31.5%

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

      7. exp-prod 31.2%

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

      8. +-commutative 31.2%

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

      9. *-commutative 31.2%

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

   3. Simplified 73.8%

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

   4. Taylor expanded in y.re around 0 23.5%

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

      1. +-commutative 23.5%

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

      2. unpow2 23.5%

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

      3. unpow2 23.5%

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

   6. Simplified 23.5%

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

   7. Taylor expanded in x.im around -inf 38.6%

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

      1. mul-1-neg 38.6%

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

   9. Simplified 38.6%

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

   if 4.99999999999999995e131 < y.re < 8.2000000000000001e195 or 1.3500000000000001e279 < y.re

   1. Initial program 50.0%

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

      1. exp-diff 41.7%

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

      2. +-rgt-identity 41.7%

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

      3. +-rgt-identity 41.7%

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

      4. exp-to-pow 41.7%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 41.7%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 41.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 41.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 41.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 41.7%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 70.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 79.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 79.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 79.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 79.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 79.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   if 8.2000000000000001e195 < y.re < 1.3500000000000001e279

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.9%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 42.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 42.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 42.9%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. 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. Taylor expanded in y.im around 0 57.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 57.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.re around 0 52.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
   8. Step-by-step derivation

      1. *-commutative 52.6%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {x.im}^{y.re} \]

      2. add-sqr-sqrt 24.0%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re} \cdot \sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)} \cdot {x.im}^{y.re} \]

      3. sqrt-unprod 19.2%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {x.im}^{y.re} \]

      4. pow2 19.2%

         \[\leadsto \sin \left(\sqrt{\color{blue}{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}}\right) \cdot {x.im}^{y.re} \]

   9. Applied egg-rr 19.2%

      \[\leadsto \sin \color{blue}{\left(\sqrt{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}\right)} \cdot {x.im}^{y.re} \]
   10. Step-by-step derivation

       1. *-commutative 19.2%

          \[\leadsto \sin \left(\sqrt{{\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}^{2}}\right) \cdot {x.im}^{y.re} \]

       2. unpow2 19.2%

          \[\leadsto \sin \left(\sqrt{\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \cdot {x.im}^{y.re} \]

       3. rem-sqrt-square 76.4%

          \[\leadsto \sin \color{blue}{\left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)} \cdot {x.im}^{y.re} \]

   11. Simplified 76.4%

       \[\leadsto \sin \color{blue}{\left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)} \cdot {x.im}^{y.re} \]
3. Recombined 5 regimes into one program.

4. Final simplification 61.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.8 \cdot 10^{-141}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-163}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+47}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+195} \lor \neg \left(y.re \leq 1.35 \cdot 10^{+279}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 13: 48.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.06 \cdot 10^{+206}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{t_1}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-74}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+164}:\\ \;\;\;\;\sin t_0 \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (exp (* (atan2 x.im x.re) y.im))))
   (if (<= y.im -1.06e+206)
     (/ (sin (* y.im (log (- x.im)))) t_1)
     (if (<= y.im 2.6e-74)
       (* t_0 (pow (hypot x.im x.re) y.re))
       (if (<= y.im 3e+164)
         (* (sin t_0) (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re))
         (/ (sin (* y.im (log (- x.re)))) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	double tmp;
	if (y_46_im <= -1.06e+206) {
		tmp = sin((y_46_im * log(-x_46_im))) / t_1;
	} else if (y_46_im <= 2.6e-74) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 3e+164) {
		tmp = sin(t_0) * pow(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = sin((y_46_im * log(-x_46_re))) / t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	double tmp;
	if (y_46_im <= -1.06e+206) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_im))) / t_1;
	} else if (y_46_im <= 2.6e-74) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 3e+164) {
		tmp = Math.sin(t_0) * Math.pow(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = Math.sin((y_46_im * Math.log(-x_46_re))) / t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	tmp = 0
	if y_46_im <= -1.06e+206:
		tmp = math.sin((y_46_im * math.log(-x_46_im))) / t_1
	elif y_46_im <= 2.6e-74:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_im <= 3e+164:
		tmp = math.sin(t_0) * math.pow(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re)
	else:
		tmp = math.sin((y_46_im * math.log(-x_46_re))) / t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = exp(Float64(atan(x_46_im, x_46_re) * y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.06e+206)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_im)))) / t_1);
	elseif (y_46_im <= 2.6e-74)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 3e+164)
		tmp = Float64(sin(t_0) * (sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) ^ y_46_re));
	else
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_re)))) / t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -1.06e+206)
		tmp = sin((y_46_im * log(-x_46_im))) / t_1;
	elseif (y_46_im <= 2.6e-74)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_im <= 3e+164)
		tmp = sin(t_0) * (sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))) ^ y_46_re);
	else
		tmp = sin((y_46_im * log(-x_46_re))) / t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -1.06e+206], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$im, 2.6e-74], 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, 3e+164], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.0599999999999999e206

