powComplex, imaginary part

Percentage Accurate: 40.5% → 75.0%

Time: 26.7s

Alternatives: 25

Speedup: 3.7×

• 36.0% 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: 40.5% 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: 75.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t\_1\\ t_3 := x.im \cdot x.im + x.re \cdot x.re\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{t\_2}{{t\_3}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 215:\\ \;\;\;\;\frac{t\_2 + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos t\_1\right)}{\frac{e^{t\_0}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot e^{y.re \cdot \log \left(\sqrt{t\_3}\right) - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (sin t_1))
        (t_3 (+ (* x.im x.im) (* x.re x.re))))
   (if (<= y.re -5.8e+39)
     (/ t_2 (pow t_3 (/ y.re -2.0)))
     (if (<= y.re 215.0)
       (/
        (+ t_2 (* y.im (* (log (hypot x.im x.re)) (cos t_1))))
        (/ (exp t_0) (pow (hypot x.re x.im) y.re)))
       (* t_2 (exp (- (* y.re (log (sqrt t_3))) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(t_1);
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -5.8e+39) {
		tmp = t_2 / pow(t_3, (y_46_re / -2.0));
	} else if (y_46_re <= 215.0) {
		tmp = (t_2 + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_1)))) / (exp(t_0) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = t_2 * exp(((y_46_re * log(sqrt(t_3))) - 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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.sin(t_1);
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -5.8e+39) {
		tmp = t_2 / Math.pow(t_3, (y_46_re / -2.0));
	} else if (y_46_re <= 215.0) {
		tmp = (t_2 + (y_46_im * (Math.log(Math.hypot(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));
	} else {
		tmp = t_2 * Math.exp(((y_46_re * Math.log(Math.sqrt(t_3))) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.sin(t_1)
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	tmp = 0
	if y_46_re <= -5.8e+39:
		tmp = t_2 / math.pow(t_3, (y_46_re / -2.0))
	elif y_46_re <= 215.0:
		tmp = (t_2 + (y_46_im * (math.log(math.hypot(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))
	else:
		tmp = t_2 * math.exp(((y_46_re * math.log(math.sqrt(t_3))) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = sin(t_1)
	t_3 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_re <= -5.8e+39)
		tmp = Float64(t_2 / (t_3 ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= 215.0)
		tmp = Float64(Float64(t_2 + Float64(y_46_im * Float64(log(hypot(x_46_im, x_46_re)) * cos(t_1)))) / Float64(exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(t_2 * exp(Float64(Float64(y_46_re * log(sqrt(t_3))) - 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 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = sin(t_1);
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	tmp = 0.0;
	if (y_46_re <= -5.8e+39)
		tmp = t_2 / (t_3 ^ (y_46_re / -2.0));
	elseif (y_46_re <= 215.0)
		tmp = (t_2 + (y_46_im * (log(hypot(x_46_im, x_46_re)) * cos(t_1)))) / (exp(t_0) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = t_2 * exp(((y_46_re * log(sqrt(t_3))) - 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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e+39], N[(t$95$2 / N[Power[t$95$3, N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 215.0], N[(N[(t$95$2 + N[(y$46$im * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 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[(t$95$2 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.80000000000000059e39

   1. Initial program 25.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 72.3%

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

   5. Simplified 72.3%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 87.2%

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

   if -5.80000000000000059e39 < y.re < 215

   1. Initial program 45.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 84.0%

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

   5. Taylor expanded in y.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 85.8%

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

   7. Simplified 85.8%

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

   if 215 < y.re

   1. Initial program 51.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 73.2%

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

   5. Simplified 73.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 215:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := x.im \cdot x.im + x.re \cdot x.re\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{t\_1}{{t\_3}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 1.82:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{e^{t\_2}} \cdot \sin \left(t\_0 + y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot e^{y.re \cdot \log \left(\sqrt{t\_3}\right) - t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (+ (* x.im x.im) (* x.re x.re))))
   (if (<= y.re -7.2e+39)
     (/ t_1 (pow t_3 (/ y.re -2.0)))
     (if (<= y.re 1.82)
       (*
        (/ (pow (hypot x.im x.re) y.re) (exp t_2))
        (sin (+ t_0 (* y.im (log (hypot x.im x.re))))))
       (* t_1 (exp (- (* y.re (log (sqrt t_3))) t_2)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -7.2e+39) {
		tmp = t_1 / pow(t_3, (y_46_re / -2.0));
	} else if (y_46_re <= 1.82) {
		tmp = (pow(hypot(x_46_im, x_46_re), y_46_re) / exp(t_2)) * sin((t_0 + (y_46_im * log(hypot(x_46_im, x_46_re)))));
	} else {
		tmp = t_1 * exp(((y_46_re * log(sqrt(t_3))) - 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_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -7.2e+39) {
		tmp = t_1 / Math.pow(t_3, (y_46_re / -2.0));
	} else if (y_46_re <= 1.82) {
		tmp = (Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re) / Math.exp(t_2)) * Math.sin((t_0 + (y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))));
	} else {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(Math.sqrt(t_3))) - t_2));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin(t_0)
	t_2 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	tmp = 0
	if y_46_re <= -7.2e+39:
		tmp = t_1 / math.pow(t_3, (y_46_re / -2.0))
	elif y_46_re <= 1.82:
		tmp = (math.pow(math.hypot(x_46_im, x_46_re), y_46_re) / math.exp(t_2)) * math.sin((t_0 + (y_46_im * math.log(math.hypot(x_46_im, x_46_re)))))
	else:
		tmp = t_1 * math.exp(((y_46_re * math.log(math.sqrt(t_3))) - t_2))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_3 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_re <= -7.2e+39)
		tmp = Float64(t_1 / (t_3 ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= 1.82)
		tmp = Float64(Float64((hypot(x_46_im, x_46_re) ^ y_46_re) / exp(t_2)) * sin(Float64(t_0 + Float64(y_46_im * log(hypot(x_46_im, x_46_re))))));
	else
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(sqrt(t_3))) - 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_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	tmp = 0.0;
	if (y_46_re <= -7.2e+39)
		tmp = t_1 / (t_3 ^ (y_46_re / -2.0));
	elseif (y_46_re <= 1.82)
		tmp = ((hypot(x_46_im, x_46_re) ^ y_46_re) / exp(t_2)) * sin((t_0 + (y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = t_1 * exp(((y_46_re * log(sqrt(t_3))) - 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$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.2e+39], N[(t$95$1 / N[Power[t$95$3, N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.82], N[(N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Exp[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(t$95$0 + N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.19999999999999969e39

   1. Initial program 25.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 72.3%

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

   5. Simplified 72.3%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 87.2%

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

   if -7.19999999999999969e39 < y.re < 1.82000000000000006

   1. Initial program 45.0%

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

   3. Taylor expanded in y.re around inf

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

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

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

   5. Simplified 84.0%

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

   if 1.82000000000000006 < y.re

   1. Initial program 51.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 73.2%

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

   5. Simplified 73.2%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 1.82:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t\_1\\ t_3 := x.im \cdot x.im + x.re \cdot x.re\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_2}{{t\_3}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 0.00085:\\ \;\;\;\;\sin \left(t\_1 + y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \frac{1}{e^{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot e^{y.re \cdot \log \left(\sqrt{t\_3}\right) - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (sin t_1))
        (t_3 (+ (* x.im x.im) (* x.re x.re))))
   (if (<= y.re -4.6e+15)
     (/ t_2 (pow t_3 (/ y.re -2.0)))
     (if (<= y.re 0.00085)
       (* (sin (+ t_1 (* y.im (log (hypot x.im x.re))))) (/ 1.0 (exp t_0)))
       (* t_2 (exp (- (* y.re (log (sqrt t_3))) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(t_1);
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_2 / pow(t_3, (y_46_re / -2.0));
	} else if (y_46_re <= 0.00085) {
		tmp = sin((t_1 + (y_46_im * log(hypot(x_46_im, x_46_re))))) * (1.0 / exp(t_0));
	} else {
		tmp = t_2 * exp(((y_46_re * log(sqrt(t_3))) - 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.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.sin(t_1);
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_2 / Math.pow(t_3, (y_46_re / -2.0));
	} else if (y_46_re <= 0.00085) {
		tmp = Math.sin((t_1 + (y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))))) * (1.0 / Math.exp(t_0));
	} else {
		tmp = t_2 * Math.exp(((y_46_re * Math.log(Math.sqrt(t_3))) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.sin(t_1)
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	tmp = 0
	if y_46_re <= -4.6e+15:
		tmp = t_2 / math.pow(t_3, (y_46_re / -2.0))
	elif y_46_re <= 0.00085:
		tmp = math.sin((t_1 + (y_46_im * math.log(math.hypot(x_46_im, x_46_re))))) * (1.0 / math.exp(t_0))
	else:
		tmp = t_2 * math.exp(((y_46_re * math.log(math.sqrt(t_3))) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = sin(t_1)
	t_3 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_re <= -4.6e+15)
		tmp = Float64(t_2 / (t_3 ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= 0.00085)
		tmp = Float64(sin(Float64(t_1 + Float64(y_46_im * log(hypot(x_46_im, x_46_re))))) * Float64(1.0 / exp(t_0)));
	else
		tmp = Float64(t_2 * exp(Float64(Float64(y_46_re * log(sqrt(t_3))) - 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 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = sin(t_1);
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.6e+15)
		tmp = t_2 / (t_3 ^ (y_46_re / -2.0));
	elseif (y_46_re <= 0.00085)
		tmp = sin((t_1 + (y_46_im * log(hypot(x_46_im, x_46_re))))) * (1.0 / exp(t_0));
	else
		tmp = t_2 * exp(((y_46_re * log(sqrt(t_3))) - 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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+15], N[(t$95$2 / N[Power[t$95$3, N[(y$46$re / -2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.00085], N[(N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.6e15

   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 75.1%

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

   5. Simplified 75.1%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 86.8%

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

   if -4.6e15 < y.re < 8.49999999999999953e-4

   1. Initial program 45.5%

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

   3. Taylor expanded in y.re around inf

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

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

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

   5. Simplified 83.4%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 81.7%

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

   if 8.49999999999999953e-4 < y.re

   1. Initial program 51.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 73.2%

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

   5. Simplified 73.2%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 0.00085:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_1}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(t\_0 + y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (sin t_0)))
   (if (<= y.re -4.6e+15)
     (/ t_1 (pow (+ (* x.im x.im) (* x.re x.re)) (/ y.re -2.0)))
     (if (<= y.re 8.2e-6)
       (*
        (sin (+ t_0 (* y.im (log (hypot x.im x.re)))))
        (/ 1.0 (exp (* (atan2 x.im x.re) y.im))))
       (* t_1 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_1 / pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= 8.2e-6) {
		tmp = sin((t_0 + (y_46_im * log(hypot(x_46_im, x_46_re))))) * (1.0 / exp((atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_1 / Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= 8.2e-6) {
		tmp = Math.sin((t_0 + (y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))))) * (1.0 / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin(t_0)
	tmp = 0
	if y_46_re <= -4.6e+15:
		tmp = t_1 / math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0))
	elif y_46_re <= 8.2e-6:
		tmp = math.sin((t_0 + (y_46_im * math.log(math.hypot(x_46_im, x_46_re))))) * (1.0 / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im)))
	else:
		tmp = t_1 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -4.6e+15)
		tmp = Float64(t_1 / (Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= 8.2e-6)
		tmp = Float64(sin(Float64(t_0 + Float64(y_46_im * log(hypot(x_46_im, x_46_re))))) * Float64(1.0 / exp(Float64(atan(x_46_im, x_46_re) * y_46_im))));
	else
		tmp = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	tmp = 0.0;
	if (y_46_re <= -4.6e+15)
		tmp = t_1 / (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (y_46_re / -2.0));
	elseif (y_46_re <= 8.2e-6)
		tmp = sin((t_0 + (y_46_im * log(hypot(x_46_im, x_46_re))))) * (1.0 / exp((atan2(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+15], N[(t$95$1 / 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], If[LessEqual[y$46$re, 8.2e-6], N[(N[Sin[N[(t$95$0 + N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.6e15

