powComplex, imaginary part

Percentage Accurate: 41.5% → 73.3%

Time: 25.1s

Alternatives: 22

Speedup: 2.6×

• 35.2% 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: 41.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sin \left(y.im \cdot t\_0\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_4 := \frac{t\_3}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\\ \mathbf{if}\;y.re \leq -430:\\ \;\;\;\;t\_2 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 0.00012:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \left(t\_0 + y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)}{t\_4}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_1}{t\_3}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+223}:\\ \;\;\;\;\frac{t\_1}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_2 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (sin (* y.im t_0)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (exp (* y.im (atan2 x.im x.re))))
        (t_4 (/ t_3 (pow (hypot x.re x.im) y.re))))
   (if (<= y.re -430.0)
     (* t_2 (pow (hypot x.im x.re) y.re))
     (if (<= y.re 0.00012)
       (/ (sin (* y.im (+ t_0 (* y.re (/ (atan2 x.im x.re) y.im))))) t_4)
       (if (<= y.re 8.2e+48)
         (/ t_1 t_3)
         (if (<= y.re 2.2e+223) (/ t_1 t_4) (* (sin t_2) (pow x.re y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_im, x_46_re));
	double t_1 = sin((y_46_im * t_0));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double t_4 = t_3 / pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -430.0) {
		tmp = t_2 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 0.00012) {
		tmp = sin((y_46_im * (t_0 + (y_46_re * (atan2(x_46_im, x_46_re) / y_46_im))))) / t_4;
	} else if (y_46_re <= 8.2e+48) {
		tmp = t_1 / t_3;
	} else if (y_46_re <= 2.2e+223) {
		tmp = t_1 / t_4;
	} else {
		tmp = sin(t_2) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.hypot(x_46_im, x_46_re));
	double t_1 = Math.sin((y_46_im * t_0));
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_4 = t_3 / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -430.0) {
		tmp = t_2 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 0.00012) {
		tmp = Math.sin((y_46_im * (t_0 + (y_46_re * (Math.atan2(x_46_im, x_46_re) / y_46_im))))) / t_4;
	} else if (y_46_re <= 8.2e+48) {
		tmp = t_1 / t_3;
	} else if (y_46_re <= 2.2e+223) {
		tmp = t_1 / t_4;
	} else {
		tmp = Math.sin(t_2) * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.hypot(x_46_im, x_46_re))
	t_1 = math.sin((y_46_im * t_0))
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	t_4 = t_3 / math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -430.0:
		tmp = t_2 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 0.00012:
		tmp = math.sin((y_46_im * (t_0 + (y_46_re * (math.atan2(x_46_im, x_46_re) / y_46_im))))) / t_4
	elif y_46_re <= 8.2e+48:
		tmp = t_1 / t_3
	elif y_46_re <= 2.2e+223:
		tmp = t_1 / t_4
	else:
		tmp = math.sin(t_2) * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = sin(Float64(y_46_im * t_0))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_4 = Float64(t_3 / (hypot(x_46_re, x_46_im) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -430.0)
		tmp = Float64(t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 0.00012)
		tmp = Float64(sin(Float64(y_46_im * Float64(t_0 + Float64(y_46_re * Float64(atan(x_46_im, x_46_re) / y_46_im))))) / t_4);
	elseif (y_46_re <= 8.2e+48)
		tmp = Float64(t_1 / t_3);
	elseif (y_46_re <= 2.2e+223)
		tmp = Float64(t_1 / t_4);
	else
		tmp = Float64(sin(t_2) * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re));
	t_1 = sin((y_46_im * t_0));
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	t_4 = t_3 / (hypot(x_46_re, x_46_im) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -430.0)
		tmp = t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 0.00012)
		tmp = sin((y_46_im * (t_0 + (y_46_re * (atan2(x_46_im, x_46_re) / y_46_im))))) / t_4;
	elseif (y_46_re <= 8.2e+48)
		tmp = t_1 / t_3;
	elseif (y_46_re <= 2.2e+223)
		tmp = t_1 / t_4;
	else
		tmp = sin(t_2) * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$im * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -430.0], N[(t$95$2 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.00012], N[(N[Sin[N[(y$46$im * N[(t$95$0 + N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[y$46$re, 8.2e+48], N[(t$95$1 / t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 2.2e+223], N[(t$95$1 / t$95$4), $MachinePrecision], N[(N[Sin[t$95$2], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -430

   1. Initial program 31.7%

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

   3. Taylor expanded in y.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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 78.0%

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

   8. Simplified 78.0%

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

   if -430 < y.re < 1.20000000000000003e-4

   1. Initial program 36.0%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 85.4%

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

   5. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

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

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

      8. associate-/l* N/A

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

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

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

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

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

      11. atan2-lowering-atan2.f64 85.4%

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

   7. Simplified 85.4%

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

   if 1.20000000000000003e-4 < y.re < 8.2000000000000005e48

   1. Initial program 62.5%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 37.5%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 88.8%

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

   7. Simplified 88.8%

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

   if 8.2000000000000005e48 < y.re < 2.2e223

   1. Initial program 44.4%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 47.2%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 66.7%

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

   7. Simplified 66.7%

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

   if 2.2e223 < y.re

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in x.im around 0

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

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

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

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

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

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

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

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

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

      5. pow-lowering-pow.f64 57.5%

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

   8. Simplified 57.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -430:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 0.00012:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) + y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+223}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 72.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := \log \left(\frac{-1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot t\_2 - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - t\_2 \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin t\_1 + y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \cos t\_1\right)}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \end{array} \]

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.im (log (hypot x.im x.re))))
        (t_2 (log (/ -1.0 x.re))))
   (if (<= x.re -5e-15)
     (*
      (exp (- (* (- 0.0 y.re) t_2) t_0))
      (sin (- (* y.re (atan2 x.im x.re)) (* t_2 y.im))))
     (/
      (+ (sin t_1) (* y.re (* (atan2 x.im x.re) (cos t_1))))
      (/ (exp t_0) (pow (hypot x.re 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_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_2 = log((-1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -5e-15) {
		tmp = exp((((0.0 - y_46_re) * t_2) - t_0)) * sin(((y_46_re * atan2(x_46_im, x_46_re)) - (t_2 * y_46_im)));
	} else {
		tmp = (sin(t_1) + (y_46_re * (atan2(x_46_im, x_46_re) * cos(t_1)))) / (exp(t_0) / pow(hypot(x_46_re, 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_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double t_2 = Math.log((-1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -5e-15) {
		tmp = Math.exp((((0.0 - y_46_re) * t_2) - t_0)) * Math.sin(((y_46_re * Math.atan2(x_46_im, x_46_re)) - (t_2 * y_46_im)));
	} else {
		tmp = (Math.sin(t_1) + (y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.cos(t_1)))) / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, 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_im * math.atan2(x_46_im, x_46_re)
	t_1 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	t_2 = math.log((-1.0 / x_46_re))
	tmp = 0
	if x_46_re <= -5e-15:
		tmp = math.exp((((0.0 - y_46_re) * t_2) - t_0)) * math.sin(((y_46_re * math.atan2(x_46_im, x_46_re)) - (t_2 * y_46_im)))
	else:
		tmp = (math.sin(t_1) + (y_46_re * (math.atan2(x_46_im, x_46_re) * math.cos(t_1)))) / (math.exp(t_0) / math.pow(math.hypot(x_46_re, 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_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_2 = log(Float64(-1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -5e-15)
		tmp = Float64(exp(Float64(Float64(Float64(0.0 - y_46_re) * t_2) - t_0)) * sin(Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) - Float64(t_2 * y_46_im))));
	else
		tmp = Float64(Float64(sin(t_1) + Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * cos(t_1)))) / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = y_46_im * log(hypot(x_46_im, x_46_re));
	t_2 = log((-1.0 / x_46_re));
	tmp = 0.0;
	if (x_46_re <= -5e-15)
		tmp = exp((((0.0 - y_46_re) * t_2) - t_0)) * sin(((y_46_re * atan2(x_46_im, x_46_re)) - (t_2 * y_46_im)));
	else
		tmp = (sin(t_1) + (y_46_re * (atan2(x_46_im, x_46_re) * cos(t_1)))) / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -5e-15], N[(N[Exp[N[(N[(N[(0.0 - y$46$re), $MachinePrecision] * t$95$2), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] - N[(t$95$2 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sin[t$95$1], $MachinePrecision] + N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.99999999999999999e-15

   1. Initial program 35.2%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

   5. Simplified 81.9%

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

   if -4.99999999999999999e-15 < 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. Step-by-step derivation

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 69.9%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 73.4%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5 \cdot 10^{-15}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot \log \left(\frac{-1}{x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) + y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_1 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \frac{t\_1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\\ \mathbf{if}\;y.re \leq -1950:\\ \;\;\;\;t\_2 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 0.00012:\\ \;\;\;\;\frac{\sin \left(t\_2 + y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}{t\_3}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+223}:\\ \;\;\;\;\frac{t\_0}{t\_3}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_2 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.im (log (hypot x.im x.re)))))
        (t_1 (exp (* y.im (atan2 x.im x.re))))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (/ t_1 (pow (hypot x.re x.im) y.re))))
   (if (<= y.re -1950.0)
     (* t_2 (pow (hypot x.im x.re) y.re))
     (if (<= y.re 0.00012)
       (/ (sin (+ t_2 (* y.im (log (hypot x.re x.im))))) t_3)
       (if (<= y.re 8.2e+48)
         (/ t_0 t_1)
         (if (<= y.re 4.5e+223) (/ t_0 t_3) (* (sin t_2) (pow x.re y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_1 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = t_1 / pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -1950.0) {
		tmp = t_2 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 0.00012) {
		tmp = sin((t_2 + (y_46_im * log(hypot(x_46_re, x_46_im))))) / t_3;
	} else if (y_46_re <= 8.2e+48) {
		tmp = t_0 / t_1;
	} else if (y_46_re <= 4.5e+223) {
		tmp = t_0 / t_3;
	} else {
		tmp = sin(t_2) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double t_1 = Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = t_1 / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -1950.0) {
		tmp = t_2 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 0.00012) {
		tmp = Math.sin((t_2 + (y_46_im * Math.log(Math.hypot(x_46_re, x_46_im))))) / t_3;
	} else if (y_46_re <= 8.2e+48) {
		tmp = t_0 / t_1;
	} else if (y_46_re <= 4.5e+223) {
		tmp = t_0 / t_3;
	} else {
		tmp = Math.sin(t_2) * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	t_1 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = t_1 / math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -1950.0:
		tmp = t_2 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 0.00012:
		tmp = math.sin((t_2 + (y_46_im * math.log(math.hypot(x_46_re, x_46_im))))) / t_3
	elif y_46_re <= 8.2e+48:
		tmp = t_0 / t_1
	elif y_46_re <= 4.5e+223:
		tmp = t_0 / t_3
	else:
		tmp = math.sin(t_2) * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	t_1 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = Float64(t_1 / (hypot(x_46_re, x_46_im) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1950.0)
		tmp = Float64(t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 0.00012)
		tmp = Float64(sin(Float64(t_2 + Float64(y_46_im * log(hypot(x_46_re, x_46_im))))) / t_3);
	elseif (y_46_re <= 8.2e+48)
		tmp = Float64(t_0 / t_1);
	elseif (y_46_re <= 4.5e+223)
		tmp = Float64(t_0 / t_3);
	else
		tmp = Float64(sin(t_2) * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	t_1 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = t_1 / (hypot(x_46_re, x_46_im) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1950.0)
		tmp = t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 0.00012)
		tmp = sin((t_2 + (y_46_im * log(hypot(x_46_re, x_46_im))))) / t_3;
	elseif (y_46_re <= 8.2e+48)
		tmp = t_0 / t_1;
	elseif (y_46_re <= 4.5e+223)
		tmp = t_0 / t_3;
	else
		tmp = sin(t_2) * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1950.0], N[(t$95$2 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.00012], N[(N[Sin[N[(t$95$2 + N[(y$46$im * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 8.2e+48], N[(t$95$0 / t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 4.5e+223], N[(t$95$0 / t$95$3), $MachinePrecision], N[(N[Sin[t$95$2], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1950