   1. Initial program 36.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 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. +-rgt-identity 27.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 27.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 27.3%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 27.3%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 27.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 27.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 27.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 27.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. 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. Taylor expanded in y.re around 0 36.4%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
   5. Step-by-step derivation

      1. +-commutative 36.4%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      2. unpow2 36.4%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      3. unpow2 36.4%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Simplified 36.4%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. Taylor expanded in x.im around -inf 54.5%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-1 \cdot x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
   8. Step-by-step derivation

      1. mul-1-neg 54.5%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   9. Simplified 54.5%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   if -1.0599999999999999e206 < y.im < 2.6000000000000001e-74

   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 39.9%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 39.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 39.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 39.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 39.9%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 39.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 39.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 39.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 39.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 78.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 51.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 51.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 51.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 59.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 59.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
   7. Step-by-step derivation

      1. *-commutative 59.8%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

      2. add-exp-log 34.1%

         \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   8. Applied egg-rr 34.1%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   9. Taylor expanded in y.re around 0 62.4%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if 2.6000000000000001e-74 < y.im < 3.00000000000000001e164

   1. Initial program 34.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 30.6%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 30.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 30.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 30.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 30.6%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 30.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 30.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 30.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 30.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 72.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 52.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. +-commutative 52.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{y.re} \]

      2. unpow2 52.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{y.re} \]

      3. unpow2 52.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{y.re} \]

   6. Simplified 52.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}} \]

   if 3.00000000000000001e164 < y.im

   1. Initial program 45.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.6%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 42.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 42.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 42.6%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 42.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 42.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 61.4%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.re around 0 45.8%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
   5. Step-by-step derivation

      1. +-commutative 45.8%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      2. unpow2 45.8%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      3. unpow2 45.8%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Simplified 45.8%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. Taylor expanded in x.re around -inf 45.2%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-1 \cdot x.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
   8. Step-by-step derivation

      1. mul-1-neg 45.2%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   9. Simplified 45.2%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.re\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.06 \cdot 10^{+206}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-74}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+164}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.re\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array} \]

Alternative 14: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+132}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (pow (hypot x.im x.re) y.re))
        (t_2 (* t_0 t_1)))
   (if (<= y.re -7.5e-106)
     t_2
     (if (<= y.re 2.4e+62)
       (/
        (* y.im (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (exp (* (atan2 x.im x.re) y.im)))
       (if (<= y.re 2e+132) t_2 (* (sin t_0) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = t_0 * t_1;
	double tmp;
	if (y_46_re <= -7.5e-106) {
		tmp = t_2;
	} else if (y_46_re <= 2.4e+62) {
		tmp = (y_46_im * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else if (y_46_re <= 2e+132) {
		tmp = t_2;
	} else {
		tmp = sin(t_0) * t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = t_0 * t_1;
	double tmp;
	if (y_46_re <= -7.5e-106) {
		tmp = t_2;
	} else if (y_46_re <= 2.4e+62) {
		tmp = (y_46_im * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else if (y_46_re <= 2e+132) {
		tmp = t_2;
	} else {
		tmp = Math.sin(t_0) * t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_2 = t_0 * t_1
	tmp = 0
	if y_46_re <= -7.5e-106:
		tmp = t_2
	elif y_46_re <= 2.4e+62:
		tmp = (y_46_im * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	elif y_46_re <= 2e+132:
		tmp = t_2
	else:
		tmp = math.sin(t_0) * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (y_46_re <= -7.5e-106)
		tmp = t_2;
	elseif (y_46_re <= 2.4e+62)
		tmp = Float64(Float64(y_46_im * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 2e+132)
		tmp = t_2;
	else
		tmp = Float64(sin(t_0) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_2 = t_0 * t_1;
	tmp = 0.0;
	if (y_46_re <= -7.5e-106)
		tmp = t_2;
	elseif (y_46_re <= 2.4e+62)
		tmp = (y_46_im * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	elseif (y_46_re <= 2e+132)
		tmp = t_2;
	else
		tmp = sin(t_0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e-106], t$95$2, If[LessEqual[y$46$re, 2.4e+62], N[(N[(y$46$im * 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[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2e+132], t$95$2, N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.5000000000000002e-106 or 2.4e62 < y.re < 1.99999999999999998e132

