powComplex, imaginary part

Percentage Accurate: 41.2% → 70.7%

Time: 22.7s

Alternatives: 27

Speedup: 3.5×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ e^{t\_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.2% 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: 70.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\frac{-1}{x.im}\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;x.im \leq -3.4 \cdot 10^{-49}:\\ \;\;\;\;e^{t\_1 \cdot \left(0 - y.re\right) - t\_0} \cdot \sin \left(t\_2 - t\_1 \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 1.3 \cdot 10^{+17}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0} \cdot \left(\sin t\_3 + y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \cos t\_3\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t\_0} \cdot \sin \left(t\_2 + y.im \cdot \log x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (log (/ -1.0 x.im)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (* y.im (log (hypot x.im x.re)))))
   (if (<= x.im -3.4e-49)
     (* (exp (- (* t_1 (- 0.0 y.re)) t_0)) (sin (- t_2 (* t_1 y.im))))
     (if (<= x.im 1.3e+17)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
        (+ (sin t_3) (* y.re (* (atan2 x.im x.re) (cos t_3)))))
       (*
        (exp (- (* y.re (log x.im)) t_0))
        (sin (+ t_2 (* y.im (log x.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = log((-1.0 / x_46_im));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	double tmp;
	if (x_46_im <= -3.4e-49) {
		tmp = exp(((t_1 * (0.0 - y_46_re)) - t_0)) * sin((t_2 - (t_1 * y_46_im)));
	} else if (x_46_im <= 1.3e+17) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (sin(t_3) + (y_46_re * (atan2(x_46_im, x_46_re) * cos(t_3))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0)) * sin((t_2 + (y_46_im * log(x_46_im))));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.log((-1.0 / x_46_im));
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if (x_46_im <= -3.4e-49) {
		tmp = Math.exp(((t_1 * (0.0 - y_46_re)) - t_0)) * Math.sin((t_2 - (t_1 * y_46_im)));
	} else if (x_46_im <= 1.3e+17) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (Math.sin(t_3) + (y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.cos(t_3))));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_0)) * Math.sin((t_2 + (y_46_im * Math.log(x_46_im))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = math.log((-1.0 / x_46_im))
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if x_46_im <= -3.4e-49:
		tmp = math.exp(((t_1 * (0.0 - y_46_re)) - t_0)) * math.sin((t_2 - (t_1 * y_46_im)))
	elif x_46_im <= 1.3e+17:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (math.sin(t_3) + (y_46_re * (math.atan2(x_46_im, x_46_re) * math.cos(t_3))))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_0)) * math.sin((t_2 + (y_46_im * math.log(x_46_im))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-1.0 / x_46_im))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -3.4e-49)
		tmp = Float64(exp(Float64(Float64(t_1 * Float64(0.0 - y_46_re)) - t_0)) * sin(Float64(t_2 - Float64(t_1 * y_46_im))));
	elseif (x_46_im <= 1.3e+17)
		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)) * Float64(sin(t_3) + Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * cos(t_3)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0)) * sin(Float64(t_2 + Float64(y_46_im * log(x_46_im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = log((-1.0 / x_46_im));
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = y_46_im * log(hypot(x_46_im, x_46_re));
	tmp = 0.0;
	if (x_46_im <= -3.4e-49)
		tmp = exp(((t_1 * (0.0 - y_46_re)) - t_0)) * sin((t_2 - (t_1 * y_46_im)));
	elseif (x_46_im <= 1.3e+17)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (sin(t_3) + (y_46_re * (atan2(x_46_im, x_46_re) * cos(t_3))));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_0)) * sin((t_2 + (y_46_im * log(x_46_im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -3.4e-49], N[(N[Exp[N[(N[(t$95$1 * N[(0.0 - y$46$re), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$2 - N[(t$95$1 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.3e+17], 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[(N[Sin[t$95$3], $MachinePrecision] + N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Cos[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$2 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.40000000000000005e-49

   1. Initial program 31.3%

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

   3. Taylor expanded in x.im around -inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \sin \color{blue}{\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{-1}{x.im}\right) \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \left(\frac{-1}{x.im}\right) \cdot \left(\mathsf{neg}\left(y.re\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \left(\frac{-1}{x.im}\right) \cdot \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{-1}{x.im}\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{-1}{x.im}\right)\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \left(\mathsf{neg}\left(y.re\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   5. Simplified 75.2%

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

   if -3.40000000000000005e-49 < x.im < 1.3e17

   1. Initial program 50.8%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(\color{blue}{y.re} \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      7. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right)\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \cos \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right)\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 70.1%

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

   if 1.3e17 < x.im

   1. Initial program 25.7%

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

   3. Taylor expanded in x.im around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \sin \color{blue}{\left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(0 - y.re \cdot \log \left(\frac{1}{x.im}\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \log \left(\frac{1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. log-rec N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \left(\mathsf{neg}\left(\log x.im\right)\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \left(0 - \log x.im\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \mathsf{\_.f64}\left(0, \log x.im\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(x.im\right)\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(x.im\right)\right)\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(x.im\right)\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.im}\right)}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.re, \mathsf{\_.f64}\left(0, \mathsf{log.f64}\left(x.im\right)\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   5. Simplified 73.8%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.4 \cdot 10^{-49}:\\ \;\;\;\;e^{\log \left(\frac{-1}{x.im}\right) \cdot \left(0 - y.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{-1}{x.im}\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 1.3 \cdot 10^{+17}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) + y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 69.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{-1}{x.im}\right)\\ \mathbf{if}\;x.im \leq -6.5 \cdot 10^{-49}:\\ \;\;\;\;e^{t\_2 \cdot \left(0 - y.re\right) - t\_0} \cdot \sin \left(t\_1 - t\_2 \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 1.5 \cdot 10^{+138}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0} \cdot \left(\sin t\_1 + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.im \cdot e^{y.re \cdot \log x.im - t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ -1.0 x.im))))
   (if (<= x.im -6.5e-49)
     (* (exp (- (* t_2 (- 0.0 y.re)) t_0)) (sin (- t_1 (* t_2 y.im))))
     (if (<= x.im 1.5e+138)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
        (+ (sin t_1) (* y.im (* (log (hypot x.im x.re)) (cos t_1)))))
       (* y.im (* (log x.im) (exp (- (* y.re (log x.im)) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log((-1.0 / x_46_im));
	double tmp;
	if (x_46_im <= -6.5e-49) {
		tmp = exp(((t_2 * (0.0 - y_46_re)) - t_0)) * sin((t_1 - (t_2 * y_46_im)));
	} else if (x_46_im <= 1.5e+138) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (sin(t_1) + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_1))));
	} else {
		tmp = y_46_im * (log(x_46_im) * exp(((y_46_re * log(x_46_im)) - t_0)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log((-1.0 / x_46_im));
	double tmp;
	if (x_46_im <= -6.5e-49) {
		tmp = Math.exp(((t_2 * (0.0 - y_46_re)) - t_0)) * Math.sin((t_1 - (t_2 * y_46_im)));
	} else if (x_46_im <= 1.5e+138) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (Math.sin(t_1) + (y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * Math.cos(t_1))));
	} else {
		tmp = y_46_im * (Math.log(x_46_im) * Math.exp(((y_46_re * Math.log(x_46_im)) - t_0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.log((-1.0 / x_46_im))
	tmp = 0
	if x_46_im <= -6.5e-49:
		tmp = math.exp(((t_2 * (0.0 - y_46_re)) - t_0)) * math.sin((t_1 - (t_2 * y_46_im)))
	elif x_46_im <= 1.5e+138:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (math.sin(t_1) + (y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * math.cos(t_1))))
	else:
		tmp = y_46_im * (math.log(x_46_im) * math.exp(((y_46_re * math.log(x_46_im)) - t_0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(Float64(-1.0 / x_46_im))
	tmp = 0.0
	if (x_46_im <= -6.5e-49)
		tmp = Float64(exp(Float64(Float64(t_2 * Float64(0.0 - y_46_re)) - t_0)) * sin(Float64(t_1 - Float64(t_2 * y_46_im))));
	elseif (x_46_im <= 1.5e+138)
		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)) * Float64(sin(t_1) + Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * cos(t_1)))));
	else
		tmp = Float64(y_46_im * Float64(log(x_46_im) * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log((-1.0 / x_46_im));
	tmp = 0.0;
	if (x_46_im <= -6.5e-49)
		tmp = exp(((t_2 * (0.0 - y_46_re)) - t_0)) * sin((t_1 - (t_2 * y_46_im)));
	elseif (x_46_im <= 1.5e+138)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * (sin(t_1) + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_1))));
	else
		tmp = y_46_im * (log(x_46_im) * exp(((y_46_re * log(x_46_im)) - t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -6.5e-49], N[(N[Exp[N[(N[(t$95$2 * N[(0.0 - y$46$re), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 - N[(t$95$2 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.5e+138], 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[(N[Sin[t$95$1], $MachinePrecision] + N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[Log[x$46$im], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -6.49999999999999968e-49

   1. Initial program 30.4%

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

   3. Taylor expanded in x.im around -inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \sin \color{blue}{\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y.re \cdot \log \left(\frac{-1}{x.im}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{-1}{x.im}\right) \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \left(\frac{-1}{x.im}\right) \cdot \left(\mathsf{neg}\left(y.re\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \left(\frac{-1}{x.im}\right) \cdot \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{-1}{x.im}\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{-1}{x.im}\right)\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \left(\mathsf{neg}\left(y.re\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   5. Simplified 74.9%

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

   if -6.49999999999999968e-49 < x.im < 1.50000000000000005e138

   1. Initial program 54.0%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)}\right) \]
   4. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)}\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\color{blue}{y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right)\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) \]

      11. accelerator-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \cos \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) \]

      12. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 66.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right)\right)\right)\right) \]

   5. Simplified 66.9%

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

   if 1.50000000000000005e138 < x.im

   1. Initial program 2.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 43.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 43.5%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 46.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 46.0%

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

   9. Taylor expanded in x.re around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log x.im\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\left(e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\log x.im}\right)\right) \]

       3. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \log \color{blue}{x.im}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \log x.im\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.im\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \log x.im\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.im\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.im\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.im\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.im\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.im\right)\right) \]

       8. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.im\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \log x.im\right)\right) \]

       9. log-lowering-log.f64 69.5%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.im\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.im\right)\right)\right) \]