   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 75.1%

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

   5. Simplified 75.1%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 86.8%

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

   if -4.6e15 < y.re < 8.1999999999999994e-6

   1. Initial program 45.8%

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

   3. Taylor expanded in y.re around inf

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

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

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

   5. Simplified 83.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\color{blue}{1}, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{sin.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right)\right)\right) \]
   7. Step-by-step derivation

   8. Simplified 82.0%

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

   if 8.1999999999999994e-6 < y.re

   1. Initial program 50.0%

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

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{t\_0}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq -2.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{t\_0}{\frac{t\_1}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (exp (* (atan2 x.im x.re) y.im))))
   (if (<= y.re -6.4e+39)
     (/ t_0 (pow (+ (* x.im x.im) (* x.re x.re)) (/ y.re -2.0)))
     (if (<= y.re -2.75e-62)
       (/ t_0 (/ t_1 (pow (hypot x.re x.im) y.re)))
       (if (<= y.re 1.55e-17)
         (/ (sin (* y.im (log (hypot x.im x.re)))) t_1)
         (* t_0 (pow (hypot x.im x.re) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	double tmp;
	if (y_46_re <= -6.4e+39) {
		tmp = t_0 / pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= -2.75e-62) {
		tmp = t_0 / (t_1 / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else if (y_46_re <= 1.55e-17) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_1;
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	double tmp;
	if (y_46_re <= -6.4e+39) {
		tmp = t_0 / Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= -2.75e-62) {
		tmp = t_0 / (t_1 / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else if (y_46_re <= 1.55e-17) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / t_1;
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	tmp = 0
	if y_46_re <= -6.4e+39:
		tmp = t_0 / math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0))
	elif y_46_re <= -2.75e-62:
		tmp = t_0 / (t_1 / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	elif y_46_re <= 1.55e-17:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / t_1
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = exp(Float64(atan(x_46_im, x_46_re) * y_46_im))
	tmp = 0.0
	if (y_46_re <= -6.4e+39)
		tmp = Float64(t_0 / (Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= -2.75e-62)
		tmp = Float64(t_0 / Float64(t_1 / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	elseif (y_46_re <= 1.55e-17)
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / t_1);
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = exp((atan2(x_46_im, x_46_re) * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -6.4e+39)
		tmp = t_0 / (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (y_46_re / -2.0));
	elseif (y_46_re <= -2.75e-62)
		tmp = t_0 / (t_1 / (hypot(x_46_re, x_46_im) ^ y_46_re));
	elseif (y_46_re <= 1.55e-17)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_1;
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -6.4e+39], N[(t$95$0 / 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], If[LessEqual[y$46$re, -2.75e-62], N[(t$95$0 / N[(t$95$1 / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.55e-17], 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$1), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.39999999999999986e39

   1. Initial program 25.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 72.3%

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

   5. Simplified 72.3%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 87.2%

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

   if -6.39999999999999986e39 < y.re < -2.75000000000000011e-62

   1. Initial program 36.7%

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

      1. exp-diff 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 77.2%

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

   5. Taylor expanded in y.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 73.7%

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

   7. Simplified 73.7%

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

   if -2.75000000000000011e-62 < y.re < 1.5499999999999999e-17

   1. Initial program 48.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 85.6%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   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 70.8%

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

   7. Simplified 70.8%

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

   if 1.5499999999999999e-17 < y.re

   1. Initial program 47.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 66.8%

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

   5. Simplified 66.8%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq -2.75 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_0}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-61}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - t\_1}\right)\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* (atan2 x.im x.re) y.im)))
   (if (<= y.re -4.6e+15)
     (/ t_0 (pow (+ (* x.im x.im) (* x.re x.re)) (/ y.re -2.0)))
     (if (<= y.re -6.6e-61)
       (* y.re (* (atan2 x.im x.re) (exp (- 0.0 t_1))))
       (if (<= y.re 7.5e-17)
         (/ (sin (* y.im (log (hypot x.im x.re)))) (exp t_1))
         (* t_0 (pow (hypot x.im x.re) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_0 / pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= -6.6e-61) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - t_1)));
	} else if (y_46_re <= 7.5e-17) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp(t_1);
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_0 / Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= -6.6e-61) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((0.0 - t_1)));
	} else if (y_46_re <= 7.5e-17) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / Math.exp(t_1);
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if y_46_re <= -4.6e+15:
		tmp = t_0 / math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0))
	elif y_46_re <= -6.6e-61:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((0.0 - t_1)))
	elif y_46_re <= 7.5e-17:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / math.exp(t_1)
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.6e+15)
		tmp = Float64(t_0 / (Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= -6.6e-61)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(0.0 - t_1))));
	elseif (y_46_re <= 7.5e-17)
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / exp(t_1));
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (y_46_re <= -4.6e+15)
		tmp = t_0 / (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (y_46_re / -2.0));
	elseif (y_46_re <= -6.6e-61)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - t_1)));
	elseif (y_46_re <= 7.5e-17)
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp(t_1);
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+15], N[(t$95$0 / 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], If[LessEqual[y$46$re, -6.6e-61], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(0.0 - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.5e-17], 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[t$95$1], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.6e15

   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 75.1%

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

   5. Simplified 75.1%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 86.8%

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

   if -4.6e15 < y.re < -6.59999999999999992e-61

   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

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

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

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

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

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

      3. atan2-lowering-atan2.f64 47.2%

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

   5. Simplified 47.2%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

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

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

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

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

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

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

      9. atan2-lowering-atan2.f64 67.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 67.9%

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

   if -6.59999999999999992e-61 < y.re < 7.49999999999999984e-17

   1. Initial program 48.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 85.6%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   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 70.8%

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

   7. Simplified 70.8%

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

   if 7.49999999999999984e-17 < y.re

   1. Initial program 47.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 66.8%

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

   5. Simplified 66.8%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq -6.6 \cdot 10^{-61}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t\_0}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -4.6e+15)
     (/ t_0 (pow (+ (* x.im x.im) (* x.re x.re)) (/ y.re -2.0)))
     (if (<= y.re 7.2e-13)
       (* y.re (* (atan2 x.im x.re) (exp (- 0.0 (* (atan2 x.im x.re) y.im)))))
       (* t_0 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_0 / pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= 7.2e-13) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_0 / Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	} else if (y_46_re <= 7.2e-13) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((0.0 - (Math.atan2(x_46_im, x_46_re) * y_46_im))));
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_re <= -4.6e+15:
		tmp = t_0 / math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0))
	elif y_46_re <= 7.2e-13:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((0.0 - (math.atan2(x_46_im, x_46_re) * y_46_im))))
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -4.6e+15)
		tmp = Float64(t_0 / (Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(y_46_re / -2.0)));
	elseif (y_46_re <= 7.2e-13)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(0.0 - Float64(atan(x_46_im, x_46_re) * y_46_im)))));
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_re <= -4.6e+15)
		tmp = t_0 / (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (y_46_re / -2.0));
	elseif (y_46_re <= 7.2e-13)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+15], N[(t$95$0 / 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], If[LessEqual[y$46$re, 7.2e-13], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(0.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.6e15

   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 75.1%

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

   5. Simplified 75.1%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 86.8%

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

   if -4.6e15 < y.re < 7.1999999999999996e-13

   1. Initial program 46.0%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 38.8%

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

   5. Simplified 38.8%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

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

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

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

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

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

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

      9. atan2-lowering-atan2.f64 57.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 57.7%

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

   if 7.1999999999999996e-13 < y.re

   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. 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 66.8%

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

   5. Simplified 66.8%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 47.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.25 \cdot 10^{-38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -8.5 \cdot 10^{-260}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.68 \cdot 10^{-136}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (hypot x.im x.re))))
        (t_1 (* (* y.re (atan2 x.im x.re)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -2.25e-38)
     t_1
     (if (<= y.re -8.5e-260)
       t_0
       (if (<= y.re 1.68e-136)
         (/ (/ (atan2 x.im x.re) (/ y.im y.re)) (/ 1.0 y.im))
         (if (<= y.re 5.5e-37) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_1 = (y_46_re * atan2(x_46_im, x_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.25e-38) {
		tmp = t_1;
	} else if (y_46_re <= -8.5e-260) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-136) {
		tmp = (atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	} else if (y_46_re <= 5.5e-37) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double t_1 = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.25e-38) {
		tmp = t_1;
	} else if (y_46_re <= -8.5e-260) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-136) {
		tmp = (Math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	} else if (y_46_re <= 5.5e-37) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	t_1 = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -2.25e-38:
		tmp = t_1
	elif y_46_re <= -8.5e-260:
		tmp = t_0
	elif y_46_re <= 1.68e-136:
		tmp = (math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im)
	elif y_46_re <= 5.5e-37:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_1 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.25e-38)
		tmp = t_1;
	elseif (y_46_re <= -8.5e-260)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-136)
		tmp = Float64(Float64(atan(x_46_im, x_46_re) / Float64(y_46_im / y_46_re)) / Float64(1.0 / y_46_im));
	elseif (y_46_re <= 5.5e-37)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	t_1 = (y_46_re * atan2(x_46_im, x_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.25e-38)
		tmp = t_1;
	elseif (y_46_re <= -8.5e-260)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-136)
		tmp = (atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	elseif (y_46_re <= 5.5e-37)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.25e-38], t$95$1, If[LessEqual[y$46$re, -8.5e-260], t$95$0, If[LessEqual[y$46$re, 1.68e-136], N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.5e-37], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.25000000000000004e-38 or 5.4999999999999998e-37 < y.re

   1. Initial program 35.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]
   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 66.5%

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

   8. Simplified 66.5%

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

   if -2.25000000000000004e-38 < y.re < -8.5000000000000003e-260 or 1.67999999999999993e-136 < y.re < 5.4999999999999998e-37

   1. Initial program 50.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 80.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   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 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 25.5%