   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 34.8%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 34.8%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 34.8%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 34.8%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 34.8%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 34.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 34.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 34.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 34.8%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 72.4%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 66.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 66.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 66.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 70.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 70.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
   7. Step-by-step derivation

      1. *-commutative 70.5%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

      2. add-exp-log 40.2%

         \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   8. Applied egg-rr 40.2%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   9. Taylor expanded in y.re around 0 77.8%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if -7.5000000000000002e-106 < y.re < 2.4e62

   1. Initial program 38.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 37.6%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 37.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 37.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 37.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 37.6%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 37.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 37.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 76.0%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.re around 0 31.4%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
   5. Step-by-step derivation

      1. +-commutative 31.4%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      2. unpow2 31.4%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      3. unpow2 31.4%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Simplified 31.4%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. Taylor expanded in y.im around 0 46.3%

      \[\leadsto \frac{\color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
   8. Step-by-step derivation

      1. unpow2 46.3%

         \[\leadsto \frac{y.im \cdot \log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      2. unpow2 46.3%

         \[\leadsto \frac{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   9. Simplified 46.3%

      \[\leadsto \frac{\color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   if 1.99999999999999998e132 < y.re

   1. Initial program 46.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.2%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 42.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 42.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 42.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   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. Taylor expanded in y.im around 0 68.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-106}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+62}:\\ \;\;\;\;\frac{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 15: 48.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.15 \cdot 10^{-106}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+195} \lor \neg \left(y.re \leq 1.55 \cdot 10^{+280}\right):\\ \;\;\;\;\sin t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|t_0\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re -4.15e-106)
     (* t_0 t_1)
     (if (<= y.re 3.1e-181)
       (/ (sin (* y.im (log x.im))) (exp (* (atan2 x.im x.re) y.im)))
       (if (or (<= y.re 4.5e+195) (not (<= y.re 1.55e+280)))
         (* (sin t_0) t_1)
         (* (sin (fabs t_0)) (pow x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.15e-106) {
		tmp = t_0 * t_1;
	} else if (y_46_re <= 3.1e-181) {
		tmp = sin((y_46_im * log(x_46_im))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else if ((y_46_re <= 4.5e+195) || !(y_46_re <= 1.55e+280)) {
		tmp = sin(t_0) * t_1;
	} else {
		tmp = sin(fabs(t_0)) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.15e-106) {
		tmp = t_0 * t_1;
	} else if (y_46_re <= 3.1e-181) {
		tmp = Math.sin((y_46_im * Math.log(x_46_im))) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else if ((y_46_re <= 4.5e+195) || !(y_46_re <= 1.55e+280)) {
		tmp = Math.sin(t_0) * t_1;
	} else {
		tmp = Math.sin(Math.abs(t_0)) * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -4.15e-106:
		tmp = t_0 * t_1
	elif y_46_re <= 3.1e-181:
		tmp = math.sin((y_46_im * math.log(x_46_im))) / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	elif (y_46_re <= 4.5e+195) or not (y_46_re <= 1.55e+280):
		tmp = math.sin(t_0) * t_1
	else:
		tmp = math.sin(math.fabs(t_0)) * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -4.15e-106)
		tmp = Float64(t_0 * t_1);
	elseif (y_46_re <= 3.1e-181)
		tmp = Float64(sin(Float64(y_46_im * log(x_46_im))) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	elseif ((y_46_re <= 4.5e+195) || !(y_46_re <= 1.55e+280))
		tmp = Float64(sin(t_0) * t_1);
	else
		tmp = Float64(sin(abs(t_0)) * (x_46_im ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -4.15e-106)
		tmp = t_0 * t_1;
	elseif (y_46_re <= 3.1e-181)
		tmp = sin((y_46_im * log(x_46_im))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	elseif ((y_46_re <= 4.5e+195) || ~((y_46_re <= 1.55e+280)))
		tmp = sin(t_0) * t_1;
	else
		tmp = sin(abs(t_0)) * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -4.15e-106], N[(t$95$0 * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 3.1e-181], N[(N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, 4.5e+195], N[Not[LessEqual[y$46$re, 1.55e+280]], $MachinePrecision]], N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision], N[(N[Sin[N[Abs[t$95$0], $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.15000000000000023e-106