   11. Simplified 69.5%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.5 \cdot 10^{-49}:\\ \;\;\;\;e^{\log \left(\frac{-1}{x.im}\right) \cdot \left(0 - y.re\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{-1}{x.im}\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 1.5 \cdot 10^{+138}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.im \cdot e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{-1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -260:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot t\_2 - t\_0} \cdot \sin \left(t\_1 - y.im \cdot t\_2\right)\\ \mathbf{elif}\;x.re \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 1.65 \cdot 10^{+208}:\\ \;\;\;\;\frac{{x.re}^{y.re}}{e^{t\_0}} \cdot \sin \left(t\_1 + y.im \cdot \log x.re\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot e^{y.re \cdot \log x.re - t\_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ -1.0 x.re))))
   (if (<= x.re -260.0)
     (* (exp (- (* (- 0.0 y.re) t_2) t_0)) (sin (- t_1 (* y.im t_2))))
     (if (<= x.re 2.2e-69)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
        (sin (* y.im (log (hypot x.im x.re)))))
       (if (<= x.re 1.65e+208)
         (* (/ (pow x.re y.re) (exp t_0)) (sin (+ t_1 (* y.im (log x.re)))))
         (* y.im (* (log x.re) (exp (- (* y.re (log x.re)) t_0)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log((-1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -260.0) {
		tmp = exp((((0.0 - y_46_re) * t_2) - t_0)) * sin((t_1 - (y_46_im * t_2)));
	} else if (x_46_re <= 2.2e-69) {
		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 if (x_46_re <= 1.65e+208) {
		tmp = (pow(x_46_re, y_46_re) / exp(t_0)) * sin((t_1 + (y_46_im * log(x_46_re))));
	} else {
		tmp = y_46_im * (log(x_46_re) * exp(((y_46_re * log(x_46_re)) - t_0)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log((-1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -260.0) {
		tmp = Math.exp((((0.0 - y_46_re) * t_2) - t_0)) * Math.sin((t_1 - (y_46_im * t_2)));
	} else if (x_46_re <= 2.2e-69) {
		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))));
	} else if (x_46_re <= 1.65e+208) {
		tmp = (Math.pow(x_46_re, y_46_re) / Math.exp(t_0)) * Math.sin((t_1 + (y_46_im * Math.log(x_46_re))));
	} else {
		tmp = y_46_im * (Math.log(x_46_re) * Math.exp(((y_46_re * Math.log(x_46_re)) - t_0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.log((-1.0 / x_46_re))
	tmp = 0
	if x_46_re <= -260.0:
		tmp = math.exp((((0.0 - y_46_re) * t_2) - t_0)) * math.sin((t_1 - (y_46_im * t_2)))
	elif x_46_re <= 2.2e-69:
		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))))
	elif x_46_re <= 1.65e+208:
		tmp = (math.pow(x_46_re, y_46_re) / math.exp(t_0)) * math.sin((t_1 + (y_46_im * math.log(x_46_re))))
	else:
		tmp = y_46_im * (math.log(x_46_re) * math.exp(((y_46_re * math.log(x_46_re)) - t_0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(Float64(-1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -260.0)
		tmp = Float64(exp(Float64(Float64(Float64(0.0 - y_46_re) * t_2) - t_0)) * sin(Float64(t_1 - Float64(y_46_im * t_2))));
	elseif (x_46_re <= 2.2e-69)
		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)))));
	elseif (x_46_re <= 1.65e+208)
		tmp = Float64(Float64((x_46_re ^ y_46_re) / exp(t_0)) * sin(Float64(t_1 + Float64(y_46_im * log(x_46_re)))));
	else
		tmp = Float64(y_46_im * Float64(log(x_46_re) * exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log((-1.0 / x_46_re));
	tmp = 0.0;
	if (x_46_re <= -260.0)
		tmp = exp((((0.0 - y_46_re) * t_2) - t_0)) * sin((t_1 - (y_46_im * t_2)));
	elseif (x_46_re <= 2.2e-69)
		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))));
	elseif (x_46_re <= 1.65e+208)
		tmp = ((x_46_re ^ y_46_re) / exp(t_0)) * sin((t_1 + (y_46_im * log(x_46_re))));
	else
		tmp = y_46_im * (log(x_46_re) * exp(((y_46_re * log(x_46_re)) - t_0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -260.0], N[(N[Exp[N[(N[(N[(0.0 - y$46$re), $MachinePrecision] * 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[x$46$re, 2.2e-69], 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], If[LessEqual[x$46$re, 1.65e+208], N[(N[(N[Power[x$46$re, y$46$re], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -260

   1. Initial program 21.8%

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

   3. Taylor expanded in x.re around -inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \sin \color{blue}{\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(-1 \cdot \left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(y.re \cdot \log \left(\frac{-1}{x.re}\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\mathsf{neg}\left(\log \left(\frac{-1}{x.re}\right) \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \left(\frac{-1}{x.re}\right) \cdot \left(\mathsf{neg}\left(y.re\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(\log \left(\frac{-1}{x.re}\right) \cdot \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\log \left(\frac{-1}{x.re}\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{-1} \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\frac{-1}{x.re}\right)\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.re\right)\right), \left(-1 \cdot y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.re\right)\right), \left(\mathsf{neg}\left(y.re\right)\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.re\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.re\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(-1 \cdot \color{blue}{\left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.re\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \sin \left(-1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.re}\right)}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.re\right)\right), \mathsf{neg.f64}\left(y.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   5. Simplified 81.5%

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

   if -260 < x.re < 2.2e-69

   1. Initial program 52.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 64.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 64.4%

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

   if 2.2e-69 < x.re < 1.65e208

   1. Initial program 48.7%

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

   3. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\sin \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. exp-diff N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{e^{y.re \cdot \log x.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right), \sin \color{blue}{\left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{y.re \cdot \log x.re}\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right), \sin \color{blue}{\left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(e^{\log x.re \cdot y.re}\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right), \sin \left(\color{blue}{y.im} \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. exp-to-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left({x.re}^{y.re}\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right), \sin \left(\color{blue}{y.im \cdot \log x.re} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right), \sin \left(\color{blue}{y.im \cdot \log x.re} + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(y.im \cdot \log x.re + \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(y.im \cdot \log x.re + \color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \sin \left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log x.re + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \left(y.im \cdot \log x.re\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \left(y.im \cdot \log x.re\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \left(y.im \cdot \log x.re\right)\right)\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(y.im, \log x.re\right)\right)\right)\right) \]

      16. log-lowering-log.f64 72.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(x.re, y.re\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right)\right)\right)\right) \]

   5. Simplified 72.0%

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

   if 1.65e208 < x.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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 44.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 44.0%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 44.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 44.1%

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

   9. Taylor expanded in x.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log x.re\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\log x.re}\right)\right) \]

       3. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \log \color{blue}{x.re}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       8. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \log x.re\right)\right) \]

       9. log-lowering-log.f64 75.4%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

   11. Simplified 75.4%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -260:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot \log \left(\frac{-1}{x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{elif}\;x.re \leq 2.2 \cdot 10^{-69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 1.65 \cdot 10^{+208}:\\ \;\;\;\;\frac{{x.re}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.re\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.re \leq -3 \cdot 10^{-68}:\\ \;\;\;\;y.im \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t\_0}{y.im}\\ \mathbf{elif}\;y.re \leq 95000000000000:\\ \;\;\;\;t\_1 \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (sin (* y.im (log (hypot x.im x.re))))))
   (if (<= y.re -3e-68)
     (* y.im (/ (* (sin (* y.re (atan2 x.im x.re))) t_0) y.im))
     (if (<= y.re 95000000000000.0)
       (* t_1 (exp (* y.im (- 0.0 (atan2 x.im x.re)))))
       (* t_1 t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -3e-68) {
		tmp = y_46_im * ((sin((y_46_re * atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	} else if (y_46_re <= 95000000000000.0) {
		tmp = t_1 * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -3e-68) {
		tmp = y_46_im * ((Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	} else if (y_46_re <= 95000000000000.0) {
		tmp = t_1 * Math.exp((y_46_im * (0.0 - Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -3e-68:
		tmp = y_46_im * ((math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * t_0) / y_46_im)
	elif y_46_re <= 95000000000000.0:
		tmp = t_1 * math.exp((y_46_im * (0.0 - math.atan2(x_46_im, x_46_re))))
	else:
		tmp = t_1 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -3e-68)
		tmp = Float64(y_46_im * Float64(Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * t_0) / y_46_im));
	elseif (y_46_re <= 95000000000000.0)
		tmp = Float64(t_1 * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re)))));
	else
		tmp = Float64(t_1 * 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 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -3e-68)
		tmp = y_46_im * ((sin((y_46_re * atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	elseif (y_46_re <= 95000000000000.0)
		tmp = t_1 * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	else
		tmp = t_1 * 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3e-68], N[(y$46$im * N[(N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 95000000000000.0], N[(t$95$1 * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3e-68

   1. Initial program 32.0%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 68.1%

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

   6. Taylor expanded in y.im around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) + \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im}\right) - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) + \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im} - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{+.f64}\left(\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im} - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right)\right) \]

   8. Simplified 68.1%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{y.im}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       3. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       5. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       6. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right)\right), y.im\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right), y.im\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right), y.im\right)\right) \]

       9. accelerator-lowering-hypot.f64 75.9%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right), y.im\right)\right) \]

   11. Simplified 75.9%

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

   if -3e-68 < y.re < 9.5e13

   1. Initial program 42.4%

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

   3. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]

      2. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]

      3. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      6. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      9. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      12. accelerator-lowering-hypot.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 61.3%

