powComplex, real part

Percentage Accurate: 41.6% → 78.1%

Time: 18.6s

Alternatives: 16

Speedup: 5.7×

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

Specification

Math

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

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)))
    (cos (+ (* 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))) * cos(((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))) * cos(((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.cos(((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.cos(((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))) * cos(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))) * cos(((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[Cos[N[(N[(t$95$0 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 41.6% accurate, 1.0× speedup

Math

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

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)))
    (cos (+ (* 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))) * cos(((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))) * cos(((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.cos(((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.cos(((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))) * cos(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))) * cos(((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[Cos[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: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\cos t\_0 \cdot t\_2\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;t\_1 \cdot \cos \left(\log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 5600:\\ \;\;\;\;t\_1 \cdot \cos \left(t\_0 + \log \left({\left(e^{0.5}\right)}^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re))
        (t_1 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_2
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -9e+20)
     (* (cos t_0) t_2)
     (if (<= y.re 6.2e-167)
       (* t_1 (cos (* (log (sqrt (pow (hypot x.im x.re) 0.25))) y.im)))
       (if (<= y.re 5600.0)
         (*
          t_1
          (cos (+ t_0 (* (log (pow (exp 0.5) (log (hypot x.re x.im)))) y.im))))
         (* (cos (* (log (sqrt (hypot x.re x.im))) y.im)) t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_2 = exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -9e+20) {
		tmp = cos(t_0) * t_2;
	} else if (y_46_re <= 6.2e-167) {
		tmp = t_1 * cos((log(sqrt(pow(hypot(x_46_im, x_46_re), 0.25))) * y_46_im));
	} else if (y_46_re <= 5600.0) {
		tmp = t_1 * cos((t_0 + (log(pow(exp(0.5), log(hypot(x_46_re, x_46_im)))) * y_46_im)));
	} else {
		tmp = cos((log(sqrt(hypot(x_46_re, x_46_im))) * y_46_im)) * t_2;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_2 = Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -9e+20) {
		tmp = Math.cos(t_0) * t_2;
	} else if (y_46_re <= 6.2e-167) {
		tmp = t_1 * Math.cos((Math.log(Math.sqrt(Math.pow(Math.hypot(x_46_im, x_46_re), 0.25))) * y_46_im));
	} else if (y_46_re <= 5600.0) {
		tmp = t_1 * Math.cos((t_0 + (Math.log(Math.pow(Math.exp(0.5), Math.log(Math.hypot(x_46_re, x_46_im)))) * y_46_im)));
	} else {
		tmp = Math.cos((Math.log(Math.sqrt(Math.hypot(x_46_re, x_46_im))) * y_46_im)) * t_2;
	}
	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_re
	t_1 = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	t_2 = math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	tmp = 0
	if y_46_re <= -9e+20:
		tmp = math.cos(t_0) * t_2
	elif y_46_re <= 6.2e-167:
		tmp = t_1 * math.cos((math.log(math.sqrt(math.pow(math.hypot(x_46_im, x_46_re), 0.25))) * y_46_im))
	elif y_46_re <= 5600.0:
		tmp = t_1 * math.cos((t_0 + (math.log(math.pow(math.exp(0.5), math.log(math.hypot(x_46_re, x_46_im)))) * y_46_im)))
	else:
		tmp = math.cos((math.log(math.sqrt(math.hypot(x_46_re, x_46_im))) * y_46_im)) * t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_2 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -9e+20)
		tmp = Float64(cos(t_0) * t_2);
	elseif (y_46_re <= 6.2e-167)
		tmp = Float64(t_1 * cos(Float64(log(sqrt((hypot(x_46_im, x_46_re) ^ 0.25))) * y_46_im)));
	elseif (y_46_re <= 5600.0)
		tmp = Float64(t_1 * cos(Float64(t_0 + Float64(log((exp(0.5) ^ log(hypot(x_46_re, x_46_im)))) * y_46_im))));
	else
		tmp = Float64(cos(Float64(log(sqrt(hypot(x_46_re, x_46_im))) * y_46_im)) * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	t_1 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	t_2 = exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	tmp = 0.0;
	if (y_46_re <= -9e+20)
		tmp = cos(t_0) * t_2;
	elseif (y_46_re <= 6.2e-167)
		tmp = t_1 * cos((log(sqrt((hypot(x_46_im, x_46_re) ^ 0.25))) * y_46_im));
	elseif (y_46_re <= 5600.0)
		tmp = t_1 * cos((t_0 + (log((exp(0.5) ^ log(hypot(x_46_re, x_46_im)))) * y_46_im)));
	else
		tmp = cos((log(sqrt(hypot(x_46_re, x_46_im))) * y_46_im)) * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9e+20], N[(N[Cos[t$95$0], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-167], N[(t$95$1 * N[Cos[N[(N[Log[N[Sqrt[N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], 0.25], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5600.0], N[(t$95$1 * N[Cos[N[(t$95$0 + N[(N[Log[N[Power[N[Exp[0.5], $MachinePrecision], N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(N[Log[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9e20