   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 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. +-rgt-identity 36.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 36.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 36.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 36.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 36.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 36.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 36.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 36.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 76.9%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 70.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 70.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 70.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 74.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 74.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
   7. Step-by-step derivation

      1. *-commutative 74.5%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

      2. add-exp-log 41.6%

         \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   8. Applied egg-rr 41.6%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   9. Taylor expanded in y.re around 0 77.7%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if -4.15000000000000023e-106 < y.re < 3.10000000000000021e-181

   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. Step-by-step derivation

      1. exp-diff 45.4%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 45.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 45.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 45.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 45.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 45.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 45.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 45.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 45.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.re around 0 42.0%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
   5. Step-by-step derivation

      1. +-commutative 42.0%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      2. unpow2 42.0%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      3. unpow2 42.0%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. Taylor expanded in x.re around 0 34.7%

      \[\leadsto \frac{\color{blue}{\sin \left(\log x.im \cdot y.im\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   if 3.10000000000000021e-181 < y.re < 4.50000000000000009e195 or 1.55e280 < y.re

   1. Initial program 36.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 32.5%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 32.5%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 32.5%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 32.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 32.5%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 32.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 32.3%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. 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. Taylor expanded in y.im around 0 41.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 41.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 41.5%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 45.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 45.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   if 4.50000000000000009e195 < y.re < 1.55e280

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.9%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 42.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.9%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 42.9%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 42.9%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 42.9%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. 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. Taylor expanded in y.im around 0 57.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 57.1%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 57.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.re around 0 52.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
   8. Step-by-step derivation

      1. *-commutative 52.6%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {x.im}^{y.re} \]

      2. add-sqr-sqrt 24.0%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re} \cdot \sqrt{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)} \cdot {x.im}^{y.re} \]

      3. sqrt-unprod 19.2%

         \[\leadsto \sin \color{blue}{\left(\sqrt{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {x.im}^{y.re} \]

      4. pow2 19.2%

         \[\leadsto \sin \left(\sqrt{\color{blue}{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}}\right) \cdot {x.im}^{y.re} \]

   9. Applied egg-rr 19.2%

      \[\leadsto \sin \color{blue}{\left(\sqrt{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2}}\right)} \cdot {x.im}^{y.re} \]
   10. Step-by-step derivation

       1. *-commutative 19.2%

          \[\leadsto \sin \left(\sqrt{{\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}^{2}}\right) \cdot {x.im}^{y.re} \]

       2. unpow2 19.2%

          \[\leadsto \sin \left(\sqrt{\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \cdot {x.im}^{y.re} \]

       3. rem-sqrt-square 76.4%

          \[\leadsto \sin \color{blue}{\left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)} \cdot {x.im}^{y.re} \]