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

   if 9.5e13 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 73.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 73.7%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 72.0%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3 \cdot 10^{-68}:\\ \;\;\;\;y.im \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{y.im}\\ \mathbf{elif}\;y.re \leq 95000000000000:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 65.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{-69}:\\ \;\;\;\;y.im \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t\_0}{y.im}\\ \mathbf{elif}\;y.re \leq 47000000000000:\\ \;\;\;\;t\_1 \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (* y.im (log (hypot x.im x.re)))))
   (if (<= y.re -4e-69)
     (* y.im (/ (* (sin (* y.re (atan2 x.im x.re))) t_0) y.im))
     (if (<= y.re 47000000000000.0)
       (* t_1 (exp (* y.im (- 0.0 (atan2 x.im x.re)))))
       (* (sin t_1) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -4e-69) {
		tmp = y_46_im * ((sin((y_46_re * atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	} else if (y_46_re <= 47000000000000.0) {
		tmp = t_1 * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	} else {
		tmp = sin(t_1) * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -4e-69) {
		tmp = y_46_im * ((Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	} else if (y_46_re <= 47000000000000.0) {
		tmp = t_1 * Math.exp((y_46_im * (0.0 - Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = Math.sin(t_1) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if y_46_re <= -4e-69:
		tmp = y_46_im * ((math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * t_0) / y_46_im)
	elif y_46_re <= 47000000000000.0:
		tmp = t_1 * math.exp((y_46_im * (0.0 - math.atan2(x_46_im, x_46_re))))
	else:
		tmp = math.sin(t_1) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -4e-69)
		tmp = Float64(y_46_im * Float64(Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * t_0) / y_46_im));
	elseif (y_46_re <= 47000000000000.0)
		tmp = Float64(t_1 * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re)))));
	else
		tmp = Float64(sin(t_1) * 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 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_1 = y_46_im * log(hypot(x_46_im, x_46_re));
	tmp = 0.0;
	if (y_46_re <= -4e-69)
		tmp = y_46_im * ((sin((y_46_re * atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	elseif (y_46_re <= 47000000000000.0)
		tmp = t_1 * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	else
		tmp = sin(t_1) * 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4e-69], N[(y$46$im * N[(N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 47000000000000.0], N[(t$95$1 * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$1], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.9999999999999999e-69

   1. Initial program 32.0%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 68.1%

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

   6. Taylor expanded in y.im around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) + \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im}\right) - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) + \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im} - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{+.f64}\left(\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im} - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right)\right) \]

   8. Simplified 68.1%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{y.im}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       3. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       5. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       6. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right)\right), y.im\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right), y.im\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right), y.im\right)\right) \]

       9. accelerator-lowering-hypot.f64 75.9%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right), y.im\right)\right) \]

   11. Simplified 75.9%

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

   if -3.9999999999999999e-69 < y.re < 4.7e13

   1. Initial program 42.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 41.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 41.9%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 36.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 36.6%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}, \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       5. atan2-lowering-atan2.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 59.3%

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

   if 4.7e13 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 73.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 73.7%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 72.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 72.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{-69}:\\ \;\;\;\;y.im \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{y.im}\\ \mathbf{elif}\;y.re \leq 47000000000000:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot t\_2\right)\\ \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+27}:\\ \;\;\;\;t\_1 \cdot {\left(t\_0 \cdot t\_0\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-112}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-131}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (sin (* y.re (atan2 x.im x.re))))
        (t_2 (pow (hypot x.im x.re) y.re))
        (t_3 (* y.im (* (log (hypot x.im x.re)) t_2))))
   (if (<= y.im -4.4e+27)
     (* t_1 (pow (* t_0 t_0) (/ (/ y.re 2.0) 2.0)))
     (if (<= y.im -7e-112) t_3 (if (<= y.im 4.6e-131) (* t_1 t_2) t_3)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_2);
	double tmp;
	if (y_46_im <= -4.4e+27) {
		tmp = t_1 * pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	} else if (y_46_im <= -7e-112) {
		tmp = t_3;
	} else if (y_46_im <= 4.6e-131) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_2 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * t_2);
	double tmp;
	if (y_46_im <= -4.4e+27) {
		tmp = t_1 * Math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	} else if (y_46_im <= -7e-112) {
		tmp = t_3;
	} else if (y_46_im <= 4.6e-131) {
		tmp = t_1 * t_2;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_2 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_3 = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * t_2)
	tmp = 0
	if y_46_im <= -4.4e+27:
		tmp = t_1 * math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0))
	elif y_46_im <= -7e-112:
		tmp = t_3
	elif y_46_im <= 4.6e-131:
		tmp = t_1 * t_2
	else:
		tmp = t_3
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_3 = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * t_2))
	tmp = 0.0
	if (y_46_im <= -4.4e+27)
		tmp = Float64(t_1 * (Float64(t_0 * t_0) ^ Float64(Float64(y_46_re / 2.0) / 2.0)));
	elseif (y_46_im <= -7e-112)
		tmp = t_3;
	elseif (y_46_im <= 4.6e-131)
		tmp = Float64(t_1 * t_2);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_2 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_3 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_2);
	tmp = 0.0;
	if (y_46_im <= -4.4e+27)
		tmp = t_1 * ((t_0 * t_0) ^ ((y_46_re / 2.0) / 2.0));
	elseif (y_46_im <= -7e-112)
		tmp = t_3;
	elseif (y_46_im <= 4.6e-131)
		tmp = t_1 * t_2;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.4e+27], N[(t$95$1 * N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(y$46$re / 2.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7e-112], t$95$3, If[LessEqual[y$46$im, 4.6e-131], N[(t$95$1 * t$95$2), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.3999999999999997e27

   1. Initial program 35.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 29.5%

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

      1. sqrt-pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)} \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{y.re}{2}\right), 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \color{blue}{\mathsf{atan2.f64}\left(x.im, x.re\right)}\right)\right)\right) \]

      15. /-lowering-/.f64 40.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.re, 2\right), 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(\color{blue}{x.im}, x.re\right)\right)\right)\right) \]

   7. Applied egg-rr 40.5%

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

   if -4.3999999999999997e27 < y.im < -6.99999999999999988e-112 or 4.60000000000000044e-131 < y.im

   1. Initial program 32.8%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 60.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 60.7%

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

   6. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \left({\left(\sqrt{{\color{blue}{x.im}}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \left({\left(\sqrt{{x.im}^{\color{blue}{2}} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right)\right) \]

      10. accelerator-lowering-hypot.f64 56.2%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right)\right) \]

   8. Simplified 56.2%

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

   if -6.99999999999999988e-112 < y.im < 4.60000000000000044e-131

   1. Initial program 48.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 73.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 73.0%

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

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.4 \cdot 10^{+27}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-112}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-131}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot t\_1\right)\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\sin t\_2 \cdot {\left(t\_0 \cdot t\_0\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{elif}\;y.im \leq -3.05 \cdot 10^{-204}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-132}:\\ \;\;\;\;t\_2 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im)))
        (t_1 (pow (hypot x.im x.re) y.re))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (* y.im (* (log (hypot x.im x.re)) t_1))))
   (if (<= y.im -3e+25)
     (* (sin t_2) (pow (* t_0 t_0) (/ (/ y.re 2.0) 2.0)))
     (if (<= y.im -3.05e-204) t_3 (if (<= y.im 1.35e-132) (* t_2 t_1) t_3)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_1);
	double tmp;
	if (y_46_im <= -3e+25) {
		tmp = sin(t_2) * pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	} else if (y_46_im <= -3.05e-204) {
		tmp = t_3;
	} else if (y_46_im <= 1.35e-132) {
		tmp = t_2 * t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = y_46_im * (Math.log(Math.hypot(x_46_im, x_46_re)) * t_1);
	double tmp;
	if (y_46_im <= -3e+25) {
		tmp = Math.sin(t_2) * Math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0));
	} else if (y_46_im <= -3.05e-204) {
		tmp = t_3;
	} else if (y_46_im <= 1.35e-132) {
		tmp = t_2 * t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = y_46_im * (math.log(math.hypot(x_46_im, x_46_re)) * t_1)
	tmp = 0
	if y_46_im <= -3e+25:
		tmp = math.sin(t_2) * math.pow((t_0 * t_0), ((y_46_re / 2.0) / 2.0))
	elif y_46_im <= -3.05e-204:
		tmp = t_3
	elif y_46_im <= 1.35e-132:
		tmp = t_2 * t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * t_1))
	tmp = 0.0
	if (y_46_im <= -3e+25)
		tmp = Float64(sin(t_2) * (Float64(t_0 * t_0) ^ Float64(Float64(y_46_re / 2.0) / 2.0)));
	elseif (y_46_im <= -3.05e-204)
		tmp = t_3;
	elseif (y_46_im <= 1.35e-132)
		tmp = Float64(t_2 * t_1);
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = y_46_im * (log(hypot(x_46_im, x_46_re)) * t_1);
	tmp = 0.0;
	if (y_46_im <= -3e+25)
		tmp = sin(t_2) * ((t_0 * t_0) ^ ((y_46_re / 2.0) / 2.0));
	elseif (y_46_im <= -3.05e-204)
		tmp = t_3;
	elseif (y_46_im <= 1.35e-132)
		tmp = t_2 * t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3e+25], N[(N[Sin[t$95$2], $MachinePrecision] * N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], N[(N[(y$46$re / 2.0), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3.05e-204], t$95$3, If[LessEqual[y$46$im, 1.35e-132], N[(t$95$2 * t$95$1), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.00000000000000006e25

   1. Initial program 35.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 29.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 29.5%

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

      1. sqrt-pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      2. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)} \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      3. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      11. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right), \left(\frac{\frac{y.re}{2}}{2}\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right), \mathsf{/.f64}\left(\left(\frac{y.re}{2}\right), 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \color{blue}{\mathsf{atan2.f64}\left(x.im, x.re\right)}\right)\right)\right) \]

      15. /-lowering-/.f64 40.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right), \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.re, 2\right), 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(\color{blue}{x.im}, x.re\right)\right)\right)\right) \]

   7. Applied egg-rr 40.5%

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

   if -3.00000000000000006e25 < y.im < -3.04999999999999987e-204 or 1.34999999999999995e-132 < y.im

   1. Initial program 37.8%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 58.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 58.5%

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

   6. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \left({\left(\sqrt{{\color{blue}{x.im}}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \left({\left(\sqrt{{x.im}^{\color{blue}{2}} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right)\right) \]

      10. accelerator-lowering-hypot.f64 56.3%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right)\right) \]

   8. Simplified 56.3%

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

   if -3.04999999999999987e-204 < y.im < 1.34999999999999995e-132

   1. Initial program 43.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 80.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. atan2-lowering-atan2.f64 80.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 80.2%