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

   10. Simplified 25.5%

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

   11. Taylor expanded in y.re around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 46.1%

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

   13. Simplified 46.1%

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

   if -8.5000000000000003e-260 < y.re < 1.67999999999999993e-136

   1. Initial program 46.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 19.7%

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

   5. Simplified 19.7%

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

   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 19.7%

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

   8. Simplified 19.7%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-div N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\frac{y.im}{y.re}\right)\right), \left(\frac{1}{y.im}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), \left(\frac{1}{y.im}\right)\right) \]

      17. /-lowering-/.f64 53.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), \mathsf{/.f64}\left(1, \color{blue}{y.im}\right)\right) \]

   10. Applied egg-rr 53.4%

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

Alternative 9: 59.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 x.im) (* x.re x.re)) (/ y.re -2.0)))))
   (if (<= y.re -4.6e+15)
     t_0
     (if (<= y.re 7.2e-13)
       (* y.re (* (atan2 x.im x.re) (exp (- 0.0 (* (atan2 x.im x.re) 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 = sin((y_46_re * atan2(x_46_im, x_46_re))) / pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_0;
	} else if (y_46_re <= 7.2e-13) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	} 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 * x_46im) + (x_46re * x_46re)) ** (y_46re / (-2.0d0)))
    if (y_46re <= (-4.6d+15)) then
        tmp = t_0
    else if (y_46re <= 7.2d-13) then
        tmp = y_46re * (atan2(x_46im, x_46re) * exp((0.0d0 - (atan2(x_46im, x_46re) * y_46im))))
    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 * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0));
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_0;
	} else if (y_46_re <= 7.2e-13) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((0.0 - (Math.atan2(x_46_im, x_46_re) * y_46_im))));
	} 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 * x_46_im) + (x_46_re * x_46_re)), (y_46_re / -2.0))
	tmp = 0
	if y_46_re <= -4.6e+15:
		tmp = t_0
	elif y_46_re <= 7.2e-13:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((0.0 - (math.atan2(x_46_im, x_46_re) * y_46_im))))
	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))) / (Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(y_46_re / -2.0)))
	tmp = 0.0
	if (y_46_re <= -4.6e+15)
		tmp = t_0;
	elseif (y_46_re <= 7.2e-13)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(0.0 - Float64(atan(x_46_im, x_46_re) * y_46_im)))));
	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 * x_46_im) + (x_46_re * x_46_re)) ^ (y_46_re / -2.0));
	tmp = 0.0;
	if (y_46_re <= -4.6e+15)
		tmp = t_0;
	elseif (y_46_re <= 7.2e-13)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	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[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]}, If[LessEqual[y$46$re, -4.6e+15], t$95$0, If[LessEqual[y$46$re, 7.2e-13], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(0.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.6e15

   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 75.1%

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

   5. Simplified 75.1%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

      7. frac-2neg N/A

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

      8. sqrt-pow2 N/A

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

      9. +-commutative N/A

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

      10. sqrt-pow2 N/A

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

      11. distribute-frac-neg2 N/A

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

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

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

   7. Applied egg-rr 86.8%

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

   if -4.6e15 < y.re < 7.1999999999999996e-13

   1. Initial program 46.0%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 38.8%

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

   5. Simplified 38.8%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

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

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

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

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

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

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

      9. atan2-lowering-atan2.f64 57.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 57.7%

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

   if 7.1999999999999996e-13 < y.re

   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. 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 66.8%

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

   5. Simplified 66.8%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

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

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

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

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

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

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

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

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

      11. sqrt-pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(\frac{1}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\color{blue}{\left(\frac{y.re}{2}\right)}}}\right)\right) \]

      12. pow-flip N/A

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

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

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

   7. Applied egg-rr 64.6%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;t\_0 \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin t\_0}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -4.6e+15)
     (* t_0 (pow (+ x.re (/ (* (* x.im x.im) 0.5) x.re)) y.re))
     (if (<= y.re 7.2e-13)
       (* y.re (* (atan2 x.im x.re) (exp (- 0.0 (* (atan2 x.im x.re) y.im)))))
       (/ (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_re <= -4.6e+15) {
		tmp = t_0 * pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 7.2e-13) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	} 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;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    if (y_46re <= (-4.6d+15)) then
        tmp = t_0 * ((x_46re + (((x_46im * x_46im) * 0.5d0) / x_46re)) ** y_46re)
    else if (y_46re <= 7.2d-13) then
        tmp = y_46re * (atan2(x_46im, x_46re) * exp((0.0d0 - (atan2(x_46im, x_46re) * y_46im))))
    else
        tmp = sin(t_0) / (((x_46im * x_46im) + (x_46re * x_46re)) ** (y_46re / (-2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = t_0 * Math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 7.2e-13) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((0.0 - (Math.atan2(x_46_im, x_46_re) * y_46_im))));
	} 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_re <= -4.6e+15:
		tmp = t_0 * math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re)
	elif y_46_re <= 7.2e-13:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((0.0 - (math.atan2(x_46_im, x_46_re) * y_46_im))))
	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_re <= -4.6e+15)
		tmp = Float64(t_0 * (Float64(x_46_re + Float64(Float64(Float64(x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re));
	elseif (y_46_re <= 7.2e-13)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(0.0 - Float64(atan(x_46_im, x_46_re) * y_46_im)))));
	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_re <= -4.6e+15)
		tmp = t_0 * ((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re);
	elseif (y_46_re <= 7.2e-13)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	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$46$re, -4.6e+15], N[(t$95$0 * N[Power[N[(x$46$re + N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e-13], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(0.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $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 3 regimes

2. if y.re < -4.6e15

   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 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{\frac{1}{2} \cdot {x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.im}^{2}\right)\right), x.re\right)\right), y.re\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.im \cdot x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

      6. *-lowering-*.f64 73.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

   8. Simplified 73.5%

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

   9. Taylor expanded in y.re around 0

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

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

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

       2. atan2-lowering-atan2.f64 81.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)}\right), y.re\right)\right) \]

   11. Simplified 81.8%

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

   if -4.6e15 < y.re < 7.1999999999999996e-13

   1. Initial program 46.0%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 38.8%

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

   5. Simplified 38.8%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

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

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

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

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

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

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

      9. atan2-lowering-atan2.f64 57.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 57.7%

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

   if 7.1999999999999996e-13 < y.re

   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. 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 66.8%

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

   5. Simplified 66.8%

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

      1. /-rgt-identity N/A

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

      2. clear-num N/A

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

      3. sqrt-pow2 N/A

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

      4. +-commutative N/A

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

      5. sqrt-pow2 N/A

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

      6. div-inv N/A

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

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

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

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

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

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

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

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

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

      11. sqrt-pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(\frac{1}{{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\color{blue}{\left(\frac{y.re}{2}\right)}}}\right)\right) \]

      12. pow-flip N/A

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

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

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

   7. Applied egg-rr 64.6%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-13}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{-2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot x.im + x.re \cdot x.re\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\left(0.5 \cdot \log t\_0\right) \cdot {t\_0}^{\left(\frac{y.re}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.im x.im) (* x.re x.re))))
   (if (<= y.re -4.6e+15)
     (*
      (* y.re (atan2 x.im x.re))
      (pow (+ x.re (/ (* (* x.im x.im) 0.5) x.re)) y.re))
     (if (<= y.re 1.65e-6)
       (* y.re (* (atan2 x.im x.re) (exp (- 0.0 (* (atan2 x.im x.re) y.im)))))
       (* y.im (* (* 0.5 (log t_0)) (pow t_0 (/ y.re 2.0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 1.65e-6) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	} else {
		tmp = y_46_im * ((0.5 * log(t_0)) * pow(t_0, (y_46_re / 2.0)));
	}
	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 = (x_46im * x_46im) + (x_46re * x_46re)
    if (y_46re <= (-4.6d+15)) then
        tmp = (y_46re * atan2(x_46im, x_46re)) * ((x_46re + (((x_46im * x_46im) * 0.5d0) / x_46re)) ** y_46re)
    else if (y_46re <= 1.65d-6) then
        tmp = y_46re * (atan2(x_46im, x_46re) * exp((0.0d0 - (atan2(x_46im, x_46re) * y_46im))))
    else
        tmp = y_46im * ((0.5d0 * log(t_0)) * (t_0 ** (y_46re / 2.0d0)))
    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 tmp;
	if (y_46_re <= -4.6e+15) {
		tmp = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 1.65e-6) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((0.0 - (Math.atan2(x_46_im, x_46_re) * y_46_im))));
	} else {
		tmp = y_46_im * ((0.5 * Math.log(t_0)) * Math.pow(t_0, (y_46_re / 2.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	tmp = 0
	if y_46_re <= -4.6e+15:
		tmp = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re)
	elif y_46_re <= 1.65e-6:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((0.0 - (math.atan2(x_46_im, x_46_re) * y_46_im))))
	else:
		tmp = y_46_im * ((0.5 * math.log(t_0)) * math.pow(t_0, (y_46_re / 2.0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_re <= -4.6e+15)
		tmp = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (Float64(x_46_re + Float64(Float64(Float64(x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re));
	elseif (y_46_re <= 1.65e-6)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(0.0 - Float64(atan(x_46_im, x_46_re) * y_46_im)))));
	else
		tmp = Float64(y_46_im * Float64(Float64(0.5 * log(t_0)) * (t_0 ^ Float64(y_46_re / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	tmp = 0.0;
	if (y_46_re <= -4.6e+15)
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * ((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re);
	elseif (y_46_re <= 1.65e-6)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((0.0 - (atan2(x_46_im, x_46_re) * y_46_im))));
	else
		tmp = y_46_im * ((0.5 * log(t_0)) * (t_0 ^ (y_46_re / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+15], N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x$46$re + N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.65e-6], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(0.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[(0.5 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.6e15

   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 75.1%

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

   5. Simplified 75.1%

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

   6. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{\frac{1}{2} \cdot {x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.im}^{2}\right)\right), x.re\right)\right), y.re\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.im \cdot x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

      6. *-lowering-*.f64 73.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

   8. Simplified 73.5%

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

   9. Taylor expanded in y.re around 0

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

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

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

       2. atan2-lowering-atan2.f64 81.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)}\right), y.re\right)\right) \]

   11. Simplified 81.8%

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

   if -4.6e15 < y.re < 1.65000000000000008e-6

   1. Initial program 46.1%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 38.9%

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

   5. Simplified 38.9%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. mul-1-neg N/A

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

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

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

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

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

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

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

      9. atan2-lowering-atan2.f64 57.2%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{*.f64}\left(-1, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   8. Simplified 57.2%

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

   if 1.65000000000000008e-6 < y.re

   1. Initial program 48.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 62.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   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 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 62.9%

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

   10. Simplified 62.9%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

   12. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\left(\left(0.5 \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right) \cdot y.im} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\left(0.5 \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot x.im + x.re \cdot x.re\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -4500000000000:\\ \;\;\;\;t\_1 \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;\sin t\_1 \cdot {\left(0 - x.re\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -8.5 \cdot 10^{-260}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-38}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\left(0.5 \cdot \log t\_0\right) \cdot {t\_0}^{\left(\frac{y.re}{2}\right)}\right)\\ \end{array} \end{array} \]