   11. Simplified 76.4%

       \[\leadsto \sin \color{blue}{\left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)} \cdot {x.im}^{y.re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.15 \cdot 10^{-106}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+195} \lor \neg \left(y.re \leq 1.55 \cdot 10^{+280}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 16: 45.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := t_1 \cdot t_2\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-106}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{t_0}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{t_0}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+133}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\sin t_1 \cdot t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (atan2 x.im x.re) y.im)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (pow (hypot x.im x.re) y.re))
        (t_3 (* t_1 t_2)))
   (if (<= y.re -2.15e-106)
     t_3
     (if (<= y.re 3.3e-181)
       (/ (sin (* y.im (log x.im))) t_0)
       (if (<= y.re 6.2e+46)
         (/ (sin (* y.im (log (- x.im)))) t_0)
         (if (<= y.re 4e+133) t_3 (* (sin t_1) t_2)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_1 * t_2;
	double tmp;
	if (y_46_re <= -2.15e-106) {
		tmp = t_3;
	} else if (y_46_re <= 3.3e-181) {
		tmp = sin((y_46_im * log(x_46_im))) / t_0;
	} else if (y_46_re <= 6.2e+46) {
		tmp = sin((y_46_im * log(-x_46_im))) / t_0;
	} else if (y_46_re <= 4e+133) {
		tmp = t_3;
	} else {
		tmp = sin(t_1) * 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.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = t_1 * t_2;
	double tmp;
	if (y_46_re <= -2.15e-106) {
		tmp = t_3;
	} else if (y_46_re <= 3.3e-181) {
		tmp = Math.sin((y_46_im * Math.log(x_46_im))) / t_0;
	} else if (y_46_re <= 6.2e+46) {
		tmp = Math.sin((y_46_im * Math.log(-x_46_im))) / t_0;
	} else if (y_46_re <= 4e+133) {
		tmp = t_3;
	} else {
		tmp = Math.sin(t_1) * t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_3 = t_1 * t_2
	tmp = 0
	if y_46_re <= -2.15e-106:
		tmp = t_3
	elif y_46_re <= 3.3e-181:
		tmp = math.sin((y_46_im * math.log(x_46_im))) / t_0
	elif y_46_re <= 6.2e+46:
		tmp = math.sin((y_46_im * math.log(-x_46_im))) / t_0
	elif y_46_re <= 4e+133:
		tmp = t_3
	else:
		tmp = math.sin(t_1) * t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(atan(x_46_im, x_46_re) * y_46_im))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_3 = Float64(t_1 * t_2)
	tmp = 0.0
	if (y_46_re <= -2.15e-106)
		tmp = t_3;
	elseif (y_46_re <= 3.3e-181)
		tmp = Float64(sin(Float64(y_46_im * log(x_46_im))) / t_0);
	elseif (y_46_re <= 6.2e+46)
		tmp = Float64(sin(Float64(y_46_im * log(Float64(-x_46_im)))) / t_0);
	elseif (y_46_re <= 4e+133)
		tmp = t_3;
	else
		tmp = Float64(sin(t_1) * 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 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_3 = t_1 * t_2;
	tmp = 0.0;
	if (y_46_re <= -2.15e-106)
		tmp = t_3;
	elseif (y_46_re <= 3.3e-181)
		tmp = sin((y_46_im * log(x_46_im))) / t_0;
	elseif (y_46_re <= 6.2e+46)
		tmp = sin((y_46_im * log(-x_46_im))) / t_0;
	elseif (y_46_re <= 4e+133)
		tmp = t_3;
	else
		tmp = sin(t_1) * 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[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, If[LessEqual[y$46$re, -2.15e-106], t$95$3, If[LessEqual[y$46$re, 3.3e-181], N[(N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+46], N[(N[Sin[N[(y$46$im * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 4e+133], t$95$3, N[(N[Sin[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.1500000000000001e-106 or 6.1999999999999995e46 < y.re < 4.0000000000000001e133

   1. Initial program 40.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 34.5%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 34.5%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 34.5%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 34.5%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 34.5%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 34.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 34.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 34.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 34.5%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 72.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 66.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 66.3%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 66.3%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 69.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 69.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
   7. Step-by-step derivation