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

4. Final simplification 57.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\\ \mathbf{elif}\;y.im \leq -3.05 \cdot 10^{-204}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-132}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-74}:\\ \;\;\;\;y.im \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t\_0}{y.im}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{+14}:\\ \;\;\;\;\left(y.im \cdot t\_1\right) \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(t\_1 \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re)) (t_1 (log (hypot x.im x.re))))
   (if (<= y.re -9.5e-74)
     (* y.im (/ (* (sin (* y.re (atan2 x.im x.re))) t_0) y.im))
     (if (<= y.re 8.6e+14)
       (* (* y.im t_1) (exp (* y.im (- 0.0 (atan2 x.im x.re)))))
       (* y.im (* t_1 t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -9.5e-74) {
		tmp = y_46_im * ((sin((y_46_re * atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	} else if (y_46_re <= 8.6e+14) {
		tmp = (y_46_im * t_1) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	} else {
		tmp = y_46_im * (t_1 * t_0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = Math.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -9.5e-74) {
		tmp = y_46_im * ((Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	} else if (y_46_re <= 8.6e+14) {
		tmp = (y_46_im * t_1) * Math.exp((y_46_im * (0.0 - Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = y_46_im * (t_1 * t_0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_1 = math.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if y_46_re <= -9.5e-74:
		tmp = y_46_im * ((math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * t_0) / y_46_im)
	elif y_46_re <= 8.6e+14:
		tmp = (y_46_im * t_1) * math.exp((y_46_im * (0.0 - math.atan2(x_46_im, x_46_re))))
	else:
		tmp = y_46_im * (t_1 * t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -9.5e-74)
		tmp = Float64(y_46_im * Float64(Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * t_0) / y_46_im));
	elseif (y_46_re <= 8.6e+14)
		tmp = Float64(Float64(y_46_im * t_1) * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re)))));
	else
		tmp = Float64(y_46_im * Float64(t_1 * 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 = hypot(x_46_im, x_46_re) ^ y_46_re;
	t_1 = log(hypot(x_46_im, x_46_re));
	tmp = 0.0;
	if (y_46_re <= -9.5e-74)
		tmp = y_46_im * ((sin((y_46_re * atan2(x_46_im, x_46_re))) * t_0) / y_46_im);
	elseif (y_46_re <= 8.6e+14)
		tmp = (y_46_im * t_1) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	else
		tmp = y_46_im * (t_1 * 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9.5e-74], N[(y$46$im * N[(N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.6e+14], N[(N[(y$46$im * t$95$1), $MachinePrecision] * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.5000000000000007e-74

   1. Initial program 32.0%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 68.1%

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

   6. Taylor expanded in y.im around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) + \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im}\right) - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right) \]

      2. associate--l+ N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right) + \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im} - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{+.f64}\left(\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im} - \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}\right)\right) \]

   8. Simplified 68.1%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}}{y.im}\right)}\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{y.im}\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       3. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       5. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), y.im\right)\right) \]

       6. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right)\right), y.im\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right), y.im\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right), y.im\right)\right) \]

       9. accelerator-lowering-hypot.f64 75.9%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right), y.im\right)\right) \]

   11. Simplified 75.9%

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

   if -9.5000000000000007e-74 < y.re < 8.6e14

   1. Initial program 42.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 41.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 41.9%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 36.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 36.6%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}, \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       5. atan2-lowering-atan2.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 59.3%

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

   if 8.6e14 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 73.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 73.7%

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

   6. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \left({\left(\sqrt{{\color{blue}{x.im}}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \left({\left(\sqrt{{x.im}^{\color{blue}{2}} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right)\right) \]

      10. accelerator-lowering-hypot.f64 66.7%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right)\right) \]

   8. Simplified 66.7%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-74}:\\ \;\;\;\;y.im \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{y.im}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{+14}:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{-69}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t\_1\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;\left(y.im \cdot t\_0\right) \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(t\_0 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re))) (t_1 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re -4.4e-69)
     (* (sin (* y.re (atan2 x.im x.re))) t_1)
     (if (<= y.re 46000000000000.0)
       (* (* y.im t_0) (exp (* y.im (- 0.0 (atan2 x.im x.re)))))
       (* y.im (* 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 = log(hypot(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 <= -4.4e-69) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * t_1;
	} else if (y_46_re <= 46000000000000.0) {
		tmp = (y_46_im * t_0) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	} else {
		tmp = y_46_im * (t_0 * t_1);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.hypot(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 <= -4.4e-69) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * t_1;
	} else if (y_46_re <= 46000000000000.0) {
		tmp = (y_46_im * t_0) * Math.exp((y_46_im * (0.0 - Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = y_46_im * (t_0 * t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.hypot(x_46_im, x_46_re))
	t_1 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -4.4e-69:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * t_1
	elif y_46_re <= 46000000000000.0:
		tmp = (y_46_im * t_0) * math.exp((y_46_im * (0.0 - math.atan2(x_46_im, x_46_re))))
	else:
		tmp = y_46_im * (t_0 * t_1)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -4.4e-69)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * t_1);
	elseif (y_46_re <= 46000000000000.0)
		tmp = Float64(Float64(y_46_im * t_0) * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re)))));
	else
		tmp = Float64(y_46_im * Float64(t_0 * t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(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 <= -4.4e-69)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * t_1;
	elseif (y_46_re <= 46000000000000.0)
		tmp = (y_46_im * t_0) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re))));
	else
		tmp = y_46_im * (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[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $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, -4.4e-69], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 46000000000000.0], N[(N[(y$46$im * t$95$0), $MachinePrecision] * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.4e-69

   1. Initial program 32.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 74.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 74.7%

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

   if -4.4e-69 < y.re < 4.6e13

   1. Initial program 42.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 41.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 41.9%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 36.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 36.6%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}, \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   10. Step-by-step derivation

       1. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

       5. atan2-lowering-atan2.f64 59.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 59.3%

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

   if 4.6e13 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 73.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 73.7%

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

   6. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right), \left({\left(\sqrt{{\color{blue}{x.im}}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \left({\left(\sqrt{{x.im}^{\color{blue}{2}} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right)\right) \]

      7. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right)\right) \]

      10. accelerator-lowering-hypot.f64 66.7%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right)\right) \]

   8. Simplified 66.7%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{-69}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;\left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 49.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;t\_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.32 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log t\_0\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-96}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_1 \cdot {t\_0}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im))) (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -4.5e-79)
     (* t_1 (pow (hypot x.im x.re) y.re))
     (if (<= y.re -1.32e-205)
       (* (* y.im (log t_0)) 0.5)
       (if (<= y.re 3.4e-96)
         (sin (* y.im (log (hypot x.im x.re))))
         (if (<= y.re 46000000000000.0)
           (* y.im (* (log x.re) (exp (* y.im (- 0.0 (atan2 x.im x.re))))))
           (* (sin t_1) (pow t_0 (/ y.re 2.0)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -1.32e-205) {
		tmp = (y_46_im * log(t_0)) * 0.5;
	} else if (y_46_re <= 3.4e-96) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else if (y_46_re <= 46000000000000.0) {
		tmp = y_46_im * (log(x_46_re) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re)))));
	} else {
		tmp = sin(t_1) * pow(t_0, (y_46_re / 2.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -1.32e-205) {
		tmp = (y_46_im * Math.log(t_0)) * 0.5;
	} else if (y_46_re <= 3.4e-96) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else if (y_46_re <= 46000000000000.0) {
		tmp = y_46_im * (Math.log(x_46_re) * Math.exp((y_46_im * (0.0 - Math.atan2(x_46_im, x_46_re)))));
	} else {
		tmp = Math.sin(t_1) * Math.pow(t_0, (y_46_re / 2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_re <= -4.5e-79:
		tmp = t_1 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= -1.32e-205:
		tmp = (y_46_im * math.log(t_0)) * 0.5
	elif y_46_re <= 3.4e-96:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	elif y_46_re <= 46000000000000.0:
		tmp = y_46_im * (math.log(x_46_re) * math.exp((y_46_im * (0.0 - math.atan2(x_46_im, x_46_re)))))
	else:
		tmp = math.sin(t_1) * math.pow(t_0, (y_46_re / 2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -4.5e-79)
		tmp = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= -1.32e-205)
		tmp = Float64(Float64(y_46_im * log(t_0)) * 0.5);
	elseif (y_46_re <= 3.4e-96)
		tmp = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))));
	elseif (y_46_re <= 46000000000000.0)
		tmp = Float64(y_46_im * Float64(log(x_46_re) * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re))))));
	else
		tmp = Float64(sin(t_1) * (t_0 ^ Float64(y_46_re / 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.5e-79)
		tmp = t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= -1.32e-205)
		tmp = (y_46_im * log(t_0)) * 0.5;
	elseif (y_46_re <= 3.4e-96)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	elseif (y_46_re <= 46000000000000.0)
		tmp = y_46_im * (log(x_46_re) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re)))));
	else
		tmp = sin(t_1) * (t_0 ^ (y_46_re / 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.5e-79], 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, -1.32e-205], N[(N[(y$46$im * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$46$re, 3.4e-96], N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 46000000000000.0], N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$1], $MachinePrecision] * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.5000000000000003e-79

   1. Initial program 32.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 74.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 74.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. atan2-lowering-atan2.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 71.6%

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

   if -4.5000000000000003e-79 < y.re < -1.31999999999999992e-205

   1. Initial program 46.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 43.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 26.2%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 41.4%

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

   if -1.31999999999999992e-205 < y.re < 3.4000000000000001e-96

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 44.3%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 42.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 42.0%

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

   9. Taylor expanded in y.re around 0

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

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

       6. accelerator-lowering-hypot.f64 42.0%

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 42.0%

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

   if 3.4000000000000001e-96 < y.re < 4.6e13

   1. Initial program 37.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 35.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 35.0%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 28.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 28.1%

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

   9. Taylor expanded in x.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log x.re\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\log x.re}\right)\right) \]

       3. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \log \color{blue}{x.re}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       8. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \log x.re\right)\right) \]

       9. log-lowering-log.f64 37.2%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

   11. Simplified 37.2%

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

   12. Taylor expanded in y.re around 0

       \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}, \mathsf{log.f64}\left(x.re\right)\right)\right) \]
   13. Step-by-step derivation

       1. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(\color{blue}{x.re}\right)\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.im \cdot \left(\mathsf{neg}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       6. atan2-lowering-atan2.f64 37.2%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

   14. Simplified 37.2%

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

   if 4.6e13 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 59.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 59.7%