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.re (atan2 x.im x.re)))
        (t_2 (* y.im (log (hypot x.im x.re)))))
   (if (<= y.re -4500000000000.0)
     (* t_1 (pow (+ x.re (/ (* (* x.im x.im) 0.5) x.re)) y.re))
     (if (<= y.re -7.4e-37)
       (* (sin t_1) (pow (- 0.0 x.re) y.re))
       (if (<= y.re -8.5e-260)
         t_2
         (if (<= y.re 1.02e-135)
           (/ (/ (atan2 x.im x.re) (/ y.im y.re)) (/ 1.0 y.im))
           (if (<= y.re 4.3e-38)
             t_2
             (if (<= y.re 1.65e-6)
               (/ y.im (/ (/ y.im y.re) (atan2 x.im x.re)))
               (* y.im (* (* 0.5 (log t_0)) (pow t_0 (/ y.re 2.0))))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = y_46_im * log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -4500000000000.0) {
		tmp = t_1 * pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= -7.4e-37) {
		tmp = sin(t_1) * pow((0.0 - x_46_re), y_46_re);
	} else if (y_46_re <= -8.5e-260) {
		tmp = t_2;
	} else if (y_46_re <= 1.02e-135) {
		tmp = (atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	} else if (y_46_re <= 4.3e-38) {
		tmp = t_2;
	} else if (y_46_re <= 1.65e-6) {
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	} else {
		tmp = y_46_im * ((0.5 * log(t_0)) * pow(t_0, (y_46_re / 2.0)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -4500000000000.0) {
		tmp = t_1 * Math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= -7.4e-37) {
		tmp = Math.sin(t_1) * Math.pow((0.0 - x_46_re), y_46_re);
	} else if (y_46_re <= -8.5e-260) {
		tmp = t_2;
	} else if (y_46_re <= 1.02e-135) {
		tmp = (Math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	} else if (y_46_re <= 4.3e-38) {
		tmp = t_2;
	} else if (y_46_re <= 1.65e-6) {
		tmp = y_46_im / ((y_46_im / y_46_re) / Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = y_46_im * ((0.5 * Math.log(t_0)) * Math.pow(t_0, (y_46_re / 2.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if y_46_re <= -4500000000000.0:
		tmp = t_1 * math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re)
	elif y_46_re <= -7.4e-37:
		tmp = math.sin(t_1) * math.pow((0.0 - x_46_re), y_46_re)
	elif y_46_re <= -8.5e-260:
		tmp = t_2
	elif y_46_re <= 1.02e-135:
		tmp = (math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im)
	elif y_46_re <= 4.3e-38:
		tmp = t_2
	elif y_46_re <= 1.65e-6:
		tmp = y_46_im / ((y_46_im / y_46_re) / math.atan2(x_46_im, x_46_re))
	else:
		tmp = y_46_im * ((0.5 * math.log(t_0)) * math.pow(t_0, (y_46_re / 2.0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -4500000000000.0)
		tmp = Float64(t_1 * (Float64(x_46_re + Float64(Float64(Float64(x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re));
	elseif (y_46_re <= -7.4e-37)
		tmp = Float64(sin(t_1) * (Float64(0.0 - x_46_re) ^ y_46_re));
	elseif (y_46_re <= -8.5e-260)
		tmp = t_2;
	elseif (y_46_re <= 1.02e-135)
		tmp = Float64(Float64(atan(x_46_im, x_46_re) / Float64(y_46_im / y_46_re)) / Float64(1.0 / y_46_im));
	elseif (y_46_re <= 4.3e-38)
		tmp = t_2;
	elseif (y_46_re <= 1.65e-6)
		tmp = Float64(y_46_im / Float64(Float64(y_46_im / y_46_re) / atan(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_im * Float64(Float64(0.5 * log(t_0)) * (t_0 ^ Float64(y_46_re / 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = y_46_im * log(hypot(x_46_im, x_46_re));
	tmp = 0.0;
	if (y_46_re <= -4500000000000.0)
		tmp = t_1 * ((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re);
	elseif (y_46_re <= -7.4e-37)
		tmp = sin(t_1) * ((0.0 - x_46_re) ^ y_46_re);
	elseif (y_46_re <= -8.5e-260)
		tmp = t_2;
	elseif (y_46_re <= 1.02e-135)
		tmp = (atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	elseif (y_46_re <= 4.3e-38)
		tmp = t_2;
	elseif (y_46_re <= 1.65e-6)
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	else
		tmp = y_46_im * ((0.5 * log(t_0)) * (t_0 ^ (y_46_re / 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * 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[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4500000000000.0], N[(t$95$1 * N[Power[N[(x$46$re + N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -7.4e-37], N[(N[Sin[t$95$1], $MachinePrecision] * N[Power[N[(0.0 - x$46$re), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -8.5e-260], t$95$2, If[LessEqual[y$46$re, 1.02e-135], N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.3e-38], t$95$2, If[LessEqual[y$46$re, 1.65e-6], N[(y$46$im / N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[(0.5 * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.re < -4.5e12

   1. Initial program 27.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 73.9%

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

   5. Simplified 73.9%

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

   6. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{\frac{1}{2} \cdot {x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.im}^{2}\right)\right), x.re\right)\right), y.re\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.im \cdot x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

      6. *-lowering-*.f64 74.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

   8. Simplified 74.0%

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

   9. Taylor expanded in y.re around 0

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

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

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

       2. atan2-lowering-atan2.f64 82.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)}\right), y.re\right)\right) \]

   11. Simplified 82.1%

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

   if -4.5e12 < y.re < -7.4e-37

   1. Initial program 35.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-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 54.5%

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

   5. Simplified 54.5%

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

   6. Taylor expanded in x.re around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 47.5%

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

   8. Simplified 47.5%

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

   if -7.4e-37 < y.re < -8.5000000000000003e-260 or 1.01999999999999994e-135 < y.re < 4.3000000000000002e-38

   1. Initial program 50.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 80.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   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 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 25.5%

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

   10. Simplified 25.5%

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

   11. Taylor expanded in y.re around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 46.1%

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

   13. Simplified 46.1%

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

   if -8.5000000000000003e-260 < y.re < 1.01999999999999994e-135

   1. Initial program 46.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 19.7%

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

   5. Simplified 19.7%

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

   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 19.7%

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

   8. Simplified 19.7%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-div N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\frac{y.im}{y.re}\right)\right), \left(\frac{1}{y.im}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), \left(\frac{1}{y.im}\right)\right) \]

      17. /-lowering-/.f64 53.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), \mathsf{/.f64}\left(1, \color{blue}{y.im}\right)\right) \]

   10. Applied egg-rr 53.4%

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

   if 4.3000000000000002e-38 < y.re < 1.65000000000000008e-6

   1. Initial program 27.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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 55.7%

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

   5. Simplified 55.7%

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

   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 48.2%

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

   8. Simplified 48.2%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-/r* N/A

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

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

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

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

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

      12. atan2-lowering-atan2.f64 57.1%

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

   10. Applied egg-rr 57.1%

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

   if 1.65000000000000008e-6 < y.re

   1. Initial program 48.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 62.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   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 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 62.9%

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

   10. Simplified 62.9%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

   12. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\left(\left(0.5 \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right) \cdot y.im} \]
3. Recombined 6 regimes into one program.

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4500000000000:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -7.4 \cdot 10^{-37}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(0 - x.re\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -8.5 \cdot 10^{-260}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-38}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\left(0.5 \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 48.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ t_1 := x.im \cdot x.im + x.re \cdot x.re\\ \mathbf{if}\;y.re \leq -0.019:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-137}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-35}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\left(0.5 \cdot \log t\_1\right) \cdot {t\_1}^{\left(\frac{y.re}{2}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ y.im (/ (/ y.im y.re) (atan2 x.im x.re))))
        (t_1 (+ (* x.im x.im) (* x.re x.re))))
   (if (<= y.re -0.019)
     (*
      (* y.re (atan2 x.im x.re))
      (pow (+ x.re (/ (* (* x.im x.im) 0.5) x.re)) y.re))
     (if (<= y.re 1.15e-137)
       t_0
       (if (<= y.re 1.25e-35)
         (* y.im (log (hypot x.im x.re)))
         (if (<= y.re 1.35e-6)
           t_0
           (* y.im (* (* 0.5 (log t_1)) (pow t_1 (/ 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_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	double t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -0.019) {
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 1.15e-137) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-35) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else if (y_46_re <= 1.35e-6) {
		tmp = t_0;
	} else {
		tmp = y_46_im * ((0.5 * log(t_1)) * pow(t_1, (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_im / ((y_46_im / y_46_re) / Math.atan2(x_46_im, x_46_re));
	double t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double tmp;
	if (y_46_re <= -0.019) {
		tmp = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 1.15e-137) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-35) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else if (y_46_re <= 1.35e-6) {
		tmp = t_0;
	} else {
		tmp = y_46_im * ((0.5 * Math.log(t_1)) * Math.pow(t_1, (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_im / ((y_46_im / y_46_re) / math.atan2(x_46_im, x_46_re))
	t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	tmp = 0
	if y_46_re <= -0.019:
		tmp = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re)
	elif y_46_re <= 1.15e-137:
		tmp = t_0
	elif y_46_re <= 1.25e-35:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	elif y_46_re <= 1.35e-6:
		tmp = t_0
	else:
		tmp = y_46_im * ((0.5 * math.log(t_1)) * math.pow(t_1, (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_im / Float64(Float64(y_46_im / y_46_re) / atan(x_46_im, x_46_re)))
	t_1 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_re <= -0.019)
		tmp = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (Float64(x_46_re + Float64(Float64(Float64(x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re));
	elseif (y_46_re <= 1.15e-137)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-35)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.35e-6)
		tmp = t_0;
	else
		tmp = Float64(y_46_im * Float64(Float64(0.5 * log(t_1)) * (t_1 ^ 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_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	tmp = 0.0;
	if (y_46_re <= -0.019)
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * ((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re);
	elseif (y_46_re <= 1.15e-137)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-35)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	elseif (y_46_re <= 1.35e-6)
		tmp = t_0;
	else
		tmp = y_46_im * ((0.5 * log(t_1)) * (t_1 ^ (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$46$im / N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -0.019], N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x$46$re + N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.15e-137], t$95$0, If[LessEqual[y$46$re, 1.25e-35], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.35e-6], t$95$0, N[(y$46$im * N[(N[(0.5 * N[Log[t$95$1], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$1, N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -0.0189999999999999995

   1. Initial program 27.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 74.0%

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

   5. Simplified 74.0%

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

   6. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{\frac{1}{2} \cdot {x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.im}^{2}\right)\right), x.re\right)\right), y.re\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.im \cdot x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

      6. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

   8. Simplified 71.0%

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

   9. Taylor expanded in y.re around 0

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

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

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

       2. atan2-lowering-atan2.f64 78.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)}\right), y.re\right)\right) \]

   11. Simplified 78.7%

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

   if -0.0189999999999999995 < y.re < 1.15000000000000004e-137 or 1.24999999999999991e-35 < y.re < 1.34999999999999999e-6