   1. Initial program 42.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -9e20 < y.re < 6.2e-167

   1. Initial program 46.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 80.1

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 86.7%

      \[\leadsto \cos \left(y.im \cdot \log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right)\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 6.2e-167 < y.re < 5600

   1. Initial program 37.4%

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

   3. Taylor expanded in y.re around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 37.4

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

   5. Applied rewrites 37.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow1/2 N/A

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

      7. pow-to-exp N/A

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

      8. rem-log-exp N/A

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

      9. pow-to-exp N/A

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

      10. log-pow N/A

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

      11. exp-prod N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-log.f64 83.8

          \[\leadsto e^{\left(-y.re\right) \cdot \left(\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.re} - \log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)\right)} \cdot \cos \left(\log \left({\left(e^{0.5}\right)}^{\color{blue}{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   7. Applied rewrites 83.8%

      \[\leadsto e^{\left(-y.re\right) \cdot \left(\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.re} - \log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)\right)} \cdot \cos \left(\log \color{blue}{\left({\left(e^{0.5}\right)}^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Taylor expanded in y.im around inf

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

   10. Applied rewrites 83.8%

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

   if 5600 < y.re

   1. Initial program 29.8%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 82.5

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

   5. Applied rewrites 82.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 5600:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left({\left(e^{0.5}\right)}^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq 8.5 \cdot 10^{+93}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left({\left(e^{0.5}\right)}^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.im\right) \cdot e^{\left(t\_0 - \frac{t\_1}{y.re}\right) \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(t\_0 \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (hypot x.re x.im)))) (t_1 (* (atan2 x.im x.re) y.im)))
   (if (<= y.re 8.5e+93)
     (*
      (cos
       (+
        (* (atan2 x.im x.re) y.re)
        (* (log (pow (exp 0.5) (log (hypot x.re x.im)))) y.im)))
      (exp (* (- t_0 (/ t_1 y.re)) y.re)))
     (*
      (cos (* t_0 y.im))
      (exp (- (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(hypot(x_46_re, x_46_im)));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= 8.5e+93) {
		tmp = cos(((atan2(x_46_im, x_46_re) * y_46_re) + (log(pow(exp(0.5), log(hypot(x_46_re, x_46_im)))) * y_46_im))) * exp(((t_0 - (t_1 / y_46_re)) * y_46_re));
	} else {
		tmp = cos((t_0 * y_46_im)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - t_1));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(Math.hypot(x_46_re, x_46_im)));
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= 8.5e+93) {
		tmp = Math.cos(((Math.atan2(x_46_im, x_46_re) * y_46_re) + (Math.log(Math.pow(Math.exp(0.5), Math.log(Math.hypot(x_46_re, x_46_im)))) * y_46_im))) * Math.exp(((t_0 - (t_1 / y_46_re)) * y_46_re));
	} else {
		tmp = Math.cos((t_0 * y_46_im)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - t_1));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(math.hypot(x_46_re, x_46_im)))
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if y_46_re <= 8.5e+93:
		tmp = math.cos(((math.atan2(x_46_im, x_46_re) * y_46_re) + (math.log(math.pow(math.exp(0.5), math.log(math.hypot(x_46_re, x_46_im)))) * y_46_im))) * math.exp(((t_0 - (t_1 / y_46_re)) * y_46_re))
	else:
		tmp = math.cos((t_0 * y_46_im)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - t_1))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(hypot(x_46_re, x_46_im)))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (y_46_re <= 8.5e+93)
		tmp = Float64(cos(Float64(Float64(atan(x_46_im, x_46_re) * y_46_re) + Float64(log((exp(0.5) ^ log(hypot(x_46_re, x_46_im)))) * y_46_im))) * exp(Float64(Float64(t_0 - Float64(t_1 / y_46_re)) * y_46_re)));
	else
		tmp = Float64(cos(Float64(t_0 * y_46_im)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(hypot(x_46_re, x_46_im)));
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (y_46_re <= 8.5e+93)
		tmp = cos(((atan2(x_46_im, x_46_re) * y_46_re) + (log((exp(0.5) ^ log(hypot(x_46_re, x_46_im)))) * y_46_im))) * exp(((t_0 - (t_1 / y_46_re)) * y_46_re));
	else
		tmp = cos((t_0 * y_46_im)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $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, 8.5e+93], N[(N[Cos[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] + N[(N[Log[N[Power[N[Exp[0.5], $MachinePrecision], N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(t$95$0 - N[(t$95$1 / y$46$re), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(t$95$0 * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < 8.5000000000000005e93

   1. Initial program 42.9%

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

   3. Taylor expanded in y.re around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. unsub-neg N/A

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

      8. lower--.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. +-commutative N/A