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. sqrt-pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{y.re}{2}\right)\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 59.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Applied egg-rr 59.7%

      \[\leadsto \color{blue}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.32 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-96}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-93}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 2.55 \cdot 10^{-26}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (* y.re (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -4.5e-79)
     t_0
     (if (<= y.re -4e-205)
       (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5)
       (if (<= y.re 6e-93)
         (sin (* y.im (log (hypot x.im x.re))))
         (if (<= y.re 2.55e-26)
           (* y.im (* (log x.re) (exp (* y.im (- 0.0 (atan2 x.im x.re))))))
           t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * atan2(x_46_im, x_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = t_0;
	} else if (y_46_re <= -4e-205) {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 6e-93) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else if (y_46_re <= 2.55e-26) {
		tmp = y_46_im * (log(x_46_re) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = t_0;
	} else if (y_46_re <= -4e-205) {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 6e-93) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else if (y_46_re <= 2.55e-26) {
		tmp = y_46_im * (Math.log(x_46_re) * Math.exp((y_46_im * (0.0 - Math.atan2(x_46_im, x_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)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -4.5e-79:
		tmp = t_0
	elif y_46_re <= -4e-205:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	elif y_46_re <= 6e-93:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	elif y_46_re <= 2.55e-26:
		tmp = y_46_im * (math.log(x_46_re) * math.exp((y_46_im * (0.0 - math.atan2(x_46_im, x_46_re)))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -4.5e-79)
		tmp = t_0;
	elseif (y_46_re <= -4e-205)
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	elseif (y_46_re <= 6e-93)
		tmp = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))));
	elseif (y_46_re <= 2.55e-26)
		tmp = Float64(y_46_im * Float64(log(x_46_re) * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_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)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.5e-79)
		tmp = t_0;
	elseif (y_46_re <= -4e-205)
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	elseif (y_46_re <= 6e-93)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	elseif (y_46_re <= 2.55e-26)
		tmp = y_46_im * (log(x_46_re) * exp((y_46_im * (0.0 - atan2(x_46_im, x_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[(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]}, If[LessEqual[y$46$re, -4.5e-79], t$95$0, If[LessEqual[y$46$re, -4e-205], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$46$re, 6e-93], N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 2.55e-26], N[(y$46$im * N[(N[Log[x$46$re], $MachinePrecision] * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.5000000000000003e-79 or 2.54999999999999995e-26 < y.re

   1. Initial program 35.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 66.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 66.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. atan2-lowering-atan2.f64 63.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 63.5%

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

   if -4.5000000000000003e-79 < y.re < -4e-205

   1. Initial program 46.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 43.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 26.2%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 41.4%

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

   if -4e-205 < y.re < 6.0000000000000003e-93

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 44.3%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 42.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 42.0%

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

   9. Taylor expanded in y.re around 0

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

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

       6. accelerator-lowering-hypot.f64 42.0%

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 42.0%

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

   if 6.0000000000000003e-93 < y.re < 2.54999999999999995e-26

   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. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 33.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 33.0%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 31.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 31.5%

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

   9. Taylor expanded in x.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log x.re\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\log x.re}\right)\right) \]

       3. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \log \color{blue}{x.re}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       8. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \log x.re\right)\right) \]

       9. log-lowering-log.f64 43.6%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

   11. Simplified 43.6%

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

   12. Taylor expanded in y.re around 0

       \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\color{blue}{\left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}, \mathsf{log.f64}\left(x.re\right)\right)\right) \]
   13. Step-by-step derivation

       1. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(\color{blue}{x.re}\right)\right)\right) \]

       2. distribute-rgt-neg-in N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.im \cdot \left(\mathsf{neg}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       3. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.im \cdot \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \left(-1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

       6. atan2-lowering-atan2.f64 43.6%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

   14. Simplified 43.6%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-93}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 2.55 \cdot 10^{-26}:\\ \;\;\;\;y.im \cdot \left(\log x.re \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;t\_0 \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 1.28 \cdot 10^{-21}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {\left(0 - x.im\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (sin t_0)))
   (if (<= y.re -48.0)
     (*
      t_0
      (pow (* x.im (+ (* (* x.re 0.5) (/ x.re (* x.im x.im))) 1.0)) y.re))
     (if (<= y.re -4.5e-79)
       t_1
       (if (<= y.re -4.5e-205)
         (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5)
         (if (<= y.re 1.28e-21)
           (sin (* y.im (log (hypot x.im x.re))))
           (* t_1 (pow (- 0.0 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 = sin(t_0);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_0 * pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	} else if (y_46_re <= -4.5e-79) {
		tmp = t_1;
	} else if (y_46_re <= -4.5e-205) {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 1.28e-21) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = t_1 * pow((0.0 - 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.sin(t_0);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_0 * Math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	} else if (y_46_re <= -4.5e-79) {
		tmp = t_1;
	} else if (y_46_re <= -4.5e-205) {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 1.28e-21) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = t_1 * Math.pow((0.0 - 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.sin(t_0)
	tmp = 0
	if y_46_re <= -48.0:
		tmp = t_0 * math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re)
	elif y_46_re <= -4.5e-79:
		tmp = t_1
	elif y_46_re <= -4.5e-205:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	elif y_46_re <= 1.28e-21:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	else:
		tmp = t_1 * math.pow((0.0 - 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 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -48.0)
		tmp = Float64(t_0 * (Float64(x_46_im * Float64(Float64(Float64(x_46_re * 0.5) * Float64(x_46_re / Float64(x_46_im * x_46_im))) + 1.0)) ^ y_46_re));
	elseif (y_46_re <= -4.5e-79)
		tmp = t_1;
	elseif (y_46_re <= -4.5e-205)
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	elseif (y_46_re <= 1.28e-21)
		tmp = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = Float64(t_1 * (Float64(0.0 - 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 = sin(t_0);
	tmp = 0.0;
	if (y_46_re <= -48.0)
		tmp = t_0 * ((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)) ^ y_46_re);
	elseif (y_46_re <= -4.5e-79)
		tmp = t_1;
	elseif (y_46_re <= -4.5e-205)
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	elseif (y_46_re <= 1.28e-21)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = t_1 * ((0.0 - 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[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -48.0], N[(t$95$0 * N[Power[N[(x$46$im * N[(N[(N[(x$46$re * 0.5), $MachinePrecision] * N[(x$46$re / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -4.5e-79], t$95$1, If[LessEqual[y$46$re, -4.5e-205], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$46$re, 1.28e-21], N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Power[N[(0.0 - x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -48

   1. Initial program 30.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 82.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 82.3%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(x.im \cdot \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)}, y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot {x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.re}^{2}\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.re}^{2}\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.re \cdot x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      9. *-lowering-*.f64 57.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 57.4%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot x.re\right) \cdot x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x.re\right) \cdot \frac{x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot x.re\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x.re \cdot \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \left(x.im \cdot x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 73.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Applied egg-rr 73.6%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       2. atan2-lowering-atan2.f64 71.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   13. Simplified 71.8%

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

   if -48 < y.re < -4.5000000000000003e-79

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 55.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 55.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 54.2%

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

   if -4.5000000000000003e-79 < y.re < -4.49999999999999956e-205

   1. Initial program 46.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 43.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 26.2%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 41.4%

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

   if -4.49999999999999956e-205 < y.re < 1.27999999999999995e-21

   1. Initial program 40.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 40.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 40.7%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 34.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 34.5%

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

   9. Taylor expanded in y.re around 0

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

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

       6. accelerator-lowering-hypot.f64 34.5%

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 34.5%

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

   if 1.27999999999999995e-21 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 55.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 55.8%

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

   6. Taylor expanded in x.im around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot x.im\right)}, y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(x.im\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 52.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(x.im\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 52.7%

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

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 1.28 \cdot 10^{-21}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(0 - x.im\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;t\_0 \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-31}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (sin t_0)))
   (if (<= y.re -48.0)
     (*
      t_0
      (pow (* x.im (+ (* (* x.re 0.5) (/ x.re (* x.im x.im))) 1.0)) y.re))
     (if (<= y.re -5e-79)
       t_1
       (if (<= y.re -5e-208)
         (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5)
         (if (<= y.re 4e-31)
           (sin (* y.im (log (hypot x.im x.re))))
           (* 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_0 * pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	} else if (y_46_re <= -5e-79) {
		tmp = t_1;
	} else if (y_46_re <= -5e-208) {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 4e-31) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = 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 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_0 * Math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	} else if (y_46_re <= -5e-79) {
		tmp = t_1;
	} else if (y_46_re <= -5e-208) {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 4e-31) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = 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 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin(t_0)
	tmp = 0
	if y_46_re <= -48.0:
		tmp = t_0 * math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re)
	elif y_46_re <= -5e-79:
		tmp = t_1
	elif y_46_re <= -5e-208:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	elif y_46_re <= 4e-31:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	else:
		tmp = 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(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -48.0)
		tmp = Float64(t_0 * (Float64(x_46_im * Float64(Float64(Float64(x_46_re * 0.5) * Float64(x_46_re / Float64(x_46_im * x_46_im))) + 1.0)) ^ y_46_re));
	elseif (y_46_re <= -5e-79)
		tmp = t_1;
	elseif (y_46_re <= -5e-208)
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	elseif (y_46_re <= 4e-31)
		tmp = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = Float64(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 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y_46_re <= -48.0)
		tmp = t_0 * ((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)) ^ y_46_re);
	elseif (y_46_re <= -5e-79)
		tmp = t_1;
	elseif (y_46_re <= -5e-208)
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	elseif (y_46_re <= 4e-31)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -48.0], N[(t$95$0 * N[Power[N[(x$46$im * N[(N[(N[(x$46$re * 0.5), $MachinePrecision] * N[(x$46$re / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5e-79], t$95$1, If[LessEqual[y$46$re, -5e-208], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$46$re, 4e-31], N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -48

   1. Initial program 30.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 82.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 82.3%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(x.im \cdot \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)}, y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot {x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.re}^{2}\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.re}^{2}\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.re \cdot x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      9. *-lowering-*.f64 57.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 57.4%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot x.re\right) \cdot x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x.re\right) \cdot \frac{x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot x.re\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x.re \cdot \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \left(x.im \cdot x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 73.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Applied egg-rr 73.6%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       2. atan2-lowering-atan2.f64 71.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   13. Simplified 71.8%