   1. Initial program 44.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 25.0%

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

   5. Simplified 25.0%

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

   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 23.7%

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

   8. Simplified 23.7%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-*r/ N/A

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

      6. clear-num N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-/r* N/A

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

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

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

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

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

      12. atan2-lowering-atan2.f64 44.3%

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

   10. Applied egg-rr 44.3%

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

   if 1.15000000000000004e-137 < y.re < 1.24999999999999991e-35

   1. Initial program 56.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 79.0%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   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 74.7%

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

   7. Simplified 74.7%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 28.8%

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

   10. Simplified 28.8%

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

   11. Taylor expanded in y.re around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 48.0%

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

   13. Simplified 48.0%

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

   if 1.34999999999999999e-6 < y.re

   1. Initial program 48.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 62.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   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 53.6%

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

   7. Simplified 53.6%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 62.9%

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

   10. Simplified 62.9%

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

       1. associate-*l* N/A

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

       2. *-commutative N/A

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

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

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

   12. Applied egg-rr 62.9%

       \[\leadsto \color{blue}{\left(\left(0.5 \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right) \cdot y.im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.019:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-137}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-35}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \left(\left(0.5 \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right) \cdot {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{y.re}{2}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.4% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(0 - x.re\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -8.5 \cdot 10^{-260}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (hypot x.im x.re))))
        (t_1 (* (* y.re (atan2 x.im x.re)) (pow (- 0.0 x.re) y.re))))
   (if (<= y.re -1.6e-37)
     t_1
     (if (<= y.re -8.5e-260)
       t_0
       (if (<= y.re 1.6e-135)
         (/ (/ (atan2 x.im x.re) (/ y.im y.re)) (/ 1.0 y.im))
         (if (<= y.re 8.5e-38) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	double t_1 = (y_46_re * atan2(x_46_im, x_46_re)) * pow((0.0 - x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.6e-37) {
		tmp = t_1;
	} else if (y_46_re <= -8.5e-260) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-135) {
		tmp = (atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	} else if (y_46_re <= 8.5e-38) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	double t_1 = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.pow((0.0 - x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.6e-37) {
		tmp = t_1;
	} else if (y_46_re <= -8.5e-260) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-135) {
		tmp = (Math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	} else if (y_46_re <= 8.5e-38) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	t_1 = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.pow((0.0 - x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -1.6e-37:
		tmp = t_1
	elif y_46_re <= -8.5e-260:
		tmp = t_0
	elif y_46_re <= 1.6e-135:
		tmp = (math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im)
	elif y_46_re <= 8.5e-38:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(hypot(x_46_im, x_46_re)))
	t_1 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (Float64(0.0 - x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.6e-37)
		tmp = t_1;
	elseif (y_46_re <= -8.5e-260)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-135)
		tmp = Float64(Float64(atan(x_46_im, x_46_re) / Float64(y_46_im / y_46_re)) / Float64(1.0 / y_46_im));
	elseif (y_46_re <= 8.5e-38)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * log(hypot(x_46_im, x_46_re));
	t_1 = (y_46_re * atan2(x_46_im, x_46_re)) * ((0.0 - x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.6e-37)
		tmp = t_1;
	elseif (y_46_re <= -8.5e-260)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-135)
		tmp = (atan2(x_46_im, x_46_re) / (y_46_im / y_46_re)) / (1.0 / y_46_im);
	elseif (y_46_re <= 8.5e-38)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[N[(0.0 - x$46$re), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.6e-37], t$95$1, If[LessEqual[y$46$re, -8.5e-260], t$95$0, If[LessEqual[y$46$re, 1.6e-135], N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 8.5e-38], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.5999999999999999e-37 or 8.50000000000000046e-38 < y.re

   1. Initial program 35.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 67.8%

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

   5. Simplified 67.8%

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

   6. Taylor expanded in x.re around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 53.3%

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

   8. Simplified 53.3%

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

   9. Taylor expanded in y.re around 0

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

       1. *-commutative N/A

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

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

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

       3. atan2-lowering-atan2.f64 54.2%

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

   11. Simplified 54.2%

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

   if -1.5999999999999999e-37 < y.re < -8.5000000000000003e-260 or 1.6e-135 < y.re < 8.50000000000000046e-38

   1. Initial program 50.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 80.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   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 69.4%

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

   7. Simplified 69.4%

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

   8. Taylor expanded in y.im around 0

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

      1. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

      7. unpow2 N/A

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

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

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

      9. unpow2 N/A

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

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

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

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

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 25.5%

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

   10. Simplified 25.5%

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

   11. Taylor expanded in y.re around 0

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

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

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

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

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 46.1%

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

   13. Simplified 46.1%

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

   if -8.5000000000000003e-260 < y.re < 1.6e-135

   1. Initial program 46.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 19.7%

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

   5. Simplified 19.7%

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

   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 19.7%

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

   8. Simplified 19.7%

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

      1. *-rgt-identity N/A

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

      2. *-inverses N/A

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

      3. associate-/l* N/A

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

      4. associate-*l/ N/A

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

      5. associate-*r/ N/A

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

      6. remove-double-div N/A

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

      7. un-div-inv N/A

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

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

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

      9. associate-*r/ N/A

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

      10. associate-*l/ N/A

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

      11. *-commutative N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \left(\frac{y.im}{y.re}\right)\right), \left(\frac{\color{blue}{1}}{y.im}\right)\right) \]

      15. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\frac{y.im}{y.re}\right)\right), \left(\frac{1}{y.im}\right)\right) \]

      16. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), \left(\frac{1}{y.im}\right)\right) \]

      17. /-lowering-/.f64 53.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), \mathsf{/.f64}\left(1, \color{blue}{y.im}\right)\right) \]

   10. Applied egg-rr 53.4%

       \[\leadsto \color{blue}{\frac{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}}{\frac{1}{y.im}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.6 \cdot 10^{-37}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(0 - x.re\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -8.5 \cdot 10^{-260}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-135}:\\ \;\;\;\;\frac{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}}{\frac{1}{y.im}}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-38}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(0 - x.re\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{y.re} \cdot \frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im}}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log (hypot x.im x.re)))))
   (if (<= y.im -3.8e+25)
     (/ 1.0 (/ (* (/ 1.0 y.re) (/ y.im (atan2 x.im x.re))) y.im))
     (if (<= y.im -3e-119)
       t_0
       (if (<= y.im 5.5e-211)
         (/ y.im (/ (/ y.im y.re) (atan2 x.im x.re)))
         (if (<= y.im 4.8e-7)
           t_0
           (* y.im (/ (atan2 x.im x.re) (/ y.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 * log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_im <= -3.8e+25) {
		tmp = 1.0 / (((1.0 / y_46_re) * (y_46_im / atan2(x_46_im, x_46_re))) / y_46_im);
	} else if (y_46_im <= -3e-119) {
		tmp = t_0;
	} else if (y_46_im <= 5.5e-211) {
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	} else if (y_46_im <= 4.8e-7) {
		tmp = t_0;
	} else {
		tmp = y_46_im * (atan2(x_46_im, x_46_re) / (y_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.log(Math.hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_im <= -3.8e+25) {
		tmp = 1.0 / (((1.0 / y_46_re) * (y_46_im / Math.atan2(x_46_im, x_46_re))) / y_46_im);
	} else if (y_46_im <= -3e-119) {
		tmp = t_0;
	} else if (y_46_im <= 5.5e-211) {
		tmp = y_46_im / ((y_46_im / y_46_re) / Math.atan2(x_46_im, x_46_re));
	} else if (y_46_im <= 4.8e-7) {
		tmp = t_0;
	} else {
		tmp = y_46_im * (Math.atan2(x_46_im, x_46_re) / (y_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.log(math.hypot(x_46_im, x_46_re))
	tmp = 0
	if y_46_im <= -3.8e+25:
		tmp = 1.0 / (((1.0 / y_46_re) * (y_46_im / math.atan2(x_46_im, x_46_re))) / y_46_im)
	elif y_46_im <= -3e-119:
		tmp = t_0
	elif y_46_im <= 5.5e-211:
		tmp = y_46_im / ((y_46_im / y_46_re) / math.atan2(x_46_im, x_46_re))
	elif y_46_im <= 4.8e-7:
		tmp = t_0
	else:
		tmp = y_46_im * (math.atan2(x_46_im, x_46_re) / (y_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 * log(hypot(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -3.8e+25)
		tmp = Float64(1.0 / Float64(Float64(Float64(1.0 / y_46_re) * Float64(y_46_im / atan(x_46_im, x_46_re))) / y_46_im));
	elseif (y_46_im <= -3e-119)
		tmp = t_0;
	elseif (y_46_im <= 5.5e-211)
		tmp = Float64(y_46_im / Float64(Float64(y_46_im / y_46_re) / atan(x_46_im, x_46_re)));
	elseif (y_46_im <= 4.8e-7)
		tmp = t_0;
	else
		tmp = Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / Float64(y_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 * log(hypot(x_46_im, x_46_re));
	tmp = 0.0;
	if (y_46_im <= -3.8e+25)
		tmp = 1.0 / (((1.0 / y_46_re) * (y_46_im / atan2(x_46_im, x_46_re))) / y_46_im);
	elseif (y_46_im <= -3e-119)
		tmp = t_0;
	elseif (y_46_im <= 5.5e-211)
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	elseif (y_46_im <= 4.8e-7)
		tmp = t_0;
	else
		tmp = y_46_im * (atan2(x_46_im, x_46_re) / (y_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[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+25], N[(1.0 / N[(N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -3e-119], t$95$0, If[LessEqual[y$46$im, 5.5e-211], N[(y$46$im / N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.8e-7], t$95$0, N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.8e25

   1. Initial program 32.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 27.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 27.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 3.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 3.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot \color{blue}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto \left(y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right) \cdot y.im \]

      6. clear-num N/A

         \[\leadsto \left(y.re \cdot \frac{1}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot y.im \]

      7. un-div-inv N/A

         \[\leadsto \frac{y.re}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot y.im \]

      8. associate-*l/ N/A

         \[\leadsto \frac{y.re \cdot y.im}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

      9. clear-num N/A

         \[\leadsto \frac{1}{\color{blue}{\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.re \cdot y.im}}} \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.re \cdot y.im}\right)}\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\left(y.re \cdot y.im\right)}\right)\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right), \left(\color{blue}{y.re} \cdot y.im\right)\right)\right) \]

      13. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \left(y.re \cdot y.im\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \left(y.im \cdot \color{blue}{y.re}\right)\right)\right) \]

      15. *-lowering-*.f64 9.4%

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{y.re}\right)\right)\right) \]

   10. Applied egg-rr 9.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im \cdot y.re}}} \]
   11. Step-by-step derivation

       1. associate-/r* N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im}}{\color{blue}{y.re}}\right)\right) \]

       2. div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im} \cdot \color{blue}{\frac{1}{y.re}}\right)\right) \]

       3. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)}{\mathsf{neg}\left(y.im\right)} \cdot \frac{\color{blue}{1}}{y.re}\right)\right) \]

       4. associate-*l/ N/A

          \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \frac{1}{y.re}}{\color{blue}{\mathsf{neg}\left(y.im\right)}}\right)\right) \]