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

      15. unpow2 N/A

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

      16. unpow2 N/A

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

      17. lower-hypot.f64 42.6

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

   5. Applied rewrites 42.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-hypot.f64 N/A

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

      6. pow1/2 N/A

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

      7. pow-to-exp N/A

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

      8. rem-log-exp N/A

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

      9. pow-to-exp N/A

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

      10. log-pow N/A

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

      11. exp-prod N/A

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

      12. lower-pow.f64 N/A

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

      13. lower-exp.f64 N/A

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

      14. lower-log.f64 83.9

          \[\leadsto e^{\left(-y.re\right) \cdot \left(\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.re} - \log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)\right)} \cdot \cos \left(\log \left({\left(e^{0.5}\right)}^{\color{blue}{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   7. Applied rewrites 83.9%

      \[\leadsto e^{\left(-y.re\right) \cdot \left(\frac{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.re} - \log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)\right)} \cdot \cos \left(\log \color{blue}{\left({\left(e^{0.5}\right)}^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   if 8.5000000000000005e93 < y.re

   1. Initial program 31.1%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 82.2

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

   5. Applied rewrites 82.2%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 8.5 \cdot 10^{+93}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + \log \left({\left(e^{0.5}\right)}^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right) \cdot y.im\right) \cdot e^{\left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) - \frac{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}{y.re}\right) \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right)\\ t_1 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot t\_2\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;t\_1 \cdot \cos \left(\log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 5600:\\ \;\;\;\;t\_0 \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* (log (sqrt (hypot x.re x.im))) y.im)))
        (t_1 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_2
         (exp
          (-
           (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
           (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -9e+20)
     (* (cos (* (atan2 x.im x.re) y.re)) t_2)
     (if (<= y.re 6.2e-167)
       (* t_1 (cos (* (log (sqrt (pow (hypot x.im x.re) 0.25))) y.im)))
       (if (<= y.re 5600.0) (* t_0 t_1) (* t_0 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cos((log(sqrt(hypot(x_46_re, x_46_im))) * y_46_im));
	double t_1 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_2 = exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -9e+20) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_2;
	} else if (y_46_re <= 6.2e-167) {
		tmp = t_1 * cos((log(sqrt(pow(hypot(x_46_im, x_46_re), 0.25))) * y_46_im));
	} else if (y_46_re <= 5600.0) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * t_2;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.cos((Math.log(Math.sqrt(Math.hypot(x_46_re, x_46_im))) * y_46_im));
	double t_1 = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_2 = Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -9e+20) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * t_2;
	} else if (y_46_re <= 6.2e-167) {
		tmp = t_1 * Math.cos((Math.log(Math.sqrt(Math.pow(Math.hypot(x_46_im, x_46_re), 0.25))) * y_46_im));
	} else if (y_46_re <= 5600.0) {
		tmp = t_0 * t_1;
	} else {
		tmp = t_0 * t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((math.log(math.sqrt(math.hypot(x_46_re, x_46_im))) * y_46_im))
	t_1 = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	t_2 = math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	tmp = 0
	if y_46_re <= -9e+20:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * t_2
	elif y_46_re <= 6.2e-167:
		tmp = t_1 * math.cos((math.log(math.sqrt(math.pow(math.hypot(x_46_im, x_46_re), 0.25))) * y_46_im))
	elif y_46_re <= 5600.0:
		tmp = t_0 * t_1
	else:
		tmp = t_0 * t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(log(sqrt(hypot(x_46_re, x_46_im))) * y_46_im))
	t_1 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_2 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -9e+20)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * t_2);
	elseif (y_46_re <= 6.2e-167)
		tmp = Float64(t_1 * cos(Float64(log(sqrt((hypot(x_46_im, x_46_re) ^ 0.25))) * y_46_im)));
	elseif (y_46_re <= 5600.0)
		tmp = Float64(t_0 * t_1);
	else
		tmp = Float64(t_0 * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((log(sqrt(hypot(x_46_re, x_46_im))) * y_46_im));
	t_1 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	t_2 = exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	tmp = 0.0;
	if (y_46_re <= -9e+20)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_2;
	elseif (y_46_re <= 6.2e-167)
		tmp = t_1 * cos((log(sqrt((hypot(x_46_im, x_46_re) ^ 0.25))) * y_46_im));
	elseif (y_46_re <= 5600.0)
		tmp = t_0 * t_1;
	else
		tmp = t_0 * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(N[Log[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9e+20], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-167], N[(t$95$1 * N[Cos[N[(N[Log[N[Sqrt[N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], 0.25], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5600.0], N[(t$95$0 * t$95$1), $MachinePrecision], N[(t$95$0 * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9e20

   1. Initial program 42.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -9e20 < y.re < 6.2e-167

   1. Initial program 46.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 80.1

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 86.7%

      \[\leadsto \cos \left(y.im \cdot \log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right)\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 6.2e-167 < y.re < 5600

   1. Initial program 37.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.3%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if 5600 < y.re