   1. Initial program 31.7%

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

   3. Taylor expanded in y.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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 71.7%

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

   5. Simplified 71.7%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 78.0%

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

   8. Simplified 78.0%

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

   if -1950 < y.re < 1.20000000000000003e-4

   1. Initial program 36.0%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 85.4%

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

   if 1.20000000000000003e-4 < y.re < 8.2000000000000005e48

   1. Initial program 62.5%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 37.5%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 88.8%

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

   7. Simplified 88.8%

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

   if 8.2000000000000005e48 < y.re < 4.5e223

   1. Initial program 44.4%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 47.2%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 66.7%

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

   7. Simplified 66.7%

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

   if 4.5e223 < y.re

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in x.im around 0

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

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

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

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

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

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

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

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

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

      5. pow-lowering-pow.f64 57.5%

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

   8. Simplified 57.5%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1950:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 0.00012:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+223}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{1}{x.re}\right)\\ t_3 := \log \left(\frac{-1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -1.45 \cdot 10^{-71}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot t\_3 - t\_0} \cdot \sin \left(t\_1 - t\_3 \cdot y.im\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot t\_2 - t\_0} \cdot \sin \left(t\_1 - y.im \cdot t\_2\right)\\ \end{array} \end{array} \]

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)))
        (t_3 (log (/ -1.0 x.re))))
   (if (<= x.re -1.45e-71)
     (* (exp (- (* (- 0.0 y.re) t_3) t_0)) (sin (- t_1 (* t_3 y.im))))
     (if (<= x.re 5e-15)
       (/
        (sin (* y.im (log (hypot x.im x.re))))
        (/ (exp t_0) (pow (hypot x.re x.im) y.re)))
       (* (exp (- (* (- 0.0 y.re) t_2) t_0)) (sin (- t_1 (* y.im t_2))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * 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 t_3 = log((-1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -1.45e-71) {
		tmp = exp((((0.0 - y_46_re) * t_3) - t_0)) * sin((t_1 - (t_3 * y_46_im)));
	} else if (x_46_re <= 5e-15) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = exp((((0.0 - y_46_re) * t_2) - t_0)) * sin((t_1 - (y_46_im * t_2)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.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 t_3 = Math.log((-1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -1.45e-71) {
		tmp = Math.exp((((0.0 - y_46_re) * t_3) - t_0)) * Math.sin((t_1 - (t_3 * y_46_im)));
	} else if (x_46_re <= 5e-15) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / (Math.exp(t_0) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = Math.exp((((0.0 - y_46_re) * t_2) - t_0)) * Math.sin((t_1 - (y_46_im * t_2)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.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))
	t_3 = math.log((-1.0 / x_46_re))
	tmp = 0
	if x_46_re <= -1.45e-71:
		tmp = math.exp((((0.0 - y_46_re) * t_3) - t_0)) * math.sin((t_1 - (t_3 * y_46_im)))
	elif x_46_re <= 5e-15:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / (math.exp(t_0) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = math.exp((((0.0 - y_46_re) * t_2) - t_0)) * math.sin((t_1 - (y_46_im * t_2)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * 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))
	t_3 = log(Float64(-1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -1.45e-71)
		tmp = Float64(exp(Float64(Float64(Float64(0.0 - y_46_re) * t_3) - t_0)) * sin(Float64(t_1 - Float64(t_3 * y_46_im))));
	elseif (x_46_re <= 5e-15)
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		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))));
	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));
	t_3 = log((-1.0 / x_46_re));
	tmp = 0.0;
	if (x_46_re <= -1.45e-71)
		tmp = exp((((0.0 - y_46_re) * t_3) - t_0)) * sin((t_1 - (t_3 * y_46_im)));
	elseif (x_46_re <= 5e-15)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = exp((((0.0 - y_46_re) * t_2) - t_0)) * sin((t_1 - (y_46_im * t_2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[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]}, Block[{t$95$3 = N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.45e-71], N[(N[Exp[N[(N[(N[(0.0 - y$46$re), $MachinePrecision] * t$95$3), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 - N[(t$95$3 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 5e-15], N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[t$95$0], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.4499999999999999e-71

   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 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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

   5. Simplified 80.6%

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

   if -1.4499999999999999e-71 < x.re < 4.99999999999999999e-15

   1. Initial program 39.3%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 73.5%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 71.1%

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

   7. Simplified 71.1%

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

   if 4.99999999999999999e-15 < x.re

   1. Initial program 30.2%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

   5. Simplified 67.8%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.45 \cdot 10^{-71}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot \log \left(\frac{-1}{x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right)\\ \mathbf{elif}\;x.re \leq 5 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot \log \left(\frac{1}{x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{1}{x.re}\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{-1}{x.re}\right)\\ t_3 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := e^{t\_3}\\ t_5 := \frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{t\_4}\\ \mathbf{if}\;x.re \leq -3.5 \cdot 10^{-73}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot t\_2 - t\_3} \cdot \sin \left(t\_1 - t\_2 \cdot y.im\right)\\ \mathbf{elif}\;x.re \leq -2.1 \cdot 10^{-201}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;x.re \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{t\_1}{\frac{t\_4}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;x.re \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot t\_0 - t\_3} \cdot \sin \left(t\_1 - y.im \cdot t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (/ 1.0 x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ -1.0 x.re)))
        (t_3 (* y.im (atan2 x.im x.re)))
        (t_4 (exp t_3))
        (t_5 (/ (sin (* y.im (log (hypot x.im x.re)))) t_4)))
   (if (<= x.re -3.5e-73)
     (* (exp (- (* (- 0.0 y.re) t_2) t_3)) (sin (- t_1 (* t_2 y.im))))
     (if (<= x.re -2.1e-201)
       t_5
       (if (<= x.re 3.2e-127)
         (/ t_1 (/ t_4 (pow (hypot x.re x.im) y.re)))
         (if (<= x.re 8.5e-10)
           t_5
           (*
            (exp (- (* (- 0.0 y.re) t_0) t_3))
            (sin (- t_1 (* y.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 = log((1.0 / 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 t_3 = y_46_im * atan2(x_46_im, x_46_re);
	double t_4 = exp(t_3);
	double t_5 = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_4;
	double tmp;
	if (x_46_re <= -3.5e-73) {
		tmp = exp((((0.0 - y_46_re) * t_2) - t_3)) * sin((t_1 - (t_2 * y_46_im)));
	} else if (x_46_re <= -2.1e-201) {
		tmp = t_5;
	} else if (x_46_re <= 3.2e-127) {
		tmp = t_1 / (t_4 / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else if (x_46_re <= 8.5e-10) {
		tmp = t_5;
	} else {
		tmp = exp((((0.0 - y_46_re) * t_0) - t_3)) * sin((t_1 - (y_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 = Math.log((1.0 / 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 t_3 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_4 = Math.exp(t_3);
	double t_5 = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / t_4;
	double tmp;
	if (x_46_re <= -3.5e-73) {
		tmp = Math.exp((((0.0 - y_46_re) * t_2) - t_3)) * Math.sin((t_1 - (t_2 * y_46_im)));
	} else if (x_46_re <= -2.1e-201) {
		tmp = t_5;
	} else if (x_46_re <= 3.2e-127) {
		tmp = t_1 / (t_4 / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else if (x_46_re <= 8.5e-10) {
		tmp = t_5;
	} else {
		tmp = Math.exp((((0.0 - y_46_re) * t_0) - t_3)) * Math.sin((t_1 - (y_46_im * t_0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log((1.0 / 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))
	t_3 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_4 = math.exp(t_3)
	t_5 = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / t_4
	tmp = 0
	if x_46_re <= -3.5e-73:
		tmp = math.exp((((0.0 - y_46_re) * t_2) - t_3)) * math.sin((t_1 - (t_2 * y_46_im)))
	elif x_46_re <= -2.1e-201:
		tmp = t_5
	elif x_46_re <= 3.2e-127:
		tmp = t_1 / (t_4 / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	elif x_46_re <= 8.5e-10:
		tmp = t_5
	else:
		tmp = math.exp((((0.0 - y_46_re) * t_0) - t_3)) * math.sin((t_1 - (y_46_im * t_0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(Float64(1.0 / 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))
	t_3 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_4 = exp(t_3)
	t_5 = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / t_4)
	tmp = 0.0
	if (x_46_re <= -3.5e-73)
		tmp = Float64(exp(Float64(Float64(Float64(0.0 - y_46_re) * t_2) - t_3)) * sin(Float64(t_1 - Float64(t_2 * y_46_im))));
	elseif (x_46_re <= -2.1e-201)
		tmp = t_5;
	elseif (x_46_re <= 3.2e-127)
		tmp = Float64(t_1 / Float64(t_4 / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	elseif (x_46_re <= 8.5e-10)
		tmp = t_5;
	else
		tmp = Float64(exp(Float64(Float64(Float64(0.0 - y_46_re) * t_0) - t_3)) * sin(Float64(t_1 - Float64(y_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 = log((1.0 / x_46_re));
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log((-1.0 / x_46_re));
	t_3 = y_46_im * atan2(x_46_im, x_46_re);
	t_4 = exp(t_3);
	t_5 = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_4;
	tmp = 0.0;
	if (x_46_re <= -3.5e-73)
		tmp = exp((((0.0 - y_46_re) * t_2) - t_3)) * sin((t_1 - (t_2 * y_46_im)));
	elseif (x_46_re <= -2.1e-201)
		tmp = t_5;
	elseif (x_46_re <= 3.2e-127)
		tmp = t_1 / (t_4 / (hypot(x_46_re, x_46_im) ^ y_46_re));
	elseif (x_46_re <= 8.5e-10)
		tmp = t_5;
	else
		tmp = exp((((0.0 - y_46_re) * t_0) - t_3)) * sin((t_1 - (y_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[Log[N[(1.0 / 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]}, Block[{t$95$3 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Exp[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, If[LessEqual[x$46$re, -3.5e-73], N[(N[Exp[N[(N[(N[(0.0 - y$46$re), $MachinePrecision] * t$95$2), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 - N[(t$95$2 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -2.1e-201], t$95$5, If[LessEqual[x$46$re, 3.2e-127], N[(t$95$1 / N[(t$95$4 / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 8.5e-10], t$95$5, N[(N[Exp[N[(N[(N[(0.0 - y$46$re), $MachinePrecision] * t$95$0), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 - N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -3.4999999999999998e-73