      1. *-commutative 69.9%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

      2. add-exp-log 39.8%

         \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   8. Applied egg-rr 39.8%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   9. Taylor expanded in y.re around 0 77.1%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if -2.1500000000000001e-106 < y.re < 3.30000000000000009e-181

   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. Step-by-step derivation

      1. exp-diff 45.4%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 45.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 45.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 45.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 45.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 45.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 45.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 45.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 45.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.re around 0 42.0%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
   5. Step-by-step derivation

      1. +-commutative 42.0%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      2. unpow2 42.0%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      3. unpow2 42.0%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Simplified 42.0%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. Taylor expanded in x.re around 0 34.7%

      \[\leadsto \frac{\color{blue}{\sin \left(\log x.im \cdot y.im\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   if 3.30000000000000009e-181 < y.re < 6.1999999999999995e46

   1. Initial program 32.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 30.4%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 30.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 30.4%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 30.4%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 30.4%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 30.4%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 30.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 30.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 30.1%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 74.5%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.re around 0 21.3%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]
   5. Step-by-step derivation

      1. +-commutative 21.3%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      2. unpow2 21.3%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

      3. unpow2 21.3%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   6. Simplified 21.3%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

   7. Taylor expanded in x.im around -inf 36.9%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-1 \cdot x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]
   8. Step-by-step derivation

      1. mul-1-neg 36.9%

         \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   9. Simplified 36.9%

      \[\leadsto \frac{\sin \left(y.im \cdot \log \color{blue}{\left(-x.im\right)}\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \]

   if 4.0000000000000001e133 < y.re

   1. Initial program 46.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.2%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 42.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 42.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 42.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   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. Taylor expanded in y.im around 0 68.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-106}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-181}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+46}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(-x.im\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+133}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 17: 45.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq 2 \cdot 10^{+132}:\\ \;\;\;\;t_0 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;\sin t_0 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re 2e+132) (* t_0 t_1) (* (sin t_0) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= 2e+132) {
		tmp = t_0 * t_1;
	} else {
		tmp = sin(t_0) * t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= 2e+132) {
		tmp = t_0 * t_1;
	} else {
		tmp = Math.sin(t_0) * t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= 2e+132:
		tmp = t_0 * t_1
	else:
		tmp = math.sin(t_0) * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= 2e+132)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(sin(t_0) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= 2e+132)
		tmp = t_0 * t_1;
	else
		tmp = sin(t_0) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, 2e+132], N[(t$95$0 * t$95$1), $MachinePrecision], N[(N[Sin[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < 1.99999999999999998e132

   1. Initial program 39.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 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. +-rgt-identity 36.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 36.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 36.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 36.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 36.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 36.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 36.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 36.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 74.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 42.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 42.3%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 42.3%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 46.2%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 46.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
   7. Step-by-step derivation

      1. *-commutative 46.2%

         \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

      2. add-exp-log 24.1%

         \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   8. Applied egg-rr 24.1%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   9. Taylor expanded in y.re around 0 49.0%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if 1.99999999999999998e132 < y.re

   1. Initial program 46.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 42.2%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 42.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 42.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 42.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 42.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 42.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   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. Taylor expanded in y.im around 0 68.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 68.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 68.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 2 \cdot 10^{+132}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 18: 36.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+14} \lor \neg \left(y.re \leq 2.55 \cdot 10^{+14}\right):\\ \;\;\;\;\sin t_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.re -1.65e+14) (not (<= y.re 2.55e+14)))
     (* (sin t_0) (pow x.im y.re))
     t_0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -1.65e+14) || !(y_46_re <= 2.55e+14)) {
		tmp = sin(t_0) * pow(x_46_im, y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    if ((y_46re <= (-1.65d+14)) .or. (.not. (y_46re <= 2.55d+14))) then
        tmp = sin(t_0) * (x_46im ** y_46re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -1.65e+14) || !(y_46_re <= 2.55e+14)) {
		tmp = Math.sin(t_0) * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if (y_46_re <= -1.65e+14) or not (y_46_re <= 2.55e+14):
		tmp = math.sin(t_0) * math.pow(x_46_im, y_46_re)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_re <= -1.65e+14) || !(y_46_re <= 2.55e+14))
		tmp = Float64(sin(t_0) * (x_46_im ^ y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if ((y_46_re <= -1.65e+14) || ~((y_46_re <= 2.55e+14)))
		tmp = sin(t_0) * (x_46_im ^ y_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -1.65e+14], N[Not[LessEqual[y$46$re, 2.55e+14]], $MachinePrecision]], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.65e14 or 2.55e14 < y.re