   1. Initial program 29.8%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 82.5

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

   5. Applied rewrites 82.5%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 5600:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;t\_0 \cdot \cos \left(\log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\log t\_1 \cdot y.im\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {t\_1}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_1 (sqrt (hypot x.re x.im))))
   (if (<= y.re -9e+20)
     (*
      (cos (* (atan2 x.im x.re) y.re))
      (exp
       (-
        (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
        (* (atan2 x.im x.re) y.im))))
     (if (<= y.re 6.2e-167)
       (* t_0 (cos (* (log (sqrt (pow (hypot x.im x.re) 0.25))) y.im)))
       (if (<= y.re 1.6e-23)
         (* (cos (* (log t_1) y.im)) t_0)
         (* 1.0 (pow t_1 y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = sqrt(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -9e+20) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 6.2e-167) {
		tmp = t_0 * cos((log(sqrt(pow(hypot(x_46_im, x_46_re), 0.25))) * y_46_im));
	} else if (y_46_re <= 1.6e-23) {
		tmp = cos((log(t_1) * y_46_im)) * t_0;
	} else {
		tmp = 1.0 * pow(t_1, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.sqrt(Math.hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -9e+20) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 6.2e-167) {
		tmp = t_0 * Math.cos((Math.log(Math.sqrt(Math.pow(Math.hypot(x_46_im, x_46_re), 0.25))) * y_46_im));
	} else if (y_46_re <= 1.6e-23) {
		tmp = Math.cos((Math.log(t_1) * y_46_im)) * t_0;
	} else {
		tmp = 1.0 * Math.pow(t_1, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	t_1 = math.sqrt(math.hypot(x_46_re, x_46_im))
	tmp = 0
	if y_46_re <= -9e+20:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_re <= 6.2e-167:
		tmp = t_0 * math.cos((math.log(math.sqrt(math.pow(math.hypot(x_46_im, x_46_re), 0.25))) * y_46_im))
	elif y_46_re <= 1.6e-23:
		tmp = math.cos((math.log(t_1) * y_46_im)) * t_0
	else:
		tmp = 1.0 * math.pow(t_1, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_1 = sqrt(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= -9e+20)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	elseif (y_46_re <= 6.2e-167)
		tmp = Float64(t_0 * cos(Float64(log(sqrt((hypot(x_46_im, x_46_re) ^ 0.25))) * y_46_im)));
	elseif (y_46_re <= 1.6e-23)
		tmp = Float64(cos(Float64(log(t_1) * y_46_im)) * t_0);
	else
		tmp = Float64(1.0 * (t_1 ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	t_1 = sqrt(hypot(x_46_re, x_46_im));
	tmp = 0.0;
	if (y_46_re <= -9e+20)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 6.2e-167)
		tmp = t_0 * cos((log(sqrt((hypot(x_46_im, x_46_re) ^ 0.25))) * y_46_im));
	elseif (y_46_re <= 1.6e-23)
		tmp = cos((log(t_1) * y_46_im)) * t_0;
	else
		tmp = 1.0 * (t_1 ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9e+20], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-167], N[(t$95$0 * N[Cos[N[(N[Log[N[Sqrt[N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], 0.25], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-23], N[(N[Cos[N[(N[Log[t$95$1], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[Power[t$95$1, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -9e20

   1. Initial program 42.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -9e20 < y.re < 6.2e-167

   1. Initial program 46.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 79.2%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 80.1

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

   7. Applied rewrites 80.1%

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

   9. Applied rewrites 86.7%

      \[\leadsto \cos \left(y.im \cdot \log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right)\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 6.2e-167 < y.re < 1.59999999999999988e-23

   1. Initial program 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if 1.59999999999999988e-23 < y.re

   1. Initial program 29.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-167}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\log \left(\sqrt{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{0.25}}\right) \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}}\right) \cdot y.im\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\log t\_1 \cdot y.im\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {t\_1}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_1 (sqrt (hypot x.re x.im))))
   (if (<= y.re -3.1e-15)
     (*
      (cos (* (atan2 x.im x.re) y.re))
      (exp
       (-
        (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
        (* (atan2 x.im x.re) y.im))))
     (if (<= y.re 2.1e-167)
       (* (cos (* (log (sqrt (sqrt (hypot x.im x.re)))) y.im)) t_0)
       (if (<= y.re 1.6e-23)
         (* (cos (* (log t_1) y.im)) t_0)
         (* 1.0 (pow t_1 y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = sqrt(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -3.1e-15) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 2.1e-167) {
		tmp = cos((log(sqrt(sqrt(hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = cos((log(t_1) * y_46_im)) * t_0;
	} else {
		tmp = 1.0 * pow(t_1, y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.sqrt(Math.hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= -3.1e-15) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.exp(((Math.log(Math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 2.1e-167) {
		tmp = Math.cos((Math.log(Math.sqrt(Math.sqrt(Math.hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = Math.cos((Math.log(t_1) * y_46_im)) * t_0;
	} else {
		tmp = 1.0 * Math.pow(t_1, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	t_1 = math.sqrt(math.hypot(x_46_re, x_46_im))
	tmp = 0
	if y_46_re <= -3.1e-15:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.exp(((math.log(math.sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_re <= 2.1e-167:
		tmp = math.cos((math.log(math.sqrt(math.sqrt(math.hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0
	elif y_46_re <= 1.6e-23:
		tmp = math.cos((math.log(t_1) * y_46_im)) * t_0
	else:
		tmp = 1.0 * math.pow(t_1, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_1 = sqrt(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= -3.1e-15)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	elseif (y_46_re <= 2.1e-167)
		tmp = Float64(cos(Float64(log(sqrt(sqrt(hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0);
	elseif (y_46_re <= 1.6e-23)
		tmp = Float64(cos(Float64(log(t_1) * y_46_im)) * t_0);
	else
		tmp = Float64(1.0 * (t_1 ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	t_1 = sqrt(hypot(x_46_re, x_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.1e-15)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 2.1e-167)
		tmp = cos((log(sqrt(sqrt(hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0;
	elseif (y_46_re <= 1.6e-23)
		tmp = cos((log(t_1) * y_46_im)) * t_0;
	else
		tmp = 1.0 * (t_1 ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3.1e-15], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-167], N[(N[Cos[N[(N[Log[N[Sqrt[N[Sqrt[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-23], N[(N[Cos[N[(N[Log[t$95$1], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * N[Power[t$95$1, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.0999999999999999e-15