   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 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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

   5. Simplified 80.6%

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

   if -3.4999999999999998e-73 < x.re < -2.10000000000000012e-201 or 3.20000000000000017e-127 < x.re < 8.4999999999999996e-10

   1. Initial program 41.7%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 74.1%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 65.4%

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

   7. Simplified 65.4%

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

   if -2.10000000000000012e-201 < x.re < 3.20000000000000017e-127

   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. Step-by-step derivation

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 72.7%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 81.2%

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

   8. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   9. Step-by-step derivation

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

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

      2. atan2-lowering-atan2.f64 69.7%

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

   10. Simplified 69.7%

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

   if 8.4999999999999996e-10 < x.re

   1. Initial program 28.6%

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

   3. Taylor expanded in 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. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      12. +-commutative N/A

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

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

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

   5. Simplified 67.8%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -3.5 \cdot 10^{-73}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot \log \left(\frac{-1}{x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{-1}{x.re}\right) \cdot y.im\right)\\ \mathbf{elif}\;x.re \leq -2.1 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;x.re \leq 3.2 \cdot 10^{-127}:\\ \;\;\;\;\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;x.re \leq 8.5 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0 - y.re\right) \cdot \log \left(\frac{1}{x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := t\_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_3 := \sin t\_1\\ \mathbf{if}\;y.re \leq -24000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{t\_0}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{t\_3}{\frac{t\_0}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+227}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.im (atan2 x.im x.re))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* t_1 (pow (hypot x.im x.re) y.re)))
        (t_3 (sin t_1)))
   (if (<= y.re -24000.0)
     t_2
     (if (<= y.re 9.8e-150)
       (/ (sin (* y.im (log (hypot x.im x.re)))) t_0)
       (if (<= y.re 7.2e+56)
         (/ t_3 (/ t_0 (pow (hypot x.re x.im) y.re)))
         (if (<= y.re 9.5e+227) t_2 (* t_3 (pow x.re y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((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 = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = sin(t_1);
	double tmp;
	if (y_46_re <= -24000.0) {
		tmp = t_2;
	} else if (y_46_re <= 9.8e-150) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_0;
	} else if (y_46_re <= 7.2e+56) {
		tmp = t_3 / (t_0 / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else if (y_46_re <= 9.5e+227) {
		tmp = t_2;
	} else {
		tmp = t_3 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((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 = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double t_3 = Math.sin(t_1);
	double tmp;
	if (y_46_re <= -24000.0) {
		tmp = t_2;
	} else if (y_46_re <= 9.8e-150) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / t_0;
	} else if (y_46_re <= 7.2e+56) {
		tmp = t_3 / (t_0 / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else if (y_46_re <= 9.5e+227) {
		tmp = t_2;
	} else {
		tmp = t_3 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((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 = t_1 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	t_3 = math.sin(t_1)
	tmp = 0
	if y_46_re <= -24000.0:
		tmp = t_2
	elif y_46_re <= 9.8e-150:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / t_0
	elif y_46_re <= 7.2e+56:
		tmp = t_3 / (t_0 / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	elif y_46_re <= 9.5e+227:
		tmp = t_2
	else:
		tmp = t_3 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(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 = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	t_3 = sin(t_1)
	tmp = 0.0
	if (y_46_re <= -24000.0)
		tmp = t_2;
	elseif (y_46_re <= 9.8e-150)
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / t_0);
	elseif (y_46_re <= 7.2e+56)
		tmp = Float64(t_3 / Float64(t_0 / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	elseif (y_46_re <= 9.5e+227)
		tmp = t_2;
	else
		tmp = Float64(t_3 * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	t_3 = sin(t_1);
	tmp = 0.0;
	if (y_46_re <= -24000.0)
		tmp = t_2;
	elseif (y_46_re <= 9.8e-150)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_0;
	elseif (y_46_re <= 7.2e+56)
		tmp = t_3 / (t_0 / (hypot(x_46_re, x_46_im) ^ y_46_re));
	elseif (y_46_re <= 9.5e+227)
		tmp = t_2;
	else
		tmp = t_3 * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[y$46$re, -24000.0], t$95$2, If[LessEqual[y$46$re, 9.8e-150], N[(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), $MachinePrecision], If[LessEqual[y$46$re, 7.2e+56], N[(t$95$3 / N[(t$95$0 / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e+227], t$95$2, N[(t$95$3 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -24000 or 7.19999999999999996e56 < y.re < 9.5000000000000005e227

   1. Initial program 36.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 61.7%

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

   5. Simplified 61.7%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 74.2%

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

   8. Simplified 74.2%

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

   if -24000 < y.re < 9.7999999999999991e-150

   1. Initial program 31.6%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 85.2%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 75.8%

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

   7. Simplified 75.8%

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

   if 9.7999999999999991e-150 < y.re < 7.19999999999999996e56

   1. Initial program 48.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 76.5%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

      3. atan2-lowering-atan2.f64 64.4%

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

   7. Simplified 64.4%

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

   if 9.5000000000000005e227 < y.re

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in x.im around 0

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

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

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

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

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

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

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

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

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

      5. pow-lowering-pow.f64 57.5%

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

   8. Simplified 57.5%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -24000:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-150}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+56}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+227}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -23000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+49}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{+224}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* t_0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -23000.0)
     t_1
     (if (<= y.re 5.4e+49)
       (/
        (sin (* y.im (log (hypot x.im x.re))))
        (exp (* y.im (atan2 x.im x.re))))
       (if (<= y.re 9.2e+224) t_1 (* (sin t_0) (pow x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -23000.0) {
		tmp = t_1;
	} else if (y_46_re <= 5.4e+49) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9.2e+224) {
		tmp = t_1;
	} else {
		tmp = sin(t_0) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -23000.0) {
		tmp = t_1;
	} else if (y_46_re <= 5.4e+49) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9.2e+224) {
		tmp = t_1;
	} else {
		tmp = Math.sin(t_0) * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -23000.0:
		tmp = t_1
	elif y_46_re <= 5.4e+49:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 9.2e+224:
		tmp = t_1
	else:
		tmp = math.sin(t_0) * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -23000.0)
		tmp = t_1;
	elseif (y_46_re <= 5.4e+49)
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 9.2e+224)
		tmp = t_1;
	else
		tmp = Float64(sin(t_0) * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -23000.0)
		tmp = t_1;
	elseif (y_46_re <= 5.4e+49)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 9.2e+224)
		tmp = t_1;
	else
		tmp = sin(t_0) * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -23000.0], t$95$1, If[LessEqual[y$46$re, 5.4e+49], N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.2e+224], t$95$1, N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -23000 or 5.4000000000000002e49 < y.re < 9.20000000000000079e224

   1. Initial program 36.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 62.5%

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

   5. Simplified 62.5%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 73.7%

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

   8. Simplified 73.7%

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

   if -23000 < y.re < 5.4000000000000002e49

   1. Initial program 37.3%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 82.0%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 69.0%