   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 33.6%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 33.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 33.6%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 33.6%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 33.6%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 33.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 33.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 33.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 33.6%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 65.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 68.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 68.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 68.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 68.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 68.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.re around 0 57.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]

   if -1.65e14 < y.re < 2.55e14

   1. Initial program 41.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 41.2%

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. +-rgt-identity 41.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. +-rgt-identity 41.2%

         \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. exp-to-pow 41.2%

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. hypot-def 41.2%

         \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      6. *-commutative 41.2%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. exp-prod 41.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. +-commutative 41.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

      9. *-commutative 41.0%

         \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

   3. Simplified 81.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

   4. Taylor expanded in y.im around 0 22.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   5. Step-by-step derivation

      1. unpow2 22.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

      2. unpow2 22.9%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

      3. hypot-def 29.8%

         \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

   6. Simplified 29.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   7. Taylor expanded in x.im around 0 12.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]

   8. Taylor expanded in y.re around 0 26.0%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+14} \lor \neg \left(y.re \leq 2.55 \cdot 10^{+14}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \end{array} \]

Alternative 19: 44.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (* y.re (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * atan2(x_46_im, x_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (y_46_re * math.atan2(x_46_im, x_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (y_46_re * atan2(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. exp-diff 37.2%

      \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. +-rgt-identity 37.2%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. +-rgt-identity 37.2%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. exp-to-pow 37.2%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. hypot-def 37.2%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. *-commutative 37.2%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   7. exp-prod 37.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. +-commutative 37.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

   9. *-commutative 37.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

3. Simplified 73.3%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

4. Taylor expanded in y.im around 0 47.0%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
5. Step-by-step derivation

   1. unpow2 47.0%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

   2. unpow2 47.0%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

   3. hypot-def 50.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

6. Simplified 50.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
7. Step-by-step derivation

   1. *-commutative 50.2%

      \[\leadsto \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   2. add-exp-log 26.1%

      \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

8. Applied egg-rr 26.1%

   \[\leadsto \sin \color{blue}{\left(e^{\log \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

9. Taylor expanded in y.re around 0 49.8%

   \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

10. Final simplification 49.8%

    \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

Alternative 20: 13.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* y.re (atan2 x.im x.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * atan2(x_46_im, x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46re * atan2(x_46im, x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * Math.atan2(x_46_im, x_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_re * math.atan2(x_46_im, x_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_re * atan(x_46_im, x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_re * atan2(x_46_im, x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Step-by-step derivation

   1. exp-diff 37.2%

      \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. +-rgt-identity 37.2%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re + 0}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. +-rgt-identity 37.2%

      \[\leadsto \frac{e^{\color{blue}{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. exp-to-pow 37.2%

      \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. hypot-def 37.2%

      \[\leadsto \frac{{\color{blue}{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. *-commutative 37.2%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   7. exp-prod 37.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\color{blue}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. +-commutative 37.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)} \]

   9. *-commutative 37.1%

      \[\leadsto \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} + \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \]

3. Simplified 73.3%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right)} \]

4. Taylor expanded in y.im around 0 47.0%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
5. Step-by-step derivation

   1. unpow2 47.0%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \]

   2. unpow2 47.0%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)}^{y.re} \]

   3. hypot-def 50.2%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]

6. Simplified 50.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

7. Taylor expanded in x.im around 0 32.3%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]

8. Taylor expanded in y.re around 0 15.2%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

9. Final simplification 15.2%

   \[\leadsto y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} \]

Reproduce

herbie shell --seed 2023240 
(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)))))