       5. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \frac{1}{y.re}\right), \color{blue}{\left(\mathsf{neg}\left(y.im\right)\right)}\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(\color{blue}{y.im}\right)\right)\right)\right) \]

       7. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{y.im}{\mathsf{neg}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

       8. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \left(\mathsf{neg}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

       9. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

       10. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

       11. atan2-lowering-atan2.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

       12. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(1, y.re\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

       13. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(1, y.re\right)\right), \left(0 - \color{blue}{y.im}\right)\right)\right) \]

       14. --lowering--.f64 32.1%

           \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(1, y.re\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{y.im}\right)\right)\right) \]

   12. Applied egg-rr 32.1%

       \[\leadsto \frac{1}{\color{blue}{\frac{\frac{y.im}{0 - \tan^{-1}_* \frac{x.im}{x.re}} \cdot \frac{1}{y.re}}{0 - y.im}}} \]

   if -3.8e25 < y.im < -3.0000000000000002e-119 or 5.49999999999999973e-211 < y.im < 4.79999999999999957e-7

   1. Initial program 45.4%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 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 88.8%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   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 86.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 86.7%

      \[\leadsto \frac{\color{blue}{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right), y.re\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right), y.re\right)\right) \]

      17. *-lowering-*.f64 51.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), y.re\right)\right) \]

   10. Simplified 51.2%

       \[\leadsto \color{blue}{\left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}} \]

   11. Taylor expanded in y.re around 0

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 47.0%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   13. Simplified 47.0%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \]

   if -3.0000000000000002e-119 < y.im < 5.49999999999999973e-211

   1. Initial program 42.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 83.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 83.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   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 45.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 45.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. *-commutative N/A

         \[\leadsto \frac{y.im \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto y.im \cdot \color{blue}{\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}} \]

      6. clear-num N/A

         \[\leadsto y.im \cdot \frac{1}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. un-div-inv N/A

         \[\leadsto \frac{y.im}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \color{blue}{\left(\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \left(\frac{\frac{y.im}{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\left(\frac{y.im}{y.re}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

      12. atan2-lowering-atan2.f64 52.0%

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   10. Applied egg-rr 52.0%

       \[\leadsto \color{blue}{\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   if 4.79999999999999957e-7 < y.im

   1. Initial program 48.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in 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 24.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 24.2%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 8.9%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 8.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot \color{blue}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto \left(y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right) \cdot y.im \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right), \color{blue}{y.im}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right), y.im\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y.re}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{y.im}\right), y.im\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{1}{\frac{y.im}{y.re}}\right), y.im\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}\right), y.im\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \left(\frac{y.im}{y.re}\right)\right), y.im\right) \]

      13. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\frac{y.im}{y.re}\right)\right), y.im\right) \]

      14. /-lowering-/.f64 27.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), y.im\right) \]

   10. Applied egg-rr 27.3%

       \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}} \cdot y.im} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+25}:\\ \;\;\;\;\frac{1}{\frac{\frac{1}{y.re} \cdot \frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im}}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-119}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-211}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -0.22:\\ \;\;\;\;t\_0 \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-36}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(0 - x.re\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -0.22)
     (* t_0 (pow (+ x.re (/ (* (* x.im x.im) 0.5) x.re)) y.re))
     (if (<= y.re 3.4e-137)
       (/ y.im (/ (/ y.im y.re) (atan2 x.im x.re)))
       (if (<= y.re 1.4e-36)
         (* y.im (log (hypot x.im x.re)))
         (* t_0 (pow (- 0.0 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_re <= -0.22) {
		tmp = t_0 * pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 3.4e-137) {
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 1.4e-36) {
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * pow((0.0 - 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_re <= -0.22) {
		tmp = t_0 * Math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re);
	} else if (y_46_re <= 3.4e-137) {
		tmp = y_46_im / ((y_46_im / y_46_re) / Math.atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 1.4e-36) {
		tmp = y_46_im * Math.log(Math.hypot(x_46_im, x_46_re));
	} else {
		tmp = t_0 * Math.pow((0.0 - 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_re <= -0.22:
		tmp = t_0 * math.pow((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)), y_46_re)
	elif y_46_re <= 3.4e-137:
		tmp = y_46_im / ((y_46_im / y_46_re) / math.atan2(x_46_im, x_46_re))
	elif y_46_re <= 1.4e-36:
		tmp = y_46_im * math.log(math.hypot(x_46_im, x_46_re))
	else:
		tmp = t_0 * math.pow((0.0 - 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_re <= -0.22)
		tmp = Float64(t_0 * (Float64(x_46_re + Float64(Float64(Float64(x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re));
	elseif (y_46_re <= 3.4e-137)
		tmp = Float64(y_46_im / Float64(Float64(y_46_im / y_46_re) / atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.4e-36)
		tmp = Float64(y_46_im * log(hypot(x_46_im, x_46_re)));
	else
		tmp = Float64(t_0 * (Float64(0.0 - 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);
	tmp = 0.0;
	if (y_46_re <= -0.22)
		tmp = t_0 * ((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re)) ^ y_46_re);
	elseif (y_46_re <= 3.4e-137)
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	elseif (y_46_re <= 1.4e-36)
		tmp = y_46_im * log(hypot(x_46_im, x_46_re));
	else
		tmp = t_0 * ((0.0 - 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]}, If[LessEqual[y$46$re, -0.22], N[(t$95$0 * N[Power[N[(x$46$re + N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.4e-137], N[(y$46$im / N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.4e-36], N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[(0.0 - x$46$re), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -0.220000000000000001

   1. Initial program 27.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 74.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 74.0%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in x.im around 0

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

      2. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \left(\frac{\frac{1}{2} \cdot {x.im}^{2}}{x.re}\right)\right), y.re\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot {x.im}^{2}\right), x.re\right)\right), y.re\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left({x.im}^{2}\right)\right), x.re\right)\right), y.re\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \left(x.im \cdot x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

      6. *-lowering-*.f64 71.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right), y.re\right)\right) \]

   8. Simplified 71.0%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re}\right)}}^{y.re} \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right), y.re\right)\right) \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{+.f64}\left(x.re, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)\right)}, y.re\right)\right) \]

       2. atan2-lowering-atan2.f64 78.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{pow.f64}\left(\mathsf{+.f64}\left(x.re, \color{blue}{\mathsf{/.f64}\left(\mathsf{*.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(x.im, x.im\right)\right), x.re\right)}\right), y.re\right)\right) \]

   11. Simplified 78.7%

       \[\leadsto \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\left(x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re}\right)}^{y.re} \]

   if -0.220000000000000001 < y.re < 3.40000000000000014e-137

   1. Initial program 46.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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 22.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 22.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   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 21.4%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 21.4%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. *-commutative N/A

         \[\leadsto \frac{y.im \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto y.im \cdot \color{blue}{\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}} \]

      6. clear-num N/A

         \[\leadsto y.im \cdot \frac{1}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. un-div-inv N/A

         \[\leadsto \frac{y.im}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \color{blue}{\left(\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \left(\frac{\frac{y.im}{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\left(\frac{y.im}{y.re}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

      12. atan2-lowering-atan2.f64 43.1%

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   10. Applied egg-rr 43.1%

       \[\leadsto \color{blue}{\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   if 3.40000000000000014e-137 < y.re < 1.4000000000000001e-36

   1. Initial program 56.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 79.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   4. Add Preprocessing

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   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 74.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right)\right), \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]

   7. Simplified 74.7%

      \[\leadsto \frac{\color{blue}{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right), y.re\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right), y.re\right)\right) \]

      17. *-lowering-*.f64 28.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), y.re\right)\right) \]

   10. Simplified 28.8%

       \[\leadsto \color{blue}{\left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}} \]

   11. Taylor expanded in y.re around 0

       \[\leadsto \color{blue}{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right) \]

       2. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right) \]

       3. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + {x.re}^{2}}\right)\right)\right) \]

       4. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right)\right) \]

       5. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right) \]

       6. hypot-lowering-hypot.f64 48.0%

          \[\leadsto \mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right)\right)\right) \]

   13. Simplified 48.0%

       \[\leadsto \color{blue}{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \]

   if 1.4000000000000001e-36 < y.re

   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. 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 65.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 65.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in x.re around -inf

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot x.re\right)}, y.re\right)\right) \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\mathsf{neg}\left(x.re\right)\right), y.re\right)\right) \]

      2. neg-lowering-neg.f64 46.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{neg.f64}\left(x.re\right), y.re\right)\right) \]

   8. Simplified 46.4%

      \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(-x.re\right)}}^{y.re} \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \mathsf{pow.f64}\left(\mathsf{neg.f64}\left(x.re\right), y.re\right)\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{neg.f64}\left(x.re\right)}, y.re\right)\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re\right), \mathsf{pow.f64}\left(\color{blue}{\mathsf{neg.f64}\left(x.re\right)}, y.re\right)\right) \]

       3. atan2-lowering-atan2.f64 42.5%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.re\right), \mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\color{blue}{x.re}\right), y.re\right)\right) \]

   11. Simplified 42.5%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \cdot {\left(-x.re\right)}^{y.re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.22:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-137}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-36}:\\ \;\;\;\;y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(0 - x.re\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 405000000:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\left(y.im \cdot \log x.re\right) \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 405000000.0)
   (/ y.im (/ (/ y.im y.re) (atan2 x.im x.re)))
   (* (* y.im (log x.re)) (pow x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 405000000.0) {
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * log(x_46_re)) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 405000000.0d0) then
        tmp = y_46im / ((y_46im / y_46re) / atan2(x_46im, x_46re))
    else
        tmp = (y_46im * log(x_46re)) * (x_46re ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 405000000.0) {
		tmp = y_46_im / ((y_46_im / y_46_re) / Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = (y_46_im * Math.log(x_46_re)) * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 405000000.0:
		tmp = y_46_im / ((y_46_im / y_46_re) / math.atan2(x_46_im, x_46_re))
	else:
		tmp = (y_46_im * math.log(x_46_re)) * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 405000000.0)
		tmp = Float64(y_46_im / Float64(Float64(y_46_im / y_46_re) / atan(x_46_im, x_46_re)));
	else
		tmp = Float64(Float64(y_46_im * log(x_46_re)) * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 405000000.0)
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	else
		tmp = (y_46_im * log(x_46_re)) * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 405000000.0], N[(y$46$im / N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 4.05e8

   1. Initial program 44.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 41.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 41.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 16.3%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 16.3%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. *-commutative N/A

         \[\leadsto \frac{y.im \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto y.im \cdot \color{blue}{\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}} \]

      6. clear-num N/A

         \[\leadsto y.im \cdot \frac{1}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. un-div-inv N/A

         \[\leadsto \frac{y.im}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \color{blue}{\left(\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \left(\frac{\frac{y.im}{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\left(\frac{y.im}{y.re}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

      12. atan2-lowering-atan2.f64 29.8%

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   10. Applied egg-rr 29.8%

       \[\leadsto \color{blue}{\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   if 4.05e8 < x.re

   1. Initial program 35.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Step-by-step derivation

      1. exp-diff 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 77.2%

      \[\leadsto \color{blue}{\frac{\sin \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}} \]
   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 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%

      \[\leadsto \frac{\color{blue}{\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}}{\frac{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}} \]