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

   7. Simplified 69.0%

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

   if 9.20000000000000079e224 < y.re

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 57.2%

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

   5. Simplified 57.2%

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

   6. Taylor expanded in x.im around 0

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

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

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

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

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

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

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

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

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

      5. pow-lowering-pow.f64 57.5%

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

   8. Simplified 57.5%

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

Alternative 8: 46.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := x.im \cdot x.im + x.re \cdot x.re\\ \mathbf{if}\;y.im \leq -320000:\\ \;\;\;\;\frac{y.im \cdot \log \left(\sqrt{t\_2}\right)}{t\_0}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-41}:\\ \;\;\;\;t\_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+93}:\\ \;\;\;\;\sin t\_1 \cdot {t\_2}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(0 - y.im\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.im (atan2 x.im x.re))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (+ (* x.im x.im) (* x.re x.re))))
   (if (<= y.im -320000.0)
     (/ (* y.im (log (sqrt t_2))) t_0)
     (if (<= y.im 1.65e-41)
       (* t_1 (pow (hypot x.im x.re) y.re))
       (if (<= y.im 1.7e+93)
         (* (sin t_1) (pow t_2 (/ y.re 2.0)))
         (/ (sin (* (log (/ -1.0 x.re)) (- 0.0 y.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 = exp((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 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_im <= -320000.0) {
		tmp = (y_46_im * log(sqrt(t_2))) / t_0;
	} else if (y_46_im <= 1.65e-41) {
		tmp = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.7e+93) {
		tmp = sin(t_1) * pow(t_2, (y_46_re / 2.0));
	} else {
		tmp = sin((log((-1.0 / x_46_re)) * (0.0 - y_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 = Math.exp((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 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_im <= -320000.0) {
		tmp = (y_46_im * Math.log(Math.sqrt(t_2))) / t_0;
	} else if (y_46_im <= 1.65e-41) {
		tmp = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.7e+93) {
		tmp = Math.sin(t_1) * Math.pow(t_2, (y_46_re / 2.0));
	} else {
		tmp = Math.sin((Math.log((-1.0 / x_46_re)) * (0.0 - y_46_im))) / t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((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 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	tmp = 0
	if y_46_im <= -320000.0:
		tmp = (y_46_im * math.log(math.sqrt(t_2))) / t_0
	elif y_46_im <= 1.65e-41:
		tmp = t_1 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_im <= 1.7e+93:
		tmp = math.sin(t_1) * math.pow(t_2, (y_46_re / 2.0))
	else:
		tmp = math.sin((math.log((-1.0 / x_46_re)) * (0.0 - y_46_im))) / t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(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 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_im <= -320000.0)
		tmp = Float64(Float64(y_46_im * log(sqrt(t_2))) / t_0);
	elseif (y_46_im <= 1.65e-41)
		tmp = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 1.7e+93)
		tmp = Float64(sin(t_1) * (t_2 ^ Float64(y_46_re / 2.0)));
	else
		tmp = Float64(sin(Float64(log(Float64(-1.0 / x_46_re)) * Float64(0.0 - y_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 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	tmp = 0.0;
	if (y_46_im <= -320000.0)
		tmp = (y_46_im * log(sqrt(t_2))) / t_0;
	elseif (y_46_im <= 1.65e-41)
		tmp = t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_im <= 1.7e+93)
		tmp = sin(t_1) * (t_2 ^ (y_46_re / 2.0));
	else
		tmp = sin((log((-1.0 / x_46_re)) * (0.0 - y_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[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -320000.0], N[(N[(y$46$im * N[Log[N[Sqrt[t$95$2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 1.65e-41], 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$46$im, 1.7e+93], N[(N[Sin[t$95$1], $MachinePrecision] * N[Power[t$95$2, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] * N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.2e5

   1. Initial program 33.6%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 47.4%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

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

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

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

      6. *-commutative N/A

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

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

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

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

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. hypot-define N/A

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

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

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

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

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

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

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

      15. atan2-lowering-atan2.f64 53.4%

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

   7. Simplified 53.4%

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

   8. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

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

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

      8. unpow2 N/A

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

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

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

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

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

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

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

      12. atan2-lowering-atan2.f64 42.2%

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

   10. Simplified 42.2%

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

   if -3.2e5 < y.im < 1.65000000000000012e-41

   1. Initial program 37.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 57.5%

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

   5. Simplified 57.5%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 60.5%

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

   8. Simplified 60.5%

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

   if 1.65000000000000012e-41 < y.im < 1.7e93

   1. Initial program 56.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

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

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

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

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

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

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

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

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 36.9%

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

   5. Simplified 36.9%

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

      1. *-commutative N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

      4. sqrt-pow2 N/A

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

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

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

      6. sqrt-pow2 N/A

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

      7. +-commutative N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      15. atan2-lowering-atan2.f64 52.9%

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

   7. Applied egg-rr 52.9%

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

   if 1.7e93 < y.im

   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. Step-by-step derivation

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 56.2%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 67.7%

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

   7. Simplified 67.7%

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

   8. Taylor expanded in x.re around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x.re}\right)\right)}\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

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

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

      4. /-lowering-/.f64 50.0%

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

   10. Simplified 50.0%

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

4. Final simplification 53.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -320000:\\ \;\;\;\;\frac{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-41}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+93}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(0 - y.im\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 46.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-36}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sin \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(0 - y.im\right)\right)}{t\_1}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sin \left(\left(0 - y.im\right) \cdot \log \left(\frac{-1}{x.im}\right)\right)}{t\_1}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+224}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (exp (* y.im (atan2 x.im x.re)))))
   (if (<= y.re -1.3e-36)
     (* t_0 (pow (sqrt (+ (* x.im x.im) (* x.re x.re))) y.re))
     (if (<= y.re 4.4e-305)
       (/ (sin (* (log (/ -1.0 x.re)) (- 0.0 y.im))) t_1)
       (if (<= y.re 2.1e-160)
         (/ (sin (* (- 0.0 y.im) (log (/ -1.0 x.im)))) t_1)
         (if (<= y.re 1.15e+224)
           (* t_0 (pow (hypot x.im x.re) y.re))
           (* (sin t_0) (pow x.re y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -1.3e-36) {
		tmp = t_0 * pow(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))), y_46_re);
	} else if (y_46_re <= 4.4e-305) {
		tmp = sin((log((-1.0 / x_46_re)) * (0.0 - y_46_im))) / t_1;
	} else if (y_46_re <= 2.1e-160) {
		tmp = sin(((0.0 - y_46_im) * log((-1.0 / x_46_im)))) / t_1;
	} else if (y_46_re <= 1.15e+224) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = sin(t_0) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -1.3e-36) {
		tmp = t_0 * Math.pow(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))), y_46_re);
	} else if (y_46_re <= 4.4e-305) {
		tmp = Math.sin((Math.log((-1.0 / x_46_re)) * (0.0 - y_46_im))) / t_1;
	} else if (y_46_re <= 2.1e-160) {
		tmp = Math.sin(((0.0 - y_46_im) * Math.log((-1.0 / x_46_im)))) / t_1;
	} else if (y_46_re <= 1.15e+224) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.sin(t_0) * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_re <= -1.3e-36:
		tmp = t_0 * math.pow(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))), y_46_re)
	elif y_46_re <= 4.4e-305:
		tmp = math.sin((math.log((-1.0 / x_46_re)) * (0.0 - y_46_im))) / t_1
	elif y_46_re <= 2.1e-160:
		tmp = math.sin(((0.0 - y_46_im) * math.log((-1.0 / x_46_im)))) / t_1
	elif y_46_re <= 1.15e+224:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.sin(t_0) * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.3e-36)
		tmp = Float64(t_0 * (sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))) ^ y_46_re));
	elseif (y_46_re <= 4.4e-305)
		tmp = Float64(sin(Float64(log(Float64(-1.0 / x_46_re)) * Float64(0.0 - y_46_im))) / t_1);
	elseif (y_46_re <= 2.1e-160)
		tmp = Float64(sin(Float64(Float64(0.0 - y_46_im) * log(Float64(-1.0 / x_46_im)))) / t_1);
	elseif (y_46_re <= 1.15e+224)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(sin(t_0) * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_re <= -1.3e-36)
		tmp = t_0 * (sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))) ^ y_46_re);
	elseif (y_46_re <= 4.4e-305)
		tmp = sin((log((-1.0 / x_46_re)) * (0.0 - y_46_im))) / t_1;
	elseif (y_46_re <= 2.1e-160)
		tmp = sin(((0.0 - y_46_im) * log((-1.0 / x_46_im)))) / t_1;
	elseif (y_46_re <= 1.15e+224)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = sin(t_0) * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e-36], N[(t$95$0 * N[Power[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.4e-305], N[(N[Sin[N[(N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] * N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-160], N[(N[Sin[N[(N[(0.0 - y$46$im), $MachinePrecision] * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[y$46$re, 1.15e+224], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.3e-36

   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. Step-by-step derivation

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 62.3%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

      10. *-commutative N/A

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

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

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

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

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

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

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

   7. Simplified 67.5%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      8. unpow2 N/A

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

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

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

      10. unpow2 N/A

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

      11. *-lowering-*.f64 74.3%

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

   10. Simplified 74.3%

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

   if -1.3e-36 < y.re < 4.39999999999999993e-305

   1. Initial program 37.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 85.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 76.2%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 76.2%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in x.re around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x.re}\right)\right)}\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \left(\mathsf{neg}\left(\log \left(\frac{-1}{x.re}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{neg.f64}\left(\log \left(\frac{-1}{x.re}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{-1}{x.re}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. /-lowering-/.f64 46.2%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.re\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Simplified 46.2%

       \[\leadsto \frac{\sin \left(y.im \cdot \color{blue}{\left(-\log \left(\frac{-1}{x.re}\right)\right)}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if 4.39999999999999993e-305 < y.re < 2.1e-160

   1. Initial program 24.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 81.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 81.3%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in x.im around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \color{blue}{\left(-1 \cdot \log \left(\frac{-1}{x.im}\right)\right)}\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \left(\mathsf{neg}\left(\log \left(\frac{-1}{x.im}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{neg.f64}\left(\log \left(\frac{-1}{x.im}\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      3. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\left(\frac{-1}{x.im}\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

      4. /-lowering-/.f64 42.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{neg.f64}\left(\mathsf{log.f64}\left(\mathsf{/.f64}\left(-1, x.im\right)\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Simplified 42.0%

       \[\leadsto \frac{\sin \left(y.im \cdot \color{blue}{\left(-\log \left(\frac{-1}{x.im}\right)\right)}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   if 2.1e-160 < y.re < 1.1500000000000001e224

   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

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 39.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 39.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 46.2%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if 1.1500000000000001e224 < y.re

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 57.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({x.re}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{x.re}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({x.re}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({x.re}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 57.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right)\right) \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-36}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-305}:\\ \;\;\;\;\frac{\sin \left(\log \left(\frac{-1}{x.re}\right) \cdot \left(0 - y.im\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-160}:\\ \;\;\;\;\frac{\sin \left(\left(0 - y.im\right) \cdot \log \left(\frac{-1}{x.im}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+224}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 46.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 2800:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+294}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 2800.0)
   (* (* y.re (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re))
   (if (<= y.im 1.2e+294)
     (/ (sin (* y.im (log x.im))) (exp (* y.im (atan2 x.im x.re))))
     (*
      y.re
      (* (atan2 x.im x.re) (+ 1.0 (* (* y.re 0.5) (log (* x.im x.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 2800.0) {
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.2e+294) {
		tmp = sin((y_46_im * log(x_46_im))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * 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 tmp;
	if (y_46_im <= 2800.0) {
		tmp = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 1.2e+294) {
		tmp = Math.sin((y_46_im * Math.log(x_46_im))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * Math.log((x_46_im * x_46_im)))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 2800.0:
		tmp = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_im <= 1.2e+294:
		tmp = math.sin((y_46_im * math.log(x_46_im))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * math.log((x_46_im * x_46_im)))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 2800.0)
		tmp = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 1.2e+294)
		tmp = Float64(sin(Float64(y_46_im * log(x_46_im))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(1.0 + Float64(Float64(y_46_re * 0.5) * log(Float64(x_46_im * 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 (y_46_im <= 2800.0)
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_im <= 1.2e+294)
		tmp = sin((y_46_im * log(x_46_im))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * x_46_im)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 2800.0], 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$46$im, 1.2e+294], N[(N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(1.0 + N[(N[(y$46$re * 0.5), $MachinePrecision] * N[Log[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < 2800