   1. Initial program 43.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 82.4

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

   5. Applied rewrites 82.4%

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

   if -3.0999999999999999e-15 < y.re < 2.10000000000000017e-167

   1. Initial program 46.5%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 81.1%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 82.1

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

   7. Applied rewrites 82.1%

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

   9. Applied rewrites 87.4%

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

   if 2.10000000000000017e-167 < y.re < 1.59999999999999988e-23

   1. Initial program 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if 1.59999999999999988e-23 < y.re

   1. Initial program 29.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.1%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}}\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\\ t_2 := {t\_1}^{y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot t\_2\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}}\right) \cdot y.im\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\log t\_1 \cdot y.im\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_1 (sqrt (hypot x.re x.im)))
        (t_2 (pow t_1 y.re)))
   (if (<= y.re -3.5e+22)
     (* (cos (* (atan2 x.im x.re) y.re)) t_2)
     (if (<= y.re 2.1e-167)
       (* (cos (* (log (sqrt (sqrt (hypot x.im x.re)))) y.im)) t_0)
       (if (<= y.re 1.6e-23) (* (cos (* (log t_1) y.im)) t_0) (* 1.0 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = sqrt(hypot(x_46_re, x_46_im));
	double t_2 = pow(t_1, y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_2;
	} else if (y_46_re <= 2.1e-167) {
		tmp = cos((log(sqrt(sqrt(hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = cos((log(t_1) * y_46_im)) * t_0;
	} else {
		tmp = 1.0 * t_2;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.sqrt(Math.hypot(x_46_re, x_46_im));
	double t_2 = Math.pow(t_1, y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * t_2;
	} else if (y_46_re <= 2.1e-167) {
		tmp = Math.cos((Math.log(Math.sqrt(Math.sqrt(Math.hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = Math.cos((Math.log(t_1) * y_46_im)) * t_0;
	} else {
		tmp = 1.0 * t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	t_1 = math.sqrt(math.hypot(x_46_re, x_46_im))
	t_2 = math.pow(t_1, y_46_re)
	tmp = 0
	if y_46_re <= -3.5e+22:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * t_2
	elif y_46_re <= 2.1e-167:
		tmp = math.cos((math.log(math.sqrt(math.sqrt(math.hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0
	elif y_46_re <= 1.6e-23:
		tmp = math.cos((math.log(t_1) * y_46_im)) * t_0
	else:
		tmp = 1.0 * t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_1 = sqrt(hypot(x_46_re, x_46_im))
	t_2 = t_1 ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -3.5e+22)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * t_2);
	elseif (y_46_re <= 2.1e-167)
		tmp = Float64(cos(Float64(log(sqrt(sqrt(hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0);
	elseif (y_46_re <= 1.6e-23)
		tmp = Float64(cos(Float64(log(t_1) * y_46_im)) * t_0);
	else
		tmp = Float64(1.0 * t_2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	t_1 = sqrt(hypot(x_46_re, x_46_im));
	t_2 = t_1 ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -3.5e+22)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_2;
	elseif (y_46_re <= 2.1e-167)
		tmp = cos((log(sqrt(sqrt(hypot(x_46_im, x_46_re)))) * y_46_im)) * t_0;
	elseif (y_46_re <= 1.6e-23)
		tmp = cos((log(t_1) * y_46_im)) * t_0;
	else
		tmp = 1.0 * t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$1, y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+22], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-167], N[(N[Cos[N[(N[Log[N[Sqrt[N[Sqrt[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-23], N[(N[Cos[N[(N[Log[t$95$1], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(1.0 * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.5e22

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if -3.5e22 < y.re < 2.10000000000000017e-167

   1. Initial program 46.4%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.5%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 79.4