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

   if -48 < y.re < -4.99999999999999999e-79

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 55.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 55.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 54.2%

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

   if -4.99999999999999999e-79 < y.re < -4.99999999999999963e-208

   1. Initial program 46.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 43.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 26.2%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 41.4%

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

   if -4.99999999999999963e-208 < y.re < 4e-31

   1. Initial program 41.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 41.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 41.6%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 35.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 35.3%

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

   9. Taylor expanded in y.re around 0

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

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

       6. accelerator-lowering-hypot.f64 35.3%

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 35.3%

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

   if 4e-31 < y.re

   1. Initial program 38.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 55.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 55.6%

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

   6. Taylor expanded in x.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({x.im}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 49.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(x.im, y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)}\right)\right) \]

   8. Simplified 49.7%

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

4. Final simplification 49.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-79}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-208}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-31}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 44.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t\_0 \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -4.6 \cdot 10^{-79}:\\ \;\;\;\;\sin t\_0\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-27}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1
         (*
          t_0
          (pow
           (* x.im (+ (* (* x.re 0.5) (/ x.re (* x.im x.im))) 1.0))
           y.re))))
   (if (<= y.re -48.0)
     t_1
     (if (<= y.re -4.6e-79)
       (sin t_0)
       (if (<= y.re -3e-205)
         (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5)
         (if (<= y.re 7e-27) (sin (* y.im (log (hypot x.im 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 = t_0 * pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_1;
	} else if (y_46_re <= -4.6e-79) {
		tmp = sin(t_0);
	} else if (y_46_re <= -3e-205) {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 7e-27) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_1;
	} else if (y_46_re <= -4.6e-79) {
		tmp = Math.sin(t_0);
	} else if (y_46_re <= -3e-205) {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 7e-27) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = 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 = t_0 * math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re)
	tmp = 0
	if y_46_re <= -48.0:
		tmp = t_1
	elif y_46_re <= -4.6e-79:
		tmp = math.sin(t_0)
	elif y_46_re <= -3e-205:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	elif y_46_re <= 7e-27:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (Float64(x_46_im * Float64(Float64(Float64(x_46_re * 0.5) * Float64(x_46_re / Float64(x_46_im * x_46_im))) + 1.0)) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -48.0)
		tmp = t_1;
	elseif (y_46_re <= -4.6e-79)
		tmp = sin(t_0);
	elseif (y_46_re <= -3e-205)
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	elseif (y_46_re <= 7e-27)
		tmp = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = t_0 * ((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -48.0)
		tmp = t_1;
	elseif (y_46_re <= -4.6e-79)
		tmp = sin(t_0);
	elseif (y_46_re <= -3e-205)
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	elseif (y_46_re <= 7e-27)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[(x$46$im * N[(N[(N[(x$46$re * 0.5), $MachinePrecision] * N[(x$46$re / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -48.0], t$95$1, If[LessEqual[y$46$re, -4.6e-79], N[Sin[t$95$0], $MachinePrecision], If[LessEqual[y$46$re, -3e-205], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$46$re, 7e-27], N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -48 or 7.0000000000000003e-27 < y.re

   1. Initial program 34.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 68.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 68.0%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(x.im \cdot \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)}, y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot {x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.re}^{2}\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.re}^{2}\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.re \cdot x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      9. *-lowering-*.f64 49.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 49.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot x.re\right) \cdot x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x.re\right) \cdot \frac{x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot x.re\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x.re \cdot \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \left(x.im \cdot x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 57.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Applied egg-rr 57.4%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       2. atan2-lowering-atan2.f64 54.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   13. Simplified 54.1%

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

   if -48 < y.re < -4.60000000000000023e-79

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 55.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 55.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 54.2%

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

   if -4.60000000000000023e-79 < y.re < -3e-205

   1. Initial program 46.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 43.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 26.2%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 41.4%

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

   if -3e-205 < y.re < 7.0000000000000003e-27

   1. Initial program 41.4%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 41.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 41.6%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 35.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 35.3%

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

   9. Taylor expanded in y.re around 0

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

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

       6. accelerator-lowering-hypot.f64 35.3%

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 35.3%

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

4. Final simplification 46.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -4.6 \cdot 10^{-79}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-27}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-112}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (* y.re (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -4.5e-79)
     t_0
     (if (<= y.re -4.5e-205)
       (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5)
       (if (<= y.re 4.4e-112) (sin (* y.im (log (hypot x.im x.re)))) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * atan2(x_46_im, x_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = t_0;
	} else if (y_46_re <= -4.5e-205) {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 4.4e-112) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = t_0;
	} else if (y_46_re <= -4.5e-205) {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	} else if (y_46_re <= 4.4e-112) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_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)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -4.5e-79:
		tmp = t_0
	elif y_46_re <= -4.5e-205:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	elif y_46_re <= 4.4e-112:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -4.5e-79)
		tmp = t_0;
	elseif (y_46_re <= -4.5e-205)
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	elseif (y_46_re <= 4.4e-112)
		tmp = sin(Float64(y_46_im * log(hypot(x_46_im, x_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)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.5e-79)
		tmp = t_0;
	elseif (y_46_re <= -4.5e-205)
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	elseif (y_46_re <= 4.4e-112)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_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[(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]}, If[LessEqual[y$46$re, -4.5e-79], t$95$0, If[LessEqual[y$46$re, -4.5e-205], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision], If[LessEqual[y$46$re, 4.4e-112], N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.5000000000000003e-79 or 4.40000000000000042e-112 < y.re

   1. Initial program 35.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 59.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 59.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. atan2-lowering-atan2.f64 57.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 57.0%

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

   if -4.5000000000000003e-79 < y.re < -4.49999999999999956e-205

   1. Initial program 46.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 43.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 26.2%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 41.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 41.4%

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

   if -4.49999999999999956e-205 < y.re < 4.40000000000000042e-112

   1. Initial program 44.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 48.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 48.2%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\color{blue}{\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)}\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

      4. accelerator-lowering-hypot.f64 44.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.im}, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   8. Simplified 44.3%

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

   9. Taylor expanded in y.re around 0

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

       1. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

       6. accelerator-lowering-hypot.f64 44.3%

          \[\leadsto \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   11. Simplified 44.3%

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

4. Final simplification 52.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-205}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-112}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 45.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t\_0 \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-90}:\\ \;\;\;\;\sin t\_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-31}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1
         (*
          t_0
          (pow
           (* x.im (+ (* (* x.re 0.5) (/ x.re (* x.im x.im))) 1.0))
           y.re))))
   (if (<= y.re -48.0)
     t_1
     (if (<= y.re -1.25e-90)
       (sin t_0)
       (if (<= y.re 3e-31) (* y.im (log (hypot x.im 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 = t_0 * pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_1;
	} else if (y_46_re <= -1.25e-90) {
		tmp = sin(t_0);
	} else if (y_46_re <= 3e-31) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re);
	double tmp;
	if (y_46_re <= -48.0) {
		tmp = t_1;
	} else if (y_46_re <= -1.25e-90) {
		tmp = Math.sin(t_0);
	} else if (y_46_re <= 3e-31) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = 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 = t_0 * math.pow((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)), y_46_re)
	tmp = 0
	if y_46_re <= -48.0:
		tmp = t_1
	elif y_46_re <= -1.25e-90:
		tmp = math.sin(t_0)
	elif y_46_re <= 3e-31:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (Float64(x_46_im * Float64(Float64(Float64(x_46_re * 0.5) * Float64(x_46_re / Float64(x_46_im * x_46_im))) + 1.0)) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -48.0)
		tmp = t_1;
	elseif (y_46_re <= -1.25e-90)
		tmp = sin(t_0);
	elseif (y_46_re <= 3e-31)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = t_0 * ((x_46_im * (((x_46_re * 0.5) * (x_46_re / (x_46_im * x_46_im))) + 1.0)) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -48.0)
		tmp = t_1;
	elseif (y_46_re <= -1.25e-90)
		tmp = sin(t_0);
	elseif (y_46_re <= 3e-31)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[(x$46$im * N[(N[(N[(x$46$re * 0.5), $MachinePrecision] * N[(x$46$re / N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -48.0], t$95$1, If[LessEqual[y$46$re, -1.25e-90], N[Sin[t$95$0], $MachinePrecision], If[LessEqual[y$46$re, 3e-31], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -48 or 2.99999999999999981e-31 < y.re

   1. Initial program 34.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 68.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 68.0%

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

   6. Taylor expanded in x.im around inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(x.im \cdot \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)}, y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \left(1 + \frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(\color{blue}{y.re}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{1}{2} \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\frac{1}{2} \cdot {x.re}^{2}}{{x.im}^{2}}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.re}^{2}\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.re}^{2}\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.re \cdot x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left({x.im}^{2}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.im \cdot x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      9. *-lowering-*.f64 49.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 49.0%

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

      1. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\frac{\left(\frac{1}{2} \cdot x.re\right) \cdot x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \left(\left(\frac{1}{2} \cdot x.re\right) \cdot \frac{x.re}{x.im \cdot x.im}\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(\frac{1}{2} \cdot x.re\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\left(x.re \cdot \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \left(\frac{x.re}{x.im \cdot x.im}\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \left(x.im \cdot x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      7. *-lowering-*.f64 57.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Applied egg-rr 57.4%

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

   11. Taylor expanded in y.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       2. atan2-lowering-atan2.f64 54.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x.re, \frac{1}{2}\right), \mathsf{/.f64}\left(x.re, \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   13. Simplified 54.1%

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

   if -48 < y.re < -1.25000000000000005e-90

   1. Initial program 40.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 52.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 52.7%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 52.0%