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 45.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 49.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 49.9%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if 2800 < y.im < 1.20000000000000005e294

   1. Initial program 43.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 59.7%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 69.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 69.3%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log x.im\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log x.im\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log x.im\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      7. atan2-lowering-atan2.f64 47.0%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.im\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Simplified 47.0%

       \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   if 1.20000000000000005e294 < y.im

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 21.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 21.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 60.0%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 60.0%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \color{blue}{y.re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 60.0%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]

   11. Taylor expanded in x.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \log \left({x.im}^{2}\right)\right)\right)}, 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot y.re\right) \cdot \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left({x.im}^{2}\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left(x.im \cdot x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       6. *-lowering-*.f64 80.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

   13. Simplified 80.0%

       \[\leadsto \left(\left(\color{blue}{\left(0.5 \cdot y.re\right) \cdot \log \left(x.im \cdot x.im\right)} + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 2800:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+294}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log x.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -13500000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-63}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.re \cdot x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-189}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{+33}:\\ \;\;\;\;y.re \cdot \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (sin (* y.re (atan2 x.im x.re))) (pow x.im y.re))))
   (if (<= y.re -13500000.0)
     t_0
     (if (<= y.re -5e-63)
       (*
        y.re
        (* (atan2 x.im x.re) (+ 1.0 (* (* y.re 0.5) (log (* x.re x.re))))))
       (if (<= y.re 1.7e-189)
         (* y.im (log (sqrt (+ (* x.im x.im) (* x.re x.re)))))
         (if (<= y.re 3.7e+33) (* y.re (log (exp (atan2 x.im x.re)))) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_im, y_46_re);
	double tmp;
	if (y_46_re <= -13500000.0) {
		tmp = t_0;
	} else if (y_46_re <= -5e-63) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_re * x_46_re)))));
	} else if (y_46_re <= 1.7e-189) {
		tmp = y_46_im * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	} else if (y_46_re <= 3.7e+33) {
		tmp = y_46_re * log(exp(atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re))) * (x_46im ** y_46re)
    if (y_46re <= (-13500000.0d0)) then
        tmp = t_0
    else if (y_46re <= (-5d-63)) then
        tmp = y_46re * (atan2(x_46im, x_46re) * (1.0d0 + ((y_46re * 0.5d0) * log((x_46re * x_46re)))))
    else if (y_46re <= 1.7d-189) then
        tmp = y_46im * log(sqrt(((x_46im * x_46im) + (x_46re * x_46re))))
    else if (y_46re <= 3.7d+33) then
        tmp = y_46re * log(exp(atan2(x_46im, x_46re)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(x_46_im, y_46_re);
	double tmp;
	if (y_46_re <= -13500000.0) {
		tmp = t_0;
	} else if (y_46_re <= -5e-63) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * Math.log((x_46_re * x_46_re)))));
	} else if (y_46_re <= 1.7e-189) {
		tmp = y_46_im * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	} else if (y_46_re <= 3.7e+33) {
		tmp = y_46_re * Math.log(Math.exp(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 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(x_46_im, y_46_re)
	tmp = 0
	if y_46_re <= -13500000.0:
		tmp = t_0
	elif y_46_re <= -5e-63:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * math.log((x_46_re * x_46_re)))))
	elif y_46_re <= 1.7e-189:
		tmp = y_46_im * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))
	elif y_46_re <= 3.7e+33:
		tmp = y_46_re * math.log(math.exp(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(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_im ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -13500000.0)
		tmp = t_0;
	elseif (y_46_re <= -5e-63)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(1.0 + Float64(Float64(y_46_re * 0.5) * log(Float64(x_46_re * x_46_re))))));
	elseif (y_46_re <= 1.7e-189)
		tmp = Float64(y_46_im * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))));
	elseif (y_46_re <= 3.7e+33)
		tmp = Float64(y_46_re * log(exp(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 = sin((y_46_re * atan2(x_46_im, x_46_re))) * (x_46_im ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -13500000.0)
		tmp = t_0;
	elseif (y_46_re <= -5e-63)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_re * x_46_re)))));
	elseif (y_46_re <= 1.7e-189)
		tmp = y_46_im * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	elseif (y_46_re <= 3.7e+33)
		tmp = y_46_re * log(exp(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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -13500000.0], t$95$0, If[LessEqual[y$46$re, -5e-63], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(1.0 + N[(N[(y$46$re * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.7e-189], N[(y$46$im * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.7e+33], N[(y$46$re * N[Log[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.35e7 or 3.6999999999999999e33 < y.re

   1. Initial program 38.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 61.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 61.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({x.im}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{x.im}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({x.im}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({x.im}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 48.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(x.im, \color{blue}{y.re}\right)\right) \]

   8. Simplified 48.7%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]

   if -1.35e7 < y.re < -5.0000000000000002e-63

   1. Initial program 26.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 34.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 34.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 27.6%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 27.6%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \color{blue}{y.re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 27.6%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]

   11. Taylor expanded in x.im around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \log \left({x.re}^{2}\right)\right)\right)}, 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot y.re\right) \cdot \log \left({x.re}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot y.re\right), \log \left({x.re}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \log \left({x.re}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left({x.re}^{2}\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left(x.re \cdot x.re\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       6. *-lowering-*.f64 35.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

   13. Simplified 35.3%

       \[\leadsto \left(\left(\color{blue}{\left(0.5 \cdot y.re\right) \cdot \log \left(x.re \cdot x.re\right)} + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re \]

   if -5.0000000000000002e-63 < y.re < 1.7000000000000001e-189

   1. Initial program 34.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 79.1%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   9. 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 26.7%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right) \]

   10. Simplified 26.7%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]

   if 1.7000000000000001e-189 < y.re < 3.6999999999999999e33

   1. Initial program 43.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 31.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 31.7%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 28.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 28.7%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. rem-log-exp N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{exp.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      4. atan2-lowering-atan2.f64 28.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{exp.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Applied egg-rr 28.7%

       \[\leadsto y.re \cdot \color{blue}{\log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 37.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -13500000:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq -5 \cdot 10^{-63}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.re \cdot x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-189}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{+33}:\\ \;\;\;\;y.re \cdot \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{-194}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{+228}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* t_0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -6e-128)
     t_1
     (if (<= y.re 2.9e-194)
       (* y.im (log (sqrt (+ (* x.im x.im) (* x.re x.re)))))
       (if (<= y.re 2.9e+228) t_1 (* (sin t_0) (pow x.re y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -6e-128) {
		tmp = t_1;
	} else if (y_46_re <= 2.9e-194) {
		tmp = y_46_im * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	} else if (y_46_re <= 2.9e+228) {
		tmp = t_1;
	} else {
		tmp = sin(t_0) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -6e-128) {
		tmp = t_1;
	} else if (y_46_re <= 2.9e-194) {
		tmp = y_46_im * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	} else if (y_46_re <= 2.9e+228) {
		tmp = t_1;
	} else {
		tmp = Math.sin(t_0) * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -6e-128:
		tmp = t_1
	elif y_46_re <= 2.9e-194:
		tmp = y_46_im * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))
	elif y_46_re <= 2.9e+228:
		tmp = t_1
	else:
		tmp = math.sin(t_0) * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -6e-128)
		tmp = t_1;
	elseif (y_46_re <= 2.9e-194)
		tmp = Float64(y_46_im * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))));
	elseif (y_46_re <= 2.9e+228)
		tmp = t_1;
	else
		tmp = Float64(sin(t_0) * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -6e-128)
		tmp = t_1;
	elseif (y_46_re <= 2.9e-194)
		tmp = y_46_im * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	elseif (y_46_re <= 2.9e+228)
		tmp = t_1;
	else
		tmp = sin(t_0) * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -6e-128], t$95$1, If[LessEqual[y$46$re, 2.9e-194], N[(y$46$im * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.9e+228], t$95$1, N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.99999999999999956e-128 or 2.8999999999999997e-194 < y.re < 2.90000000000000002e228

   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

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 47.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 47.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 53.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 53.3%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if -5.99999999999999956e-128 < y.re < 2.8999999999999997e-194

   1. Initial program 38.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 89.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 81.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 81.4%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   9. 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 28.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right) \]

   10. Simplified 28.4%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]

   if 2.90000000000000002e228 < y.re

   1. Initial program 42.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 57.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 57.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({x.re}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{x.re}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({x.re}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({x.re}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 57.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right)\right) \]

   8. Simplified 57.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 46.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq 4 \cdot 10^{-44}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.im 4e-44)
     (* t_0 (pow (hypot x.im x.re) y.re))
     (* (sin t_0) (pow (+ (* x.im x.im) (* x.re x.re)) (/ 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 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= 4e-44) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = sin(t_0) * pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (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 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= 4e-44) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.sin(t_0) * Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / 2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_im <= 4e-44:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.sin(t_0) * math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (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(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= 4e-44)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(sin(t_0) * (Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ 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 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_im <= 4e-44)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = sin(t_0) * (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, 4e-44], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 3.99999999999999981e-44

   1. Initial program 35.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 46.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 46.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 50.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 50.5%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if 3.99999999999999981e-44 < y.im

   1. Initial program 41.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 27.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 27.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. sqrt-pow2 N/A

         \[\leadsto {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      3. +-commutative N/A

         \[\leadsto {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)} \cdot \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. sqrt-pow2 N/A

         \[\leadsto {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right), \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      6. sqrt-pow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(x.im \cdot x.im + x.re \cdot x.re\right), \left(\frac{y.re}{2}\right)\right), \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \left(\frac{y.re}{2}\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      13. sin-lowering-sin.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      15. atan2-lowering-atan2.f64 38.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right), \mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Applied egg-rr 38.4%

      \[\leadsto \color{blue}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 4 \cdot 10^{-44}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 46.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq 5 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.im 5e-7)
     (* t_0 (pow (hypot x.im x.re) y.re))
     (* t_0 (pow (sqrt (+ (* x.im x.im) (* x.re x.re))) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= 5e-7) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_0 * pow(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= 5e-7) {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_0 * Math.pow(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_im <= 5e-7:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = t_0 * math.pow(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= 5e-7)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(t_0 * (sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_im <= 5e-7)
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = t_0 * (sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, 5e-7], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 4.99999999999999977e-7