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

   7. Applied rewrites 79.4%

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

   9. Applied rewrites 85.2%

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

   if 2.10000000000000017e-167 < y.re < 1.59999999999999988e-23

   1. Initial program 39.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.9%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if 1.59999999999999988e-23 < y.re

   1. Initial program 29.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.1%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-167}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\sqrt{\mathsf{hypot}\left(x.im, x.re\right)}}\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;\cos \left(\log \left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot t\_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (sqrt (hypot x.re x.im)) y.re)))
   (if (<= y.re -3.5e+22)
     (* (cos (* (atan2 x.im x.re) y.re)) t_0)
     (if (<= y.re 1.6e-23)
       (* 1.0 (exp (* (- y.im) (atan2 x.im x.re))))
       (* 1.0 t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(sqrt(hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.sqrt(Math.hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = Math.cos((Math.atan2(x_46_im, x_46_re) * y_46_re)) * t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = 1.0 * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.sqrt(math.hypot(x_46_re, x_46_im)), y_46_re)
	tmp = 0
	if y_46_re <= -3.5e+22:
		tmp = math.cos((math.atan2(x_46_im, x_46_re) * y_46_re)) * t_0
	elif y_46_re <= 1.6e-23:
		tmp = 1.0 * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = 1.0 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -3.5e+22)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * t_0);
	elseif (y_46_re <= 1.6e-23)
		tmp = Float64(1.0 * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * 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 = sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -3.5e+22)
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * t_0;
	elseif (y_46_re <= 1.6e-23)
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = 1.0 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+22], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-23], N[(1.0 * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.5e22

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if -3.5e22 < y.re < 1.59999999999999988e-23

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 80.6

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

   7. Applied rewrites 80.6%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 81.4%

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

   if 1.59999999999999988e-23 < y.re

   1. Initial program 29.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.1%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.5e+22)
   (*
    (pow (fma 0.5 (/ (* x.re x.re) x.im) x.im) y.re)
    (cos (* (atan2 x.im x.re) y.re)))
   (if (<= y.re 1.6e-23)
     (* 1.0 (exp (* (- y.im) (atan2 x.im x.re))))
     (* 1.0 (pow (sqrt (hypot x.re x.im)) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = pow(fma(0.5, ((x_46_re * x_46_re) / x_46_im), x_46_im), y_46_re) * cos((atan2(x_46_im, x_46_re) * y_46_re));
	} else if (y_46_re <= 1.6e-23) {
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * pow(sqrt(hypot(x_46_re, x_46_im)), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.5e+22)
		tmp = Float64((fma(0.5, Float64(Float64(x_46_re * x_46_re) / x_46_im), x_46_im) ^ y_46_re) * cos(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	elseif (y_46_re <= 1.6e-23)
		tmp = Float64(1.0 * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.5e+22], N[(N[Power[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision] * N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.6e-23], N[(1.0 * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.5e22

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 76.7%

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

   if -3.5e22 < y.re < 1.59999999999999988e-23

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 80.6

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

   7. Applied rewrites 80.6%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 81.4%

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

   if 1.59999999999999988e-23 < y.re

   1. Initial program 29.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 72.1%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (hypot x.re x.im)) y.re))))
   (if (<= y.re -3.5e+22)
     t_0
     (if (<= y.re 1.6e-23) (* 1.0 (exp (* (- y.im) (atan2 x.im x.re)))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(Math.sqrt(Math.hypot(x_46_re, x_46_im)), y_46_re);
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-23) {
		tmp = 1.0 * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(math.sqrt(math.hypot(x_46_re, x_46_im)), y_46_re)
	tmp = 0
	if y_46_re <= -3.5e+22:
		tmp = t_0
	elif y_46_re <= 1.6e-23:
		tmp = 1.0 * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.5e+22)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-23)
		tmp = Float64(1.0 * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (sqrt(hypot(x_46_re, x_46_im)) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.5e+22)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-23)
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+22], t$95$0, If[LessEqual[y$46$re, 1.6e-23], N[(1.0 * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.5e22 or 1.59999999999999988e-23 < y.re

   1. Initial program 35.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 74.1%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot 1 \]

   if -3.5e22 < y.re < 1.59999999999999988e-23

   1. Initial program 44.8%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 80.6

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

   7. Applied rewrites 80.6%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 81.4%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-23}:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 74.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re) 1.0)))
   (if (<= y.re -3.5e+22)
     t_0
     (if (<= y.re 6.0) (* 1.0 (exp (* (- y.im) (atan2 x.im x.re)))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(fma(0.5, ((x_46_im * x_46_im) / x_46_re), x_46_re), y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -3.5e+22) {
		tmp = t_0;
	} else if (y_46_re <= 6.0) {
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((fma(0.5, Float64(Float64(x_46_im * x_46_im) / x_46_re), x_46_re) ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -3.5e+22)
		tmp = t_0;
	elseif (y_46_re <= 6.0)
		tmp = Float64(1.0 * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e+22], t$95$0, If[LessEqual[y$46$re, 6.0], N[(1.0 * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.5e22 or 6 < y.re