   8. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)} \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \color{blue}{\left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right), \left({\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re}\right)\right) \]

      4. log-lowering-log.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) \]

      11. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right), \color{blue}{y.re}\right)\right) \]

      12. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right), y.re\right)\right) \]

      13. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      14. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left(x.im \cdot x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      15. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right), y.re\right)\right) \]

      16. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right), y.re\right)\right) \]

      17. *-lowering-*.f64 39.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right), y.re\right)\right) \]

   10. Simplified 39.2%

       \[\leadsto \color{blue}{\left(y.im \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right) \cdot {\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}} \]

   11. Taylor expanded in x.im around 0

       \[\leadsto \color{blue}{y.im \cdot \left(\log x.re \cdot {x.re}^{y.re}\right)} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \left(y.im \cdot \log x.re\right) \cdot \color{blue}{{x.re}^{y.re}} \]

       2. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(y.im \cdot \log x.re\right), \color{blue}{\left({x.re}^{y.re}\right)}\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \log x.re\right), \left({\color{blue}{x.re}}^{y.re}\right)\right) \]

       4. log-lowering-log.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right), \left({x.re}^{y.re}\right)\right) \]

       5. pow-lowering-pow.f64 60.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(y.im, \mathsf{log.f64}\left(x.re\right)\right), \mathsf{pow.f64}\left(x.re, \color{blue}{y.re}\right)\right) \]

   13. Simplified 60.8%

       \[\leadsto \color{blue}{\left(y.im \cdot \log x.re\right) \cdot {x.re}^{y.re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 21.1% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{-35}:\\ \;\;\;\;y.re \cdot \frac{y.im}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.2e-35)
   (* y.re (/ y.im (/ y.im (atan2 x.im x.re))))
   (/ y.im (/ (/ y.im y.re) (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.2e-35) {
		tmp = y_46_re * (y_46_im / (y_46_im / atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.2d-35)) then
        tmp = y_46re * (y_46im / (y_46im / atan2(x_46im, x_46re)))
    else
        tmp = y_46im / ((y_46im / y_46re) / atan2(x_46im, x_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.2e-35) {
		tmp = y_46_re * (y_46_im / (y_46_im / Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_im / ((y_46_im / y_46_re) / Math.atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.2e-35:
		tmp = y_46_re * (y_46_im / (y_46_im / math.atan2(x_46_im, x_46_re)))
	else:
		tmp = y_46_im / ((y_46_im / y_46_re) / math.atan2(x_46_im, x_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.2e-35)
		tmp = Float64(y_46_re * Float64(y_46_im / Float64(y_46_im / atan(x_46_im, x_46_re))));
	else
		tmp = Float64(y_46_im / Float64(Float64(y_46_im / y_46_re) / atan(x_46_im, x_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.2e-35)
		tmp = y_46_re * (y_46_im / (y_46_im / atan2(x_46_im, x_46_re)));
	else
		tmp = y_46_im / ((y_46_im / y_46_re) / atan2(x_46_im, x_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.2e-35], N[(y$46$re * N[(y$46$im / N[(y$46$im / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$im / N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.19999999999999994e-35

   1. Initial program 29.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 69.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 69.7%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 17.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 17.7%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(1 \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\left(y.im \cdot \frac{1}{y.im}\right) \cdot \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(y.im \cdot \color{blue}{\left(\frac{1}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(y.im \cdot \frac{1}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\frac{y.im}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \color{blue}{\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(y.im, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

      8. atan2-lowering-atan2.f64 22.6%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   10. Applied egg-rr 22.6%

       \[\leadsto y.re \cdot \color{blue}{\frac{y.im}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   if -2.19999999999999994e-35 < y.re

   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 31.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 31.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 15.5%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 15.5%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. *-commutative N/A

         \[\leadsto \frac{y.im \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto y.im \cdot \color{blue}{\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}} \]

      6. clear-num N/A

         \[\leadsto y.im \cdot \frac{1}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. un-div-inv N/A

         \[\leadsto \frac{y.im}{\color{blue}{\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \color{blue}{\left(\frac{y.im}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

      9. associate-/r* N/A

         \[\leadsto \mathsf{/.f64}\left(y.im, \left(\frac{\frac{y.im}{y.re}}{\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\left(\frac{y.im}{y.re}\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

      12. atan2-lowering-atan2.f64 32.3%

          \[\leadsto \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   10. Applied egg-rr 32.3%

       \[\leadsto \color{blue}{\frac{y.im}{\frac{\frac{y.im}{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 21.2% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-37}:\\ \;\;\;\;y.re \cdot \frac{y.im}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8e-37)
   (* y.re (/ y.im (/ y.im (atan2 x.im x.re))))
   (* y.im (/ (atan2 x.im x.re) (/ y.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8e-37) {
		tmp = y_46_re * (y_46_im / (y_46_im / atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_im * (atan2(x_46_im, x_46_re) / (y_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) :: tmp
    if (y_46re <= (-8d-37)) then
        tmp = y_46re * (y_46im / (y_46im / atan2(x_46im, x_46re)))
    else
        tmp = y_46im * (atan2(x_46im, x_46re) / (y_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 tmp;
	if (y_46_re <= -8e-37) {
		tmp = y_46_re * (y_46_im / (y_46_im / Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_im * (Math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -8e-37:
		tmp = y_46_re * (y_46_im / (y_46_im / math.atan2(x_46_im, x_46_re)))
	else:
		tmp = y_46_im * (math.atan2(x_46_im, x_46_re) / (y_46_im / y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8e-37)
		tmp = Float64(y_46_re * Float64(y_46_im / Float64(y_46_im / atan(x_46_im, x_46_re))));
	else
		tmp = Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / Float64(y_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)
	tmp = 0.0;
	if (y_46_re <= -8e-37)
		tmp = y_46_re * (y_46_im / (y_46_im / atan2(x_46_im, x_46_re)));
	else
		tmp = y_46_im * (atan2(x_46_im, x_46_re) / (y_46_im / y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8e-37], N[(y$46$re * N[(y$46$im / N[(y$46$im / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.00000000000000053e-37

   1. Initial program 29.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 69.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 69.7%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   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 17.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 17.7%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(1 \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      2. rgt-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\left(y.im \cdot \frac{1}{y.im}\right) \cdot \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(y.im \cdot \color{blue}{\left(\frac{1}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(y.im \cdot \frac{1}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}}\right)\right) \]

      5. un-div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\frac{y.im}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \color{blue}{\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)}\right)\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(y.im, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

      8. atan2-lowering-atan2.f64 22.6%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right) \]

   10. Applied egg-rr 22.6%

       \[\leadsto y.re \cdot \color{blue}{\frac{y.im}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   if -8.00000000000000053e-37 < y.re

   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 31.6%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 31.6%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 15.5%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 15.5%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. associate-*l/ N/A

         \[\leadsto \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot \color{blue}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto \left(y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right) \cdot y.im \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right), \color{blue}{y.im}\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right), y.im\right) \]

      8. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y.re}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), y.im\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{y.im}\right), y.im\right) \]

      10. clear-num N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{1}{\frac{y.im}{y.re}}\right), y.im\right) \]

      11. un-div-inv N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}\right), y.im\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, \left(\frac{y.im}{y.re}\right)\right), y.im\right) \]

      13. atan2-lowering-atan2.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \left(\frac{y.im}{y.re}\right)\right), y.im\right) \]

      14. /-lowering-/.f64 32.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), \mathsf{/.f64}\left(y.im, y.re\right)\right), y.im\right) \]

   10. Applied egg-rr 32.3%

       \[\leadsto \color{blue}{\frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}} \cdot y.im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-37}:\\ \;\;\;\;y.re \cdot \frac{y.im}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{\frac{y.im}{y.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 19.9% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 1450:\\ \;\;\;\;\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 1450.0)
   (* (* (atan2 x.im x.re) y.im) (/ y.re y.im))
   (* y.re (* y.im (/ (atan2 x.im x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1450.0) {
		tmp = (atan2(x_46_im, x_46_re) * y_46_im) * (y_46_re / y_46_im);
	} else {
		tmp = y_46_re * (y_46_im * (atan2(x_46_im, x_46_re) / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 1450.0d0) then
        tmp = (atan2(x_46im, x_46re) * y_46im) * (y_46re / y_46im)
    else
        tmp = y_46re * (y_46im * (atan2(x_46im, x_46re) / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1450.0) {
		tmp = (Math.atan2(x_46_im, x_46_re) * y_46_im) * (y_46_re / y_46_im);
	} else {
		tmp = y_46_re * (y_46_im * (Math.atan2(x_46_im, x_46_re) / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 1450.0:
		tmp = (math.atan2(x_46_im, x_46_re) * y_46_im) * (y_46_re / y_46_im)
	else:
		tmp = y_46_re * (y_46_im * (math.atan2(x_46_im, x_46_re) / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 1450.0)
		tmp = Float64(Float64(atan(x_46_im, x_46_re) * y_46_im) * Float64(y_46_re / y_46_im));
	else
		tmp = Float64(y_46_re * Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 1450.0)
		tmp = (atan2(x_46_im, x_46_re) * y_46_im) * (y_46_re / y_46_im);
	else
		tmp = y_46_re * (y_46_im * (atan2(x_46_im, x_46_re) / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 1450.0], N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1450

   1. Initial program 44.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. 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 41.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 41.5%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   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 16.4%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 16.4%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

      2. *-inverses N/A

         \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

      4. *-commutative N/A

         \[\leadsto \frac{y.im \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im} \]

      5. associate-*r/ N/A

         \[\leadsto y.im \cdot \color{blue}{\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}} \]

      6. *-commutative N/A

         \[\leadsto y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}{y.im} \]

      7. associate-/l* N/A

         \[\leadsto y.im \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\frac{y.re}{y.im}}\right) \]

      8. associate-*r* N/A

         \[\leadsto \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{\frac{y.re}{y.im}} \]

      9. *-commutative N/A

         \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) \cdot \frac{\color{blue}{y.re}}{y.im} \]

      10. *-commutative N/A

          \[\leadsto \frac{y.re}{y.im} \cdot \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\frac{y.re}{y.im}\right), \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)}\right) \]

      12. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.re, y.im\right), \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.im\right)\right) \]

      13. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.re, y.im\right), \left(y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.re, y.im\right), \mathsf{*.f64}\left(y.im, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

      15. atan2-lowering-atan2.f64 28.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.re, y.im\right), \mathsf{*.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

   10. Applied egg-rr 28.9%

       \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   if 1450 < x.re

   1. Initial program 36.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 48.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

   5. Simplified 48.1%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

      2. atan2-lowering-atan2.f64 15.7%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

   8. Simplified 15.7%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   9. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{1}\right)\right) \]

      2. lft-mult-inverse N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\frac{1}{y.im} \cdot \color{blue}{y.im}\right)\right)\right) \]

      3. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{1}{y.im}\right) \cdot \color{blue}{y.im}\right)\right) \]

      4. div-inv N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot y.im\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right), \color{blue}{y.im}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im\right), y.im\right)\right) \]