   1. Initial program 36.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 45.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 45.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{hypot.f64}\left(x.im, x.re\right)}, y.re\right)\right) \]

      2. atan2-lowering-atan2.f64 49.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, \color{blue}{x.re}\right), y.re\right)\right) \]

   8. Simplified 49.6%

      \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]

   if 4.99999999999999977e-7 < y.im

   1. Initial program 41.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 57.4%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) + y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\color{blue}{\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\color{blue}{\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)}\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{atan2.f64}\left(x.im, x.re\right)}, y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(y.re \cdot \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \left(\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      12. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

      13. cos-lowering-cos.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{cos.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 57.4%

      \[\leadsto \frac{\color{blue}{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) + y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right), y.re\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right), y.re\right)\right) \]

      11. *-lowering-*.f64 36.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), y.re\right)\right) \]

   10. Simplified 36.8%

       \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 34.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{-14}:\\ \;\;\;\;t\_0 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.im -3.1e-5)
     (*
      y.re
      (* (atan2 x.im x.re) (+ 1.0 (* (* y.re 0.5) (log (* x.im x.im))))))
     (if (<= x.im 9e-14) (* t_0 (pow x.re y.re)) (* t_0 (pow x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -3.1e-5) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * x_46_im)))));
	} else if (x_46_im <= 9e-14) {
		tmp = t_0 * pow(x_46_re, y_46_re);
	} else {
		tmp = t_0 * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46im <= (-3.1d-5)) then
        tmp = y_46re * (atan2(x_46im, x_46re) * (1.0d0 + ((y_46re * 0.5d0) * log((x_46im * x_46im)))))
    else if (x_46im <= 9d-14) then
        tmp = t_0 * (x_46re ** y_46re)
    else
        tmp = t_0 * (x_46im ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -3.1e-5) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * Math.log((x_46_im * x_46_im)))));
	} else if (x_46_im <= 9e-14) {
		tmp = t_0 * Math.pow(x_46_re, y_46_re);
	} else {
		tmp = t_0 * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -3.1e-5:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * math.log((x_46_im * x_46_im)))))
	elif x_46_im <= 9e-14:
		tmp = t_0 * math.pow(x_46_re, y_46_re)
	else:
		tmp = t_0 * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -3.1e-5)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(1.0 + Float64(Float64(y_46_re * 0.5) * log(Float64(x_46_im * x_46_im))))));
	elseif (x_46_im <= 9e-14)
		tmp = Float64(t_0 * (x_46_re ^ y_46_re));
	else
		tmp = Float64(t_0 * (x_46_im ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -3.1e-5)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * x_46_im)))));
	elseif (x_46_im <= 9e-14)
		tmp = t_0 * (x_46_re ^ y_46_re);
	else
		tmp = t_0 * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -3.1e-5], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(1.0 + N[(N[(y$46$re * 0.5), $MachinePrecision] * N[Log[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 9e-14], N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.10000000000000014e-5

   1. Initial program 23.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 36.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 36.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 29.4%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 29.4%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \color{blue}{y.re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 29.4%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]

   11. Taylor expanded in x.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \log \left({x.im}^{2}\right)\right)\right)}, 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot y.re\right) \cdot \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left({x.im}^{2}\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left(x.im \cdot x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       6. *-lowering-*.f64 31.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

   13. Simplified 31.4%

       \[\leadsto \left(\left(\color{blue}{\left(0.5 \cdot y.re\right) \cdot \log \left(x.im \cdot x.im\right)} + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re \]

   if -3.10000000000000014e-5 < x.im < 8.9999999999999995e-14

   1. Initial program 47.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 39.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 39.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({x.re}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{x.re}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({x.re}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({x.re}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 32.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right)\right) \]

   8. Simplified 32.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}} \]

   if 8.9999999999999995e-14 < x.im

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 46.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 46.4%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({x.im}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{x.im}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({x.im}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({x.im}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 46.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(x.im, \color{blue}{y.re}\right)\right) \]

   8. Simplified 46.4%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.1 \cdot 10^{-5}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{-14}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 23.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot x.im + x.re \cdot x.re\\ t_1 := y.im \cdot \log \left(\sqrt{t\_0}\right)\\ \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;y.re \cdot \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq 0.006:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\left(y.re \cdot 0.5\right) \cdot \log t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.im x.im) (* x.re x.re))) (t_1 (* y.im (log (sqrt t_0)))))
   (if (<= y.im -2.05e-73)
     t_1
     (if (<= y.im 1.25e-123)
       (* y.re (log (exp (atan2 x.im x.re))))
       (if (<= y.im 0.006)
         t_1
         (if (<= y.im 2.9e+72)
           (*
            (* y.re y.re)
            (*
             (atan2 x.im x.re)
             (log (+ x.im (/ (* (* x.re x.re) 0.5) x.im)))))
           (* y.re (* (atan2 x.im x.re) (* (* y.re 0.5) (log t_0))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = y_46_im * log(sqrt(t_0));
	double tmp;
	if (y_46_im <= -2.05e-73) {
		tmp = t_1;
	} else if (y_46_im <= 1.25e-123) {
		tmp = y_46_re * log(exp(atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 0.006) {
		tmp = t_1;
	} else if (y_46_im <= 2.9e+72) {
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * log(t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * x_46im) + (x_46re * x_46re)
    t_1 = y_46im * log(sqrt(t_0))
    if (y_46im <= (-2.05d-73)) then
        tmp = t_1
    else if (y_46im <= 1.25d-123) then
        tmp = y_46re * log(exp(atan2(x_46im, x_46re)))
    else if (y_46im <= 0.006d0) then
        tmp = t_1
    else if (y_46im <= 2.9d+72) then
        tmp = (y_46re * y_46re) * (atan2(x_46im, x_46re) * log((x_46im + (((x_46re * x_46re) * 0.5d0) / x_46im))))
    else
        tmp = y_46re * (atan2(x_46im, x_46re) * ((y_46re * 0.5d0) * log(t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = y_46_im * Math.log(Math.sqrt(t_0));
	double tmp;
	if (y_46_im <= -2.05e-73) {
		tmp = t_1;
	} else if (y_46_im <= 1.25e-123) {
		tmp = y_46_re * Math.log(Math.exp(Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 0.006) {
		tmp = t_1;
	} else if (y_46_im <= 2.9e+72) {
		tmp = (y_46_re * y_46_re) * (Math.atan2(x_46_im, x_46_re) * Math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * Math.log(t_0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	t_1 = y_46_im * math.log(math.sqrt(t_0))
	tmp = 0
	if y_46_im <= -2.05e-73:
		tmp = t_1
	elif y_46_im <= 1.25e-123:
		tmp = y_46_re * math.log(math.exp(math.atan2(x_46_im, x_46_re)))
	elif y_46_im <= 0.006:
		tmp = t_1
	elif y_46_im <= 2.9e+72:
		tmp = (y_46_re * y_46_re) * (math.atan2(x_46_im, x_46_re) * math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))))
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * math.log(t_0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	t_1 = Float64(y_46_im * log(sqrt(t_0)))
	tmp = 0.0
	if (y_46_im <= -2.05e-73)
		tmp = t_1;
	elseif (y_46_im <= 1.25e-123)
		tmp = Float64(y_46_re * log(exp(atan(x_46_im, x_46_re))));
	elseif (y_46_im <= 0.006)
		tmp = t_1;
	elseif (y_46_im <= 2.9e+72)
		tmp = Float64(Float64(y_46_re * y_46_re) * Float64(atan(x_46_im, x_46_re) * log(Float64(x_46_im + Float64(Float64(Float64(x_46_re * x_46_re) * 0.5) / x_46_im)))));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(Float64(y_46_re * 0.5) * log(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 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	t_1 = y_46_im * log(sqrt(t_0));
	tmp = 0.0;
	if (y_46_im <= -2.05e-73)
		tmp = t_1;
	elseif (y_46_im <= 1.25e-123)
		tmp = y_46_re * log(exp(atan2(x_46_im, x_46_re)));
	elseif (y_46_im <= 0.006)
		tmp = t_1;
	elseif (y_46_im <= 2.9e+72)
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * log(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[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.05e-73], t$95$1, If[LessEqual[y$46$im, 1.25e-123], N[(y$46$re * N[Log[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 0.006], t$95$1, If[LessEqual[y$46$im, 2.9e+72], N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Log[N[(x$46$im + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(N[(y$46$re * 0.5), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.05000000000000008e-73 or 1.25000000000000007e-123 < y.im < 0.0060000000000000001

   1. Initial program 31.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   9. 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 26.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right) \]

   10. Simplified 26.4%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]

   if -2.05000000000000008e-73 < y.im < 1.25000000000000007e-123

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 31.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. rem-log-exp N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

      3. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{exp.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      4. atan2-lowering-atan2.f64 37.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{exp.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   10. Applied egg-rr 37.7%

       \[\leadsto y.re \cdot \color{blue}{\log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)} \]

   if 0.0060000000000000001 < y.im < 2.90000000000000017e72

   1. Initial program 60.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 29.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 29.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 8.4%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 8.4%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   9. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{{y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y.re}^{2}\right), \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       5. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. atan2-lowering-atan2.f64 8.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   11. Simplified 8.0%

       \[\leadsto \color{blue}{\left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   12. Taylor expanded in x.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(x.im + \frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)}\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{\frac{1}{2} \cdot {x.re}^{2}}{x.im}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.re}^{2}\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.re}^{2}\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.re \cdot x.re\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       6. *-lowering-*.f64 40.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

   14. Simplified 40.8%

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \color{blue}{\left(x.im + \frac{0.5 \cdot \left(x.re \cdot x.re\right)}{x.im}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if 2.90000000000000017e72 < y.im

   1. Initial program 35.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 26.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 27.1%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 27.1%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \color{blue}{y.re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 27.1%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]

   11. Taylor expanded in y.re around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot y.re\right) \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot y.re\right), \log \left({x.im}^{2} + {x.re}^{2}\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \log \left({x.im}^{2} + {x.re}^{2}\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       9. *-lowering-*.f64 29.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