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

   if -1.25000000000000005e-90 < y.re < 2.99999999999999981e-31

   1. Initial program 42.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 46.2%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 32.9%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 32.9%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -48:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-90}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-31}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot \left(\left(x.re \cdot 0.5\right) \cdot \frac{x.re}{x.im \cdot x.im} + 1\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 33.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot x.re + x.im \cdot x.im\\ \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;{x.re}^{y.re} \cdot \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;y.re \leq -1.52 \cdot 10^{-90}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-97}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left({\left(t\_0 \cdot t\_0\right)}^{0.25}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re x.re) (* x.im x.im))))
   (if (<= y.re -210000.0)
     (* (pow x.re y.re) (* y.im (log x.re)))
     (if (<= y.re -1.52e-90)
       (sin (* y.re (atan2 x.im x.re)))
       (if (<= y.re 6.5e-97)
         (* y.im (log (hypot x.im x.re)))
         (* y.im (log (pow (* t_0 t_0) 0.25))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = pow(x_46_re, y_46_re) * (y_46_im * log(x_46_re));
	} else if (y_46_re <= -1.52e-90) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 6.5e-97) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = y_46_im * log(pow((t_0 * t_0), 0.25));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = Math.pow(x_46_re, y_46_re) * (y_46_im * Math.log(x_46_re));
	} else if (y_46_re <= -1.52e-90) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 6.5e-97) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = y_46_im * Math.log(Math.pow((t_0 * t_0), 0.25));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	tmp = 0
	if y_46_re <= -210000.0:
		tmp = math.pow(x_46_re, y_46_re) * (y_46_im * math.log(x_46_re))
	elif y_46_re <= -1.52e-90:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 6.5e-97:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = y_46_im * math.log(math.pow((t_0 * t_0), 0.25))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (y_46_re <= -210000.0)
		tmp = Float64((x_46_re ^ y_46_re) * Float64(y_46_im * log(x_46_re)));
	elseif (y_46_re <= -1.52e-90)
		tmp = sin(Float64(y_46_re * atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 6.5e-97)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_im * log((Float64(t_0 * t_0) ^ 0.25)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	tmp = 0.0;
	if (y_46_re <= -210000.0)
		tmp = (x_46_re ^ y_46_re) * (y_46_im * log(x_46_re));
	elseif (y_46_re <= -1.52e-90)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 6.5e-97)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = y_46_im * log(((t_0 * t_0) ^ 0.25));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -210000.0], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.52e-90], N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 6.5e-97], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[Log[N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], 0.25], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.1e5

   1. Initial program 29.1%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 83.8%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 83.8%

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

   9. Taylor expanded in x.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log x.re\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\log x.re}\right)\right) \]

       3. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \log \color{blue}{x.re}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       8. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \log x.re\right)\right) \]

       9. log-lowering-log.f64 36.4%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

   11. Simplified 36.4%

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

   12. Taylor expanded in y.im around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log x.re\right), \color{blue}{\left({x.re}^{y.re}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log x.re\right), \left({\color{blue}{x.re}}^{y.re}\right)\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right), \left({x.re}^{y.re}\right)\right) \]

       5. pow-lowering-pow.f64 34.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right)\right) \]

   14. Simplified 34.6%

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

   if -2.1e5 < y.re < -1.51999999999999991e-90

   1. Initial program 42.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 54.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 54.4%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 50.3%

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

   if -1.51999999999999991e-90 < y.re < 6.5000000000000004e-97

   1. Initial program 42.4%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 50.1%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 37.6%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 37.6%

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

   if 6.5000000000000004e-97 < y.re

   1. Initial program 38.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 31.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 6.0%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 6.0%

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

      1. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)\right)\right) \]

      3. sqr-pow N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{\frac{1}{2}}{2}\right)} \cdot {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)\right) \]

      4. pow-prod-down N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left({\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{1}{2}}{2}\right)}\right)\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(x.im \cdot x.im + x.re \cdot x.re\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      12. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \left(\frac{\frac{1}{2}}{2}\right)\right)\right)\right) \]

      15. metadata-eval 29.7%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), \frac{1}{4}\right)\right)\right) \]

   10. Applied egg-rr 29.7%

       \[\leadsto y.im \cdot \log \color{blue}{\left({\left(\left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)}^{0.25}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;{x.re}^{y.re} \cdot \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;y.re \leq -1.52 \cdot 10^{-90}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-97}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left({\left(\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)}^{0.25}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 30.8% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;{x.re}^{y.re} \cdot \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -210000.0)
   (* (pow x.re y.re) (* y.im (log x.re)))
   (if (<= y.re -5.2e-91)
     (sin (* y.re (atan2 x.im x.re)))
     (if (<= y.re 1.02e-44)
       (* y.im (log (hypot x.im x.re)))
       (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = pow(x_46_re, y_46_re) * (y_46_im * log(x_46_re));
	} else if (y_46_re <= -5.2e-91) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1.02e-44) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = Math.pow(x_46_re, y_46_re) * (y_46_im * Math.log(x_46_re));
	} else if (y_46_re <= -5.2e-91) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1.02e-44) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -210000.0:
		tmp = math.pow(x_46_re, y_46_re) * (y_46_im * math.log(x_46_re))
	elif y_46_re <= -5.2e-91:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 1.02e-44:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -210000.0)
		tmp = Float64((x_46_re ^ y_46_re) * Float64(y_46_im * log(x_46_re)));
	elseif (y_46_re <= -5.2e-91)
		tmp = sin(Float64(y_46_re * atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.02e-44)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -210000.0)
		tmp = (x_46_re ^ y_46_re) * (y_46_im * log(x_46_re));
	elseif (y_46_re <= -5.2e-91)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.02e-44)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -210000.0], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-91], N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.02e-44], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.1e5

   1. Initial program 29.1%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   4. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right)\right) \]

      6. accelerator-lowering-hypot.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right) \]

   5. Simplified 83.8%

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

   6. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 83.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right)\right)\right), y.re\right), \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   8. Simplified 83.8%

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

   9. Taylor expanded in x.im around 0

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

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log x.re\right)}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\left(e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\log x.re}\right)\right) \]

       3. exp-lowering-exp.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\left(y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \log \color{blue}{x.re}\right)\right) \]

       4. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       5. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \log x.re\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       6. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \log x.re\right)\right) \]

       8. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \log x.re\right)\right) \]

       9. log-lowering-log.f64 36.4%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{log.f64}\left(x.re\right)\right)\right) \]

   11. Simplified 36.4%

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

   12. Taylor expanded in y.im around 0

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

       1. associate-*r* N/A

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

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log x.re\right), \color{blue}{\left({x.re}^{y.re}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log x.re\right), \left({\color{blue}{x.re}}^{y.re}\right)\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right), \left({x.re}^{y.re}\right)\right) \]

       5. pow-lowering-pow.f64 34.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right)\right) \]

   14. Simplified 34.6%

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

   if -2.1e5 < y.re < -5.20000000000000028e-91

   1. Initial program 42.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 54.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 54.4%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 50.3%

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

   if -5.20000000000000028e-91 < y.re < 1.0199999999999999e-44

   1. Initial program 42.2%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 47.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 33.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 33.4%

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

   if 1.0199999999999999e-44 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 30.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 4.1%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 27.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 27.3%

       \[\leadsto \color{blue}{\left(y.im \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot 0.5} \]
3. Recombined 4 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;{x.re}^{y.re} \cdot \left(y.im \cdot \log x.re\right)\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-44}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 25.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-45}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.5e-91)
   (sin (* y.re (atan2 x.im x.re)))
   (if (<= y.re 9e-45)
     (* y.im (log (hypot x.im x.re)))
     (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.5e-91) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9e-45) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.5e-91) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9e-45) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -7.5e-91:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 9e-45:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7.5e-91)
		tmp = sin(Float64(y_46_re * atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 9e-45)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -7.5e-91)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 9e-45)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.5e-91], N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 9e-45], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.50000000000000051e-91

   1. Initial program 33.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 72.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 72.6%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{1}, \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 24.6%

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

   if -7.50000000000000051e-91 < y.re < 8.9999999999999997e-45

   1. Initial program 42.2%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 47.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 33.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 33.4%

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

   if 8.9999999999999997e-45 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 30.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 4.1%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 27.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 27.3%

       \[\leadsto \color{blue}{\left(y.im \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-91}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-45}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 25.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{-90}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-44}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1e-90)
   (* y.re (atan2 x.im x.re))
   (if (<= y.re 1.05e-44)
     (* y.im (log (hypot x.im x.re)))
     (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1e-90) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else if (y_46_re <= 1.05e-44) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1e-90) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else if (y_46_re <= 1.05e-44) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1e-90:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	elif y_46_re <= 1.05e-44:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1e-90)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	elseif (y_46_re <= 1.05e-44)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1e-90)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	elseif (y_46_re <= 1.05e-44)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1e-90], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05e-44], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.99999999999999995e-91

   1. Initial program 33.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 72.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 72.6%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 24.3%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 24.3%

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

   if -9.99999999999999995e-91 < y.re < 1.05000000000000001e-44

   1. Initial program 42.2%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 47.4%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 33.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 33.4%

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

   if 1.05000000000000001e-44 < y.re

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 30.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 4.1%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 4.1%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 27.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 27.3%

       \[\leadsto \color{blue}{\left(y.im \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{-90}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-44}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 20.5% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.5e-79)
   (* y.re (atan2 x.im x.re))
   (* (* y.im (log (+ (* x.re x.re) (* x.im x.im)))) 0.5)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else {
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.5d-79)) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else
        tmp = (y_46im * log(((x_46re * x_46re) + (x_46im * x_46im)))) * 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.5e-79) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else {
		tmp = (y_46_im * Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.5e-79:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	else:
		tmp = (y_46_im * math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.5e-79)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	else
		tmp = Float64(Float64(y_46_im * log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * 0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.5e-79)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	else
		tmp = (y_46_im * log(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.5e-79], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.5000000000000003e-79

   1. Initial program 32.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 74.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 74.1%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 24.0%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 24.0%

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

   if -4.5000000000000003e-79 < y.re

   1. Initial program 41.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 40.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 21.3%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 21.3%

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

      1. pow1/2 N/A

         \[\leadsto y.im \cdot \log \left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right) \]

      2. +-commutative N/A

         \[\leadsto y.im \cdot \log \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right) \]

      3. log-pow N/A

         \[\leadsto y.im \cdot \left(\frac{1}{2} \cdot \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right) \]

      4. *-commutative N/A

         \[\leadsto y.im \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \color{blue}{\frac{1}{2}}\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot \color{blue}{\frac{1}{2}} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \color{blue}{\frac{1}{2}}\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right), \frac{1}{2}\right) \]

      8. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right)\right)\right), \frac{1}{2}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right), \frac{1}{2}\right) \]

      10. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

      12. *-lowering-*.f64 27.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \frac{1}{2}\right) \]