      7. atan2-lowering-atan2.f64 23.0%

         \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right), y.im\right)\right) \]

   10. Applied egg-rr 23.0%

       \[\leadsto y.re \cdot \color{blue}{\left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot y.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 1450:\\ \;\;\;\;\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right) \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 21.6% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\frac{1}{y.re} \cdot \frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im}} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ 1.0 (/ (* (/ 1.0 y.re) (/ y.im (atan2 x.im x.re))) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 / (((1.0 / y_46_re) * (y_46_im / atan2(x_46_im, x_46_re))) / y_46_im);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0 / (((1.0d0 / y_46re) * (y_46im / atan2(x_46im, x_46re))) / y_46im)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 / (((1.0 / y_46_re) * (y_46_im / Math.atan2(x_46_im, x_46_re))) / y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 / (((1.0 / y_46_re) * (y_46_im / math.atan2(x_46_im, x_46_re))) / y_46_im)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 / Float64(Float64(Float64(1.0 / y_46_re) * Float64(y_46_im / atan(x_46_im, x_46_re))) / y_46_im))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 / (((1.0 / y_46_re) * (y_46_im / atan2(x_46_im, x_46_re))) / y_46_im);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 / N[(N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. 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 43.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

5. Simplified 43.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

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 16.2%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 16.2%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{1} \]

   2. *-inverses N/A

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{y.im}{\color{blue}{y.im}} \]

   3. associate-/l* N/A

      \[\leadsto \frac{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im}{\color{blue}{y.im}} \]

   4. associate-*l/ N/A

      \[\leadsto \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot \color{blue}{y.im} \]

   5. associate-*r/ N/A

      \[\leadsto \left(y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right) \cdot y.im \]

   6. clear-num N/A

      \[\leadsto \left(y.re \cdot \frac{1}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot y.im \]

   7. un-div-inv N/A

      \[\leadsto \frac{y.re}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot y.im \]

   8. associate-*l/ N/A

      \[\leadsto \frac{y.re \cdot y.im}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   9. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.re \cdot y.im}}} \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.re \cdot y.im}\right)}\right) \]

   11. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right), \color{blue}{\left(y.re \cdot y.im\right)}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \tan^{-1}_* \frac{x.im}{x.re}\right), \left(\color{blue}{y.re} \cdot y.im\right)\right)\right) \]

   13. atan2-lowering-atan2.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \left(y.re \cdot y.im\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \left(y.im \cdot \color{blue}{y.re}\right)\right)\right) \]

   15. *-lowering-*.f64 13.8%

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), \mathsf{*.f64}\left(y.im, \color{blue}{y.re}\right)\right)\right) \]

10. Applied egg-rr 13.8%

    \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im \cdot y.re}}} \]
11. Step-by-step derivation

    1. associate-/r* N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im}}{\color{blue}{y.re}}\right)\right) \]

    2. div-inv N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im} \cdot \color{blue}{\frac{1}{y.re}}\right)\right) \]

    3. frac-2neg N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)}{\mathsf{neg}\left(y.im\right)} \cdot \frac{\color{blue}{1}}{y.re}\right)\right) \]

    4. associate-*l/ N/A

       \[\leadsto \mathsf{/.f64}\left(1, \left(\frac{\left(\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \frac{1}{y.re}}{\color{blue}{\mathsf{neg}\left(y.im\right)}}\right)\right) \]

    5. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \cdot \frac{1}{y.re}\right), \color{blue}{\left(\mathsf{neg}\left(y.im\right)\right)}\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\mathsf{neg}\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(\color{blue}{y.im}\right)\right)\right)\right) \]

    7. distribute-neg-frac2 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{y.im}{\mathsf{neg}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)}\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

    8. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \left(\mathsf{neg}\left(\tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

    9. neg-sub0 N/A

       \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

    10. --lowering--.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

    11. atan2-lowering-atan2.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left(\frac{1}{y.re}\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

    12. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(1, y.re\right)\right), \left(\mathsf{neg}\left(y.im\right)\right)\right)\right) \]

    13. neg-sub0 N/A

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(1, y.re\right)\right), \left(0 - \color{blue}{y.im}\right)\right)\right) \]

    14. --lowering--.f64 28.5%

        \[\leadsto \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, \mathsf{\_.f64}\left(0, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{/.f64}\left(1, y.re\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{y.im}\right)\right)\right) \]

12. Applied egg-rr 28.5%

    \[\leadsto \frac{1}{\color{blue}{\frac{\frac{y.im}{0 - \tan^{-1}_* \frac{x.im}{x.re}} \cdot \frac{1}{y.re}}{0 - y.im}}} \]

13. Final simplification 28.5%

    \[\leadsto \frac{1}{\frac{\frac{1}{y.re} \cdot \frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}{y.im}} \]
14. Add Preprocessing

Alternative 22: 20.7% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ y.im \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* y.im (/ (* y.re (atan2 x.im x.re)) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * ((y_46_re * atan2(x_46_im, x_46_re)) / y_46_im);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46im * ((y_46re * atan2(x_46im, x_46re)) / y_46im)
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_im * ((y_46_re * Math.atan2(x_46_im, x_46_re)) / y_46_im);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_im * ((y_46_re * math.atan2(x_46_im, x_46_re)) / y_46_im)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_im * Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) / y_46_im))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_im * ((y_46_re * atan2(x_46_im, x_46_re)) / y_46_im);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$im * N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. 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 43.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

5. Simplified 43.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

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 16.2%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 16.2%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Step-by-step derivation

   1. remove-double-div N/A

      \[\leadsto y.re \cdot \frac{1}{\color{blue}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   2. div-inv N/A

      \[\leadsto \frac{y.re}{\color{blue}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y.re, \color{blue}{\left(\frac{1}{\tan^{-1}_* \frac{x.im}{x.re}}\right)}\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(y.re, \mathsf{/.f64}\left(1, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   5. atan2-lowering-atan2.f64 16.2%

      \[\leadsto \mathsf{/.f64}\left(y.re, \mathsf{/.f64}\left(1, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right) \]

10. Applied egg-rr 16.2%

    \[\leadsto \color{blue}{\frac{y.re}{\frac{1}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]
11. Step-by-step derivation

    1. associate-/r/ N/A

       \[\leadsto \frac{y.re}{1} \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \]

    2. /-rgt-identity N/A

       \[\leadsto y.re \cdot \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re} \]

    3. *-lft-identity N/A

       \[\leadsto 1 \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

    4. *-inverses N/A

       \[\leadsto \frac{y.im}{y.im} \cdot \left(\color{blue}{y.re} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

    5. associate-*l/ N/A

       \[\leadsto \frac{y.im \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\color{blue}{y.im}} \]

    6. associate-/l* N/A

       \[\leadsto y.im \cdot \color{blue}{\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}} \]

    7. *-commutative N/A

       \[\leadsto y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}{y.im} \]

    8. associate-*l/ N/A

       \[\leadsto y.im \cdot \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot \color{blue}{y.re}\right) \]

    9. *-commutative N/A

       \[\leadsto \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot y.re\right) \cdot \color{blue}{y.im} \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot y.re\right), \color{blue}{y.im}\right) \]

    11. associate-*l/ N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}{y.im}\right), y.im\right) \]

    12. *-commutative N/A

        \[\leadsto \mathsf{*.f64}\left(\left(\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right), y.im\right) \]

    13. /-lowering-/.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), y.im\right), y.im\right) \]

    14. *-lowering-*.f64 N/A

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}\right), y.im\right), y.im\right) \]

    15. atan2-lowering-atan2.f64 27.0%

        \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right), y.im\right), y.im\right) \]

12. Applied egg-rr 27.0%

    \[\leadsto \color{blue}{\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot y.im} \]

13. Final simplification 27.0%

    \[\leadsto y.im \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} \]
14. Add Preprocessing

Alternative 23: 15.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ y.re \cdot \frac{y.im}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* y.re (/ y.im (/ y.im (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * (y_46_im / (y_46_im / 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 * (y_46im / (y_46im / 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 * (y_46_im / (y_46_im / 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 * (y_46_im / (y_46_im / 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 * Float64(y_46_im / Float64(y_46_im / 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 * (y_46_im / (y_46_im / 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[(y$46$im / N[(y$46$im / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. 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 43.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

5. Simplified 43.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

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 16.2%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 16.2%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Step-by-step derivation

   1. *-lft-identity N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(1 \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right) \]

   2. rgt-mult-inverse N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(\left(y.im \cdot \frac{1}{y.im}\right) \cdot \tan^{-1}_* \frac{\color{blue}{x.im}}{x.re}\right)\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(y.im \cdot \color{blue}{\left(\frac{1}{y.im} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\right) \]

   4. associate-/r/ N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(y.im \cdot \frac{1}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}}\right)\right) \]

   5. un-div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(\frac{y.im}{\color{blue}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \color{blue}{\left(\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(y.im, \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

   8. atan2-lowering-atan2.f64 18.7%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{/.f64}\left(y.im, \mathsf{/.f64}\left(y.im, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right)\right)\right) \]

10. Applied egg-rr 18.7%

    \[\leadsto y.re \cdot \color{blue}{\frac{y.im}{\frac{y.im}{\tan^{-1}_* \frac{x.im}{x.re}}}} \]
11. Add Preprocessing

Alternative 24: 14.9% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* y.re (* y.im (/ (atan2 x.im x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * (y_46_im * (atan2(x_46_im, x_46_re) / y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46re * (y_46im * (atan2(x_46im, x_46re) / y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * (y_46_im * (Math.atan2(x_46_im, x_46_re) / y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return y_46_re * (y_46_im * (math.atan2(x_46_im, x_46_re) / y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_re * Float64(y_46_im * Float64(atan(x_46_im, x_46_re) / y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_re * (y_46_im * (atan2(x_46_im, x_46_re) / y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(y$46$re * N[(y$46$im * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.0%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. 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 43.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

5. Simplified 43.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

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 16.2%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 16.2%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Step-by-step derivation

   1. *-rgt-identity N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{1}\right)\right) \]

   2. lft-mult-inverse N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(\frac{1}{y.im} \cdot \color{blue}{y.im}\right)\right)\right) \]

   3. associate-*l* N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{1}{y.im}\right) \cdot \color{blue}{y.im}\right)\right) \]

   4. div-inv N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot y.im\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right), \color{blue}{y.im}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im\right), y.im\right)\right) \]

   7. atan2-lowering-atan2.f64 18.4%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right), y.im\right)\right) \]

10. Applied egg-rr 18.4%

    \[\leadsto y.re \cdot \color{blue}{\left(\frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im} \cdot y.im\right)} \]

11. Final simplification 18.4%

    \[\leadsto y.re \cdot \left(y.im \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right) \]
12. Add Preprocessing

Alternative 25: 13.7% 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 42.0%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. 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 43.2%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sin.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.im, x.re\right), y.re\right)\right) \]

5. Simplified 43.2%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}} \]

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 16.2%

      \[\leadsto \mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, \color{blue}{x.re}\right)\right) \]

8. Simplified 16.2%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024159 
(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)))))