   13. Simplified 29.0%

       \[\leadsto \left(\color{blue}{\left(\left(0.5 \cdot y.re\right) \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-123}:\\ \;\;\;\;y.re \cdot \log \left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq 0.006:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 22.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot x.im + x.re \cdot x.re\\ t_1 := y.im \cdot \log \left(\sqrt{t\_0}\right)\\ \mathbf{if}\;y.im \leq -4.3 \cdot 10^{-77}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.im \leq 4300000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\left(y.re \cdot 0.5\right) \cdot \log t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.im x.im) (* x.re x.re))) (t_1 (* y.im (log (sqrt t_0)))))
   (if (<= y.im -4.3e-77)
     t_1
     (if (<= y.im 8.5e-125)
       (* y.re (atan2 x.im x.re))
       (if (<= y.im 4300000.0)
         t_1
         (if (<= y.im 2.9e+72)
           (*
            (* y.re y.re)
            (*
             (atan2 x.im x.re)
             (log (+ x.im (/ (* (* x.re x.re) 0.5) x.im)))))
           (* y.re (* (atan2 x.im x.re) (* (* y.re 0.5) (log t_0))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = y_46_im * log(sqrt(t_0));
	double tmp;
	if (y_46_im <= -4.3e-77) {
		tmp = t_1;
	} else if (y_46_im <= 8.5e-125) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else if (y_46_im <= 4300000.0) {
		tmp = t_1;
	} else if (y_46_im <= 2.9e+72) {
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * log(t_0)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * x_46im) + (x_46re * x_46re)
    t_1 = y_46im * log(sqrt(t_0))
    if (y_46im <= (-4.3d-77)) then
        tmp = t_1
    else if (y_46im <= 8.5d-125) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else if (y_46im <= 4300000.0d0) then
        tmp = t_1
    else if (y_46im <= 2.9d+72) then
        tmp = (y_46re * y_46re) * (atan2(x_46im, x_46re) * log((x_46im + (((x_46re * x_46re) * 0.5d0) / x_46im))))
    else
        tmp = y_46re * (atan2(x_46im, x_46re) * ((y_46re * 0.5d0) * log(t_0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = y_46_im * Math.log(Math.sqrt(t_0));
	double tmp;
	if (y_46_im <= -4.3e-77) {
		tmp = t_1;
	} else if (y_46_im <= 8.5e-125) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else if (y_46_im <= 4300000.0) {
		tmp = t_1;
	} else if (y_46_im <= 2.9e+72) {
		tmp = (y_46_re * y_46_re) * (Math.atan2(x_46_im, x_46_re) * Math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * Math.log(t_0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	t_1 = y_46_im * math.log(math.sqrt(t_0))
	tmp = 0
	if y_46_im <= -4.3e-77:
		tmp = t_1
	elif y_46_im <= 8.5e-125:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	elif y_46_im <= 4300000.0:
		tmp = t_1
	elif y_46_im <= 2.9e+72:
		tmp = (y_46_re * y_46_re) * (math.atan2(x_46_im, x_46_re) * math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))))
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * math.log(t_0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	t_1 = Float64(y_46_im * log(sqrt(t_0)))
	tmp = 0.0
	if (y_46_im <= -4.3e-77)
		tmp = t_1;
	elseif (y_46_im <= 8.5e-125)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	elseif (y_46_im <= 4300000.0)
		tmp = t_1;
	elseif (y_46_im <= 2.9e+72)
		tmp = Float64(Float64(y_46_re * y_46_re) * Float64(atan(x_46_im, x_46_re) * log(Float64(x_46_im + Float64(Float64(Float64(x_46_re * x_46_re) * 0.5) / x_46_im)))));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(Float64(y_46_re * 0.5) * log(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 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	t_1 = y_46_im * log(sqrt(t_0));
	tmp = 0.0;
	if (y_46_im <= -4.3e-77)
		tmp = t_1;
	elseif (y_46_im <= 8.5e-125)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	elseif (y_46_im <= 4300000.0)
		tmp = t_1;
	elseif (y_46_im <= 2.9e+72)
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * ((y_46_re * 0.5) * log(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[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[Log[N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.3e-77], t$95$1, If[LessEqual[y$46$im, 8.5e-125], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4300000.0], t$95$1, If[LessEqual[y$46$im, 2.9e+72], N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Log[N[(x$46$im + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(N[(y$46$re * 0.5), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.3000000000000002e-77 or 8.5000000000000002e-125 < y.im < 4.3e6

   1. Initial program 31.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   9. 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 26.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right) \]

   10. Simplified 26.4%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]

   if -4.3000000000000002e-77 < y.im < 8.5000000000000002e-125

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 31.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 4.3e6 < y.im < 2.90000000000000017e72

   1. Initial program 60.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 29.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 29.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 8.4%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 8.4%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   9. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{{y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y.re}^{2}\right), \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       5. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. atan2-lowering-atan2.f64 8.0%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   11. Simplified 8.0%

       \[\leadsto \color{blue}{\left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   12. Taylor expanded in x.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(x.im + \frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)}\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{\frac{1}{2} \cdot {x.re}^{2}}{x.im}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.re}^{2}\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.re}^{2}\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.re \cdot x.re\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       6. *-lowering-*.f64 40.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

   14. Simplified 40.8%

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \color{blue}{\left(x.im + \frac{0.5 \cdot \left(x.re \cdot x.re\right)}{x.im}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if 2.90000000000000017e72 < y.im

   1. Initial program 35.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 26.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 26.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 27.1%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 27.1%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \color{blue}{y.re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 27.1%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]

   11. Taylor expanded in y.re around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)}, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\left(\frac{1}{2} \cdot y.re\right) \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot y.re\right), \log \left({x.im}^{2} + {x.re}^{2}\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \log \left({x.im}^{2} + {x.re}^{2}\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       6. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       9. *-lowering-*.f64 29.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

   13. Simplified 29.0%

       \[\leadsto \left(\color{blue}{\left(\left(0.5 \cdot y.re\right) \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.3 \cdot 10^{-77}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.im \leq 4300000:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{+72}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 21.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{if}\;y.im \leq -1.04 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-124}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.im \leq 820000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (sqrt (+ (* x.im x.im) (* x.re x.re)))))))
   (if (<= y.im -1.04e-79)
     t_0
     (if (<= y.im 2e-124)
       (* y.re (atan2 x.im x.re))
       (if (<= y.im 820000.0)
         t_0
         (if (<= y.im 4.4e+79)
           (*
            (* y.re y.re)
            (*
             (atan2 x.im x.re)
             (log (+ x.im (/ (* (* x.re x.re) 0.5) x.im)))))
           (*
            y.re
            (*
             (atan2 x.im x.re)
             (+ 1.0 (* (* y.re 0.5) (log (* x.im 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 * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if (y_46_im <= -1.04e-79) {
		tmp = t_0;
	} else if (y_46_im <= 2e-124) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else if (y_46_im <= 820000.0) {
		tmp = t_0;
	} else if (y_46_im <= 4.4e+79) {
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * 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) :: t_0
    real(8) :: tmp
    t_0 = y_46im * log(sqrt(((x_46im * x_46im) + (x_46re * x_46re))))
    if (y_46im <= (-1.04d-79)) then
        tmp = t_0
    else if (y_46im <= 2d-124) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else if (y_46im <= 820000.0d0) then
        tmp = t_0
    else if (y_46im <= 4.4d+79) then
        tmp = (y_46re * y_46re) * (atan2(x_46im, x_46re) * log((x_46im + (((x_46re * x_46re) * 0.5d0) / x_46im))))
    else
        tmp = y_46re * (atan2(x_46im, x_46re) * (1.0d0 + ((y_46re * 0.5d0) * log((x_46im * 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 t_0 = y_46_im * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if (y_46_im <= -1.04e-79) {
		tmp = t_0;
	} else if (y_46_im <= 2e-124) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else if (y_46_im <= 820000.0) {
		tmp = t_0;
	} else if (y_46_im <= 4.4e+79) {
		tmp = (y_46_re * y_46_re) * (Math.atan2(x_46_im, x_46_re) * Math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * Math.log((x_46_im * 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.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))
	tmp = 0
	if y_46_im <= -1.04e-79:
		tmp = t_0
	elif y_46_im <= 2e-124:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	elif y_46_im <= 820000.0:
		tmp = t_0
	elif y_46_im <= 4.4e+79:
		tmp = (y_46_re * y_46_re) * (math.atan2(x_46_im, x_46_re) * math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))))
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * math.log((x_46_im * 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 * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -1.04e-79)
		tmp = t_0;
	elseif (y_46_im <= 2e-124)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	elseif (y_46_im <= 820000.0)
		tmp = t_0;
	elseif (y_46_im <= 4.4e+79)
		tmp = Float64(Float64(y_46_re * y_46_re) * Float64(atan(x_46_im, x_46_re) * log(Float64(x_46_im + Float64(Float64(Float64(x_46_re * x_46_re) * 0.5) / x_46_im)))));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(1.0 + Float64(Float64(y_46_re * 0.5) * log(Float64(x_46_im * 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 * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	tmp = 0.0;
	if (y_46_im <= -1.04e-79)
		tmp = t_0;
	elseif (y_46_im <= 2e-124)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	elseif (y_46_im <= 820000.0)
		tmp = t_0;
	elseif (y_46_im <= 4.4e+79)
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * 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[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.04e-79], t$95$0, If[LessEqual[y$46$im, 2e-124], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 820000.0], t$95$0, If[LessEqual[y$46$im, 4.4e+79], N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Log[N[(x$46$im + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(1.0 + N[(N[(y$46$re * 0.5), $MachinePrecision] * N[Log[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.04e-79 or 1.99999999999999987e-124 < y.im < 8.2e5

   1. Initial program 31.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 64.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 53.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 53.6%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   9. 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 26.4%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right) \]

   10. Simplified 26.4%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]

   if -1.04e-79 < y.im < 1.99999999999999987e-124

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 31.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 8.2e5 < y.im < 4.3999999999999998e79

   1. Initial program 58.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   4. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 37.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 37.4%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 19.2%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 19.2%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   9. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{{y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y.re}^{2}\right), \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       5. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. atan2-lowering-atan2.f64 18.8%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   11. Simplified 18.8%

       \[\leadsto \color{blue}{\left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   12. Taylor expanded in x.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\color{blue}{\left(x.im + \frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)}\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]
   13. Step-by-step derivation

       1. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{1}{2} \cdot \frac{{x.re}^{2}}{x.im}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       2. associate-*r/ N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \left(\frac{\frac{1}{2} \cdot {x.re}^{2}}{x.im}\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       3. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.re}^{2}\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.re}^{2}\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.re \cdot x.re\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