   1. Initial program 36.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 68.1%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 69.9%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   if -3.5e22 < y.re < 6

   1. Initial program 43.9%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-neg N/A

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

      5. exp-sum N/A

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

      6. associate-*l* N/A

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

      7. lower-*.f64 N/A

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

   4. Applied rewrites 79.0%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. neg-mul-1 N/A

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

      11. lower-exp.f64 N/A

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

      12. distribute-lft-neg-in N/A

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

      13. lower-*.f64 N/A

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

      14. lower-neg.f64 N/A

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

      15. lower-atan2.f64 81.0

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

   7. Applied rewrites 81.0%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 81.7%

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

Alternative 11: 58.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 0.00205:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow (fma 0.5 (/ (* x.im x.im) x.re) x.re) y.re) 1.0)))
   (if (<= y.re -4e-8) t_0 (if (<= y.re 0.00205) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(fma(0.5, ((x_46_im * x_46_im) / x_46_re), x_46_re), y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -4e-8) {
		tmp = t_0;
	} else if (y_46_re <= 0.00205) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((fma(0.5, Float64(Float64(x_46_im * x_46_im) / x_46_re), x_46_re) ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -4e-8)
		tmp = t_0;
	elseif (y_46_re <= 0.00205)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision] + x$46$re), $MachinePrecision], y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -4e-8], t$95$0, If[LessEqual[y$46$re, 0.00205], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.0000000000000001e-8 or 0.00205000000000000017 < y.re

   1. Initial program 36.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 66.0%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 67.8%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   if -4.0000000000000001e-8 < y.re < 0.00205000000000000017

   1. Initial program 44.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 55.4%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 54.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.2 \cdot 10^{-136}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 2.4 \cdot 10^{-250}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -1.2e-136)
   (* (pow (- x.re) y.re) 1.0)
   (if (<= x.re 2.4e-250)
     (* (pow (- x.im) y.re) 1.0)
     (* (pow x.re y.re) 1.0))))

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 <= -1.2e-136) {
		tmp = pow(-x_46_re, y_46_re) * 1.0;
	} else if (x_46_re <= 2.4e-250) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	}
	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) :: tmp
    if (x_46re <= (-1.2d-136)) then
        tmp = (-x_46re ** y_46re) * 1.0d0
    else if (x_46re <= 2.4d-250) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else
        tmp = (x_46re ** y_46re) * 1.0d0
    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 tmp;
	if (x_46_re <= -1.2e-136) {
		tmp = Math.pow(-x_46_re, y_46_re) * 1.0;
	} else if (x_46_re <= 2.4e-250) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -1.2e-136:
		tmp = math.pow(-x_46_re, y_46_re) * 1.0
	elif x_46_re <= 2.4e-250:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	else:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.2e-136)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * 1.0);
	elseif (x_46_re <= 2.4e-250)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	else
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	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 <= -1.2e-136)
		tmp = (-x_46_re ^ y_46_re) * 1.0;
	elseif (x_46_re <= 2.4e-250)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	else
		tmp = (x_46_re ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -1.2e-136], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 2.4e-250], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.1999999999999999e-136

   1. Initial program 47.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 37.3%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 35.3%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.re around -inf

       \[\leadsto {\left(-1 \cdot x.re\right)}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 64.8%

       \[\leadsto {\left(-x.re\right)}^{y.re} \cdot 1 \]

   if -1.1999999999999999e-136 < x.re < 2.3999999999999999e-250

   1. Initial program 47.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 55.2

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 37.6%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 48.5%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.im around -inf

       \[\leadsto {\left(-1 \cdot x.im\right)}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 61.3%

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

   if 2.3999999999999999e-250 < x.re

   1. Initial program 32.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 57.2

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 56.0%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 55.1%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 56.6%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 54.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.12 \cdot 10^{-221}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.im \leq 7.8 \cdot 10^{-74}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -1.12e-221)
   (* (pow (- x.im) y.re) 1.0)
   (if (<= x.im 7.8e-74) (* (pow x.re y.re) 1.0) (* (pow x.im y.re) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1.12e-221) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 7.8e-74) {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re) * 1.0;
	}
	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) :: tmp
    if (x_46im <= (-1.12d-221)) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else if (x_46im <= 7.8d-74) then
        tmp = (x_46re ** y_46re) * 1.0d0
    else
        tmp = (x_46im ** y_46re) * 1.0d0
    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 tmp;
	if (x_46_im <= -1.12e-221) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 7.8e-74) {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -1.12e-221:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	elif x_46_im <= 7.8e-74:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.12e-221)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	elseif (x_46_im <= 7.8e-74)
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -1.12e-221)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	elseif (x_46_im <= 7.8e-74)
		tmp = (x_46_re ^ y_46_re) * 1.0;
	else
		tmp = (x_46_im ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -1.12e-221], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$im, 7.8e-74], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.12000000000000003e-221