   10. Applied egg-rr 27.5%

       \[\leadsto \color{blue}{\left(y.im \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.5 \cdot 10^{-79}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 16.2% accurate, 7.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 2.6 \cdot 10^{-185}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 2.6e-185)
   (* y.re (atan2 x.im x.re))
   (* y.im (log (+ x.re (/ (* (* x.im x.im) 0.5) x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 2.6e-185) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else {
		tmp = y_46_im * log((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 2.6d-185) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else
        tmp = y_46im * log((x_46re + (((x_46im * x_46im) * 0.5d0) / x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 2.6e-185) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else {
		tmp = y_46_im * Math.log((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 2.6e-185:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	else:
		tmp = y_46_im * math.log((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 2.6e-185)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	else
		tmp = Float64(y_46_im * log(Float64(x_46_re + Float64(Float64(Float64(x_46_im * x_46_im) * 0.5) / x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 2.6e-185)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	else
		tmp = y_46_im * log((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 2.6e-185], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[Log[N[(x$46$re + N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 2.59999999999999985e-185

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      3. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. accelerator-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      9. atan2-lowering-atan2.f64 45.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   5. Simplified 45.3%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 17.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 17.7%

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

   if 2.59999999999999985e-185 < x.re

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 43.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 18.0%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 18.0%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\color{blue}{\left(x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)}\right)\right) \]
   10. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{\frac{1}{2} \cdot {x.im}^{2}}{x.re}\right)\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.im}^{2}\right), x.re\right)\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.im}^{2}\right)\right), x.re\right)\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.im \cdot x.im\right)\right), x.re\right)\right)\right)\right) \]

       6. *-lowering-*.f64 21.5%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right)\right)\right) \]

   11. Simplified 21.5%

       \[\leadsto y.im \cdot \log \color{blue}{\left(x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 2.6 \cdot 10^{-185}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log \left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 11.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -23000:\\ \;\;\;\;y.im \cdot \log \left(0 - x.im\right)\\ \mathbf{elif}\;x.im \leq 3 \cdot 10^{-116}:\\ \;\;\;\;y.im \cdot \log x.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log x.im\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -23000.0)
   (* y.im (log (- 0.0 x.im)))
   (if (<= x.im 3e-116) (* y.im (log x.re)) (* y.im (log x.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -23000.0) {
		tmp = y_46_im * log((0.0 - x_46_im));
	} else if (x_46_im <= 3e-116) {
		tmp = y_46_im * log(x_46_re);
	} else {
		tmp = y_46_im * log(x_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-23000.0d0)) then
        tmp = y_46im * log((0.0d0 - x_46im))
    else if (x_46im <= 3d-116) then
        tmp = y_46im * log(x_46re)
    else
        tmp = y_46im * log(x_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -23000.0) {
		tmp = y_46_im * Math.log((0.0 - x_46_im));
	} else if (x_46_im <= 3e-116) {
		tmp = y_46_im * Math.log(x_46_re);
	} else {
		tmp = y_46_im * Math.log(x_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -23000.0:
		tmp = y_46_im * math.log((0.0 - x_46_im))
	elif x_46_im <= 3e-116:
		tmp = y_46_im * math.log(x_46_re)
	else:
		tmp = y_46_im * math.log(x_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -23000.0)
		tmp = Float64(y_46_im * log(Float64(0.0 - x_46_im)));
	elseif (x_46_im <= 3e-116)
		tmp = Float64(y_46_im * log(x_46_re));
	else
		tmp = Float64(y_46_im * log(x_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -23000.0)
		tmp = y_46_im * log((0.0 - x_46_im));
	elseif (x_46_im <= 3e-116)
		tmp = y_46_im * log(x_46_re);
	else
		tmp = y_46_im * log(x_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -23000.0], N[(y$46$im * N[Log[N[(0.0 - x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 3e-116], N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -23000

   1. Initial program 29.8%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 49.6%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 13.5%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 13.5%

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

   9. Taylor expanded in x.im around -inf

      \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x.im\right)}\right)\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(x.im\right)\right)\right)\right) \]

       2. neg-lowering-neg.f64 12.3%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{neg.f64}\left(x.im\right)\right)\right) \]

   11. Simplified 12.3%

       \[\leadsto y.im \cdot \log \color{blue}{\left(-x.im\right)} \]

   if -23000 < x.im < 3.00000000000000026e-116

   1. Initial program 45.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 51.3%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 18.7%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 18.7%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log x.re} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log x.re}\right) \]

       2. log-lowering-log.f64 11.5%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right) \]

   11. Simplified 11.5%

       \[\leadsto \color{blue}{y.im \cdot \log x.re} \]

   if 3.00000000000000026e-116 < x.im

   1. Initial program 36.6%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 47.1%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 17.1%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 17.1%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{y.im \cdot \log x.im} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log x.im}\right) \]

       2. log-lowering-log.f64 11.9%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right) \]

   11. Simplified 11.9%

       \[\leadsto \color{blue}{y.im \cdot \log x.im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 11.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -23000:\\ \;\;\;\;y.im \cdot \log \left(0 - x.im\right)\\ \mathbf{elif}\;x.im \leq 3 \cdot 10^{-116}:\\ \;\;\;\;y.im \cdot \log x.re\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log x.im\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 11.6% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;y.im \cdot \log \left(0 - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -1e-310) (* y.im (log (- 0.0 x.re))) (* y.im (log x.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1e-310) {
		tmp = y_46_im * log((0.0 - x_46_re));
	} else {
		tmp = y_46_im * log(x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-1d-310)) then
        tmp = y_46im * log((0.0d0 - x_46re))
    else
        tmp = y_46im * log(x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1e-310) {
		tmp = y_46_im * Math.log((0.0 - x_46_re));
	} else {
		tmp = y_46_im * Math.log(x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -1e-310:
		tmp = y_46_im * math.log((0.0 - x_46_re))
	else:
		tmp = y_46_im * math.log(x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -1e-310)
		tmp = Float64(y_46_im * log(Float64(0.0 - x_46_re)));
	else
		tmp = Float64(y_46_im * log(x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -1e-310)
		tmp = y_46_im * log((0.0 - x_46_re));
	else
		tmp = y_46_im * log(x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -1e-310], N[(y$46$im * N[Log[N[(0.0 - x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -9.999999999999969e-311

   1. Initial program 39.0%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 55.7%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 16.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 16.4%

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

   9. Taylor expanded in x.re around -inf

      \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}\right)\right) \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right)\right)\right) \]

       2. neg-lowering-neg.f64 10.6%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{neg.f64}\left(x.re\right)\right)\right) \]

   11. Simplified 10.6%

       \[\leadsto y.im \cdot \log \color{blue}{\left(-x.re\right)} \]

   if -9.999999999999969e-311 < x.re

   1. Initial program 37.6%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 42.0%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 17.3%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 17.3%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log x.re} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log x.re}\right) \]

       2. log-lowering-log.f64 14.1%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right) \]

   11. Simplified 14.1%

       \[\leadsto \color{blue}{y.im \cdot \log x.re} \]
3. Recombined 2 regimes into one program.

4. Final simplification 12.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;y.im \cdot \log \left(0 - x.re\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log x.re\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 9.4% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7.9 \cdot 10^{-305}:\\ \;\;\;\;y.im \cdot \log x.im\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \log x.re\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -7.9e-305) (* y.im (log x.im)) (* y.im (log x.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -7.9e-305) {
		tmp = y_46_im * log(x_46_im);
	} else {
		tmp = y_46_im * log(x_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-7.9d-305)) then
        tmp = y_46im * log(x_46im)
    else
        tmp = y_46im * log(x_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -7.9e-305) {
		tmp = y_46_im * Math.log(x_46_im);
	} else {
		tmp = y_46_im * Math.log(x_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -7.9e-305:
		tmp = y_46_im * math.log(x_46_im)
	else:
		tmp = y_46_im * math.log(x_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -7.9e-305)
		tmp = Float64(y_46_im * log(x_46_im));
	else
		tmp = Float64(y_46_im * log(x_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -7.9e-305)
		tmp = y_46_im * log(x_46_im);
	else
		tmp = y_46_im * log(x_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -7.9e-305], N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -7.90000000000000024e-305

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 56.1%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 16.5%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 16.5%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{y.im \cdot \log x.im} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log x.im}\right) \]

       2. log-lowering-log.f64 6.1%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right) \]

   11. Simplified 6.1%

       \[\leadsto \color{blue}{y.im \cdot \log x.im} \]

   if -7.90000000000000024e-305 < x.re

   1. Initial program 38.2%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

   5. Simplified 41.6%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

      4. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

      5. accelerator-lowering-hypot.f64 17.1%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   8. Simplified 17.1%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log x.re} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log x.re}\right) \]

       2. log-lowering-log.f64 14.0%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right) \]

   11. Simplified 14.0%

       \[\leadsto \color{blue}{y.im \cdot \log x.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 13.4% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* y.re (atan2 x.im x.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * atan2(x_46_im, x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46re * atan2(x_46im, x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * Math.atan2(x_46_im, x_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_re * math.atan2(x_46_im, x_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_re * atan(x_46_im, x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_re * atan2(x_46_im, x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.4%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

   3. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), y.re\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. accelerator-lowering-hypot.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   7. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   9. atan2-lowering-atan2.f64 44.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

5. Simplified 44.2%

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

6. Taylor expanded in y.re around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   2. atan2-lowering-atan2.f64 15.7%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 15.7%

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

Alternative 27: 5.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ y.im \cdot \log x.im \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* y.im (log x.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * log(x_46_im);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46im * log(x_46im)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * Math.log(x_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_im * math.log(x_46_im)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_im * log(x_46_im))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_im * log(x_46_im);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

3. Taylor expanded in y.im around 0

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

   1. +-commutative N/A

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

   2. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)}\right) \]

5. Simplified 49.4%

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

6. Taylor expanded in y.re around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

   2. log-lowering-log.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

   3. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

   5. accelerator-lowering-hypot.f64 16.8%

      \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

8. Simplified 16.8%

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

9. Taylor expanded in x.re around 0

   \[\leadsto \color{blue}{y.im \cdot \log x.im} \]
10. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log x.im}\right) \]

    2. log-lowering-log.f64 5.1%

       \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right) \]

11. Simplified 5.1%

    \[\leadsto \color{blue}{y.im \cdot \log x.im} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(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)))))