       6. *-lowering-*.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{+.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.re, x.re\right)\right), x.im\right)\right)\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right) \]

   14. Simplified 47.8%

       \[\leadsto \left(y.re \cdot y.re\right) \cdot \left(\log \color{blue}{\left(x.im + \frac{0.5 \cdot \left(x.re \cdot x.re\right)}{x.im}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if 4.3999999999999998e79 < y.im

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 23.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 23.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 24.3%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 24.3%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \color{blue}{y.re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 24.3%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]

   11. Taylor expanded in x.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \log \left({x.im}^{2}\right)\right)\right)}, 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot y.re\right) \cdot \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left({x.im}^{2}\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left(x.im \cdot x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       6. *-lowering-*.f64 22.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

   13. Simplified 22.9%

       \[\leadsto \left(\left(\color{blue}{\left(0.5 \cdot y.re\right) \cdot \log \left(x.im \cdot x.im\right)} + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re \]
3. Recombined 4 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.04 \cdot 10^{-79}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-124}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.im \leq 820000:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{+79}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 21.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{-80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-124}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.im \leq 530000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (sqrt (+ (* x.im x.im) (* x.re x.re)))))))
   (if (<= y.im -7.5e-80)
     t_0
     (if (<= y.im 8.2e-124)
       (* y.re (atan2 x.im x.re))
       (if (<= y.im 530000000.0)
         t_0
         (*
          y.re
          (*
           (atan2 x.im x.re)
           (+ 1.0 (* (* y.re 0.5) (log (* x.im 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 * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if (y_46_im <= -7.5e-80) {
		tmp = t_0;
	} else if (y_46_im <= 8.2e-124) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else if (y_46_im <= 530000000.0) {
		tmp = t_0;
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * 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) :: t_0
    real(8) :: tmp
    t_0 = y_46im * log(sqrt(((x_46im * x_46im) + (x_46re * x_46re))))
    if (y_46im <= (-7.5d-80)) then
        tmp = t_0
    else if (y_46im <= 8.2d-124) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else if (y_46im <= 530000000.0d0) then
        tmp = t_0
    else
        tmp = y_46re * (atan2(x_46im, x_46re) * (1.0d0 + ((y_46re * 0.5d0) * log((x_46im * 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 t_0 = y_46_im * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if (y_46_im <= -7.5e-80) {
		tmp = t_0;
	} else if (y_46_im <= 8.2e-124) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else if (y_46_im <= 530000000.0) {
		tmp = t_0;
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * Math.log((x_46_im * 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.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))
	tmp = 0
	if y_46_im <= -7.5e-80:
		tmp = t_0
	elif y_46_im <= 8.2e-124:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	elif y_46_im <= 530000000.0:
		tmp = t_0
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * math.log((x_46_im * 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 * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -7.5e-80)
		tmp = t_0;
	elseif (y_46_im <= 8.2e-124)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	elseif (y_46_im <= 530000000.0)
		tmp = t_0;
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * Float64(1.0 + Float64(Float64(y_46_re * 0.5) * log(Float64(x_46_im * 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 * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	tmp = 0.0;
	if (y_46_im <= -7.5e-80)
		tmp = t_0;
	elseif (y_46_im <= 8.2e-124)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	elseif (y_46_im <= 530000000.0)
		tmp = t_0;
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (1.0 + ((y_46_re * 0.5) * log((x_46_im * 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[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.5e-80], t$95$0, If[LessEqual[y$46$im, 8.2e-124], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 530000000.0], t$95$0, N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(1.0 + N[(N[(y$46$re * 0.5), $MachinePrecision] * N[Log[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.49999999999999999e-80 or 8.2000000000000008e-124 < y.im < 5.3e8

   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. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 64.1%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 54.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 54.0%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   9. 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 25.7%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right) \]

   10. Simplified 25.7%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]

   if -7.49999999999999999e-80 < y.im < 8.2000000000000008e-124

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 31.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 5.3e8 < y.im

   1. Initial program 39.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 26.3%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 26.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 24.1%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 24.1%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \color{blue}{y.re} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \color{blue}{y.re}\right) \]

   10. Applied egg-rr 24.1%

       \[\leadsto \color{blue}{\left(\left(y.re \cdot \left(\log \left(x.im \cdot x.im + x.re \cdot x.re\right) \cdot 0.5\right) + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re} \]

   11. Taylor expanded in x.re around 0

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \log \left({x.im}^{2}\right)\right)\right)}, 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(\left(\frac{1}{2} \cdot y.re\right) \cdot \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{2} \cdot y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \log \left({x.im}^{2}\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left({x.im}^{2}\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\left(x.im \cdot x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

       6. *-lowering-*.f64 21.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, y.re\right), \mathsf{log.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right)\right)\right), 1\right), \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.re\right) \]

   13. Simplified 21.5%

       \[\leadsto \left(\left(\color{blue}{\left(0.5 \cdot y.re\right) \cdot \log \left(x.im \cdot x.im\right)} + 1\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{-80}:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-124}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{elif}\;y.im \leq 530000000:\\ \;\;\;\;y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(1 + \left(y.re \cdot 0.5\right) \cdot \log \left(x.im \cdot x.im\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 21.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-79}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{-123}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (sqrt (+ (* x.im x.im) (* x.re x.re)))))))
   (if (<= y.im -1.7e-79)
     t_0
     (if (<= y.im 2.3e-123) (* y.re (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_im * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e-79) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-123) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_46im * log(sqrt(((x_46im * x_46im) + (x_46re * x_46re))))
    if (y_46im <= (-1.7d-79)) then
        tmp = t_0
    else if (y_46im <= 2.3d-123) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	double tmp;
	if (y_46_im <= -1.7e-79) {
		tmp = t_0;
	} else if (y_46_im <= 2.3e-123) {
		tmp = y_46_re * 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_im * math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))
	tmp = 0
	if y_46_im <= -1.7e-79:
		tmp = t_0
	elif y_46_im <= 2.3e-123:
		tmp = y_46_re * 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(y_46_im * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -1.7e-79)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-123)
		tmp = Float64(y_46_re * 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_im * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))));
	tmp = 0.0;
	if (y_46_im <= -1.7e-79)
		tmp = t_0;
	elseif (y_46_im <= 2.3e-123)
		tmp = y_46_re * 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[(y$46$im * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.7e-79], t$95$0, If[LessEqual[y$46$im, 2.3e-123], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.69999999999999988e-79 or 2.29999999999999987e-123 < y.im

   1. Initial program 35.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \color{blue}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      2. associate-*l/ N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      3. associate-/l* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re} \cdot \color{blue}{\frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \color{blue}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

      5. associate-/r/ N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{\color{blue}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}}} \]

      6. exp-diff N/A

         \[\leadsto \frac{\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im - \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}} \]

   3. Simplified 61.2%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left(e^{\color{blue}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      7. hypot-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      8. hypot-lowering-hypot.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      9. exp-lowering-exp.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

      11. atan2-lowering-atan2.f64 59.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right) \]

   7. Simplified 59.0%

      \[\leadsto \color{blue}{\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   9. 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. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right) \]

      4. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right) \]

      8. *-lowering-*.f64 20.0%

         \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right) \]

   10. Simplified 20.0%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]

   if -1.69999999999999988e-79 < y.im < 2.29999999999999987e-123

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 69.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 69.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 31.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 31.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 16.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 1.5:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 1.5)
   (* y.re (atan2 x.im x.re))
   (* (* y.re y.re) (* (atan2 x.im x.re) (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 (y_46_im <= 1.5) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else {
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * 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 (y_46im <= 1.5d0) then
        tmp = y_46re * atan2(x_46im, x_46re)
    else
        tmp = (y_46re * y_46re) * (atan2(x_46im, x_46re) * 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 (y_46_im <= 1.5) {
		tmp = y_46_re * Math.atan2(x_46_im, x_46_re);
	} else {
		tmp = (y_46_re * y_46_re) * (Math.atan2(x_46_im, x_46_re) * 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 y_46_im <= 1.5:
		tmp = y_46_re * math.atan2(x_46_im, x_46_re)
	else:
		tmp = (y_46_re * y_46_re) * (math.atan2(x_46_im, x_46_re) * 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 (y_46_im <= 1.5)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	else
		tmp = Float64(Float64(y_46_re * y_46_re) * Float64(atan(x_46_im, x_46_re) * 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 (y_46_im <= 1.5)
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	else
		tmp = (y_46_re * y_46_re) * (atan2(x_46_im, x_46_re) * 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[y$46$im, 1.5], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 1.5

   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.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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 45.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 45.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 18.1%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 18.1%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 1.5 < y.im

   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. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      2. sin-lowering-sin.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

      9. hypot-lowering-hypot.f64 26.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 26.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{\left(y.re \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}\right)\right) \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\color{blue}{y.re} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right)\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      9. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

      14. atan2-lowering-atan2.f64 23.1%

          \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{+.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right)\right) \]

   8. Simplified 23.1%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.re \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   9. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{{y.re}^{2} \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y.re}^{2}\right), \color{blue}{\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       5. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

       6. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. unpow2 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. atan2-lowering-atan2.f64 24.4%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   11. Simplified 24.4%

       \[\leadsto \color{blue}{\left(y.re \cdot y.re\right) \cdot \left(\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   12. Taylor expanded in x.re around 0

       \[\leadsto \color{blue}{{y.re}^{2} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   13. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({y.re}^{2}\right), \color{blue}{\left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

       2. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot y.re\right), \left(\color{blue}{\log x.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \left(\color{blue}{\log x.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\log x.im, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

       5. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(x.im\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

       6. atan2-lowering-atan2.f64 14.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, y.re\right), \mathsf{*.f64}\left(\mathsf{log.f64}\left(x.im\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   14. Simplified 14.2%

       \[\leadsto \color{blue}{\left(y.re \cdot y.re\right) \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 1.5:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \log x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 13.8% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* y.re (atan2 x.im x.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * atan2(x_46_im, x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46re * atan2(x_46im, x_46re)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * Math.atan2(x_46_im, x_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_re * math.atan2(x_46_im, x_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_re * atan(x_46_im, x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_re * atan2(x_46_im, x_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 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.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. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

   2. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left({\left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)}^{y.re}\right)\right) \]

   4. atan2-lowering-atan2.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

   5. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right), y.re\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

   8. hypot-define N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re\right)\right) \]

   9. hypot-lowering-hypot.f64 40.8%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

5. Simplified 40.8%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

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 14.1%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 14.1%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024158 
(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)))))