   1. Initial program 46.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 36.8%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 36.8%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.im around -inf

       \[\leadsto {\left(-1 \cdot x.im\right)}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 56.3%

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

   if -1.12000000000000003e-221 < x.im < 7.8000000000000003e-74

   1. Initial program 44.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 71.9

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

   5. Applied rewrites 71.9%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 59.2%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 60.8%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 64.0%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]

   if 7.8000000000000003e-74 < x.im

   1. Initial program 29.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 47.2%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 48.5%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.re around 0

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 54.6%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.re}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -0.0015:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.68 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow x.re y.re) 1.0)))
   (if (<= y.re -0.0015) t_0 (if (<= y.re 1.68e-18) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(x_46_re, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -0.0015) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-18) {
		tmp = 1.0;
	} 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 = (x_46re ** y_46re) * 1.0d0
    if (y_46re <= (-0.0015d0)) then
        tmp = t_0
    else if (y_46re <= 1.68d-18) then
        tmp = 1.0d0
    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.pow(x_46_re, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -0.0015) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-18) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(x_46_re, y_46_re) * 1.0
	tmp = 0
	if y_46_re <= -0.0015:
		tmp = t_0
	elif y_46_re <= 1.68e-18:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((x_46_re ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -0.0015)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-18)
		tmp = 1.0;
	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 = (x_46_re ^ y_46_re) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -0.0015)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-18)
		tmp = 1.0;
	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[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -0.0015], t$95$0, If[LessEqual[y$46$re, 1.68e-18], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.0015 or 1.6799999999999999e-18 < y.re

   1. Initial program 35.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 64.7%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 66.4%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.im around 0

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 61.3%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]

   if -0.0015 < y.re < 1.6799999999999999e-18

   1. Initial program 45.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. unpow2 N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 54.4

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

   5. Applied rewrites 54.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 54.4%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -3.9:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.68 \cdot 10^{-18}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow x.im y.re) 1.0)))
   (if (<= y.re -3.9) t_0 (if (<= y.re 1.68e-18) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -3.9) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-18) {
		tmp = 1.0;
	} 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 = (x_46im ** y_46re) * 1.0d0
    if (y_46re <= (-3.9d0)) then
        tmp = t_0
    else if (y_46re <= 1.68d-18) then
        tmp = 1.0d0
    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.pow(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -3.9) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-18) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(x_46_im, y_46_re) * 1.0
	tmp = 0
	if y_46_re <= -3.9:
		tmp = t_0
	elif y_46_re <= 1.68e-18:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64((x_46_im ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -3.9)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-18)
		tmp = 1.0;
	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 = (x_46_im ^ y_46_re) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -3.9)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-18)
		tmp = 1.0;
	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[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -3.9], t$95$0, If[LessEqual[y$46$re, 1.68e-18], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.89999999999999991 or 1.6799999999999999e-18 < y.re

   1. Initial program 35.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. +-commutative N/A

         \[\leadsto {\left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. unpow2 N/A

         \[\leadsto {\left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lower-hypot.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(x.re, x.im\right)}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      10. *-commutative N/A

          \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      12. lower-atan2.f64 71.6

          \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   5. Applied rewrites 71.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in x.im around 0

      \[\leadsto {\left(x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{x.re}\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 64.9%

      \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 66.6%

       \[\leadsto {\left(\mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re}, x.re\right)\right)}^{y.re} \cdot 1 \]

   12. Taylor expanded in x.re around 0

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
   13. Step-by-step derivation

   14. Applied rewrites 54.8%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]

   if -3.89999999999999991 < y.re < 1.6799999999999999e-18

   1. Initial program 45.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. +-commutative N/A

         \[\leadsto {\left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. unpow2 N/A

         \[\leadsto {\left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lower-hypot.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(x.re, x.im\right)}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      10. *-commutative N/A

          \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

      12. lower-atan2.f64 54.1

          \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

   5. Applied rewrites 54.1%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 54.1%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 25.8% accurate, 680.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

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
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;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 40.8%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing

3. Taylor expanded in y.im around 0

   \[\leadsto \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. lower-pow.f64 N/A

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   4. lower-sqrt.f64 N/A

      \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   5. +-commutative N/A

      \[\leadsto {\left(\sqrt{\color{blue}{{x.re}^{2} + {x.im}^{2}}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   6. unpow2 N/A

      \[\leadsto {\left(\sqrt{\color{blue}{x.re \cdot x.re} + {x.im}^{2}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. unpow2 N/A

      \[\leadsto {\left(\sqrt{x.re \cdot x.re + \color{blue}{x.im \cdot x.im}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. lower-hypot.f64 N/A

      \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(x.re, x.im\right)}}\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \color{blue}{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   10. *-commutative N/A

       \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   11. lower-*.f64 N/A

       \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

   12. lower-atan2.f64 62.1

       \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right) \]

5. Applied rewrites 62.1%

   \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{y.re} \cdot \cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]

6. Taylor expanded in y.re around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 31.5%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024250 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))