powComplex, real part

Percentage Accurate: 40.2% → 79.7%

Time: 16.6s

Alternatives: 11

Speedup: 4.0×

• 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: 40.2% accurate, 1.0× speedup

Math

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

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: 79.7% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
}

Julia

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

Wolfram

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

Derivation

1. Initial program 34.9%

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

3. Simplified 82.0%

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

4. Final simplification 82.0%

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

Alternative 2: 76.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 79.1%

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

   2. Taylor expanded in y.im around 0 82.0%

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

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

   1. Initial program 0.0%

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

      1. exp-diff 0.0%

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

      2. exp-to-pow 0.0%

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

      3. hypot-def 0.0%

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

      4. *-commutative 0.0%

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

      5. exp-prod 0.0%

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

      6. +-commutative 0.0%

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

      7. *-commutative 0.0%

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

      8. fma-def 0.0%

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

      9. hypot-def 76.3%

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

   3. Simplified 76.3%

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

   4. Taylor expanded in y.re around 0 0.0%

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

      1. unpow2 0.0%

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

      2. unpow2 0.0%

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

      3. hypot-def 78.4%

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

   6. Simplified 78.4%

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

4. Final simplification 80.0%

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

Alternative 3: 74.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -3.6e-68)
     t_0
     (if (<= y.re 5.8e+27)
       (*
        (cos (fma y.re (atan2 x.im x.re) (* (log (hypot x.re x.im)) y.im)))
        (exp (* (atan2 x.im x.re) (- y.im))))
       (* (cos (* y.re (atan2 x.im x.re))) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -3.6e-68) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e+27) {
		tmp = cos(fma(y_46_re, atan2(x_46_im, x_46_re), (log(hypot(x_46_re, x_46_im)) * y_46_im))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -3.6e-68)
		tmp = t_0;
	elseif (y_46_re <= 5.8e+27)
		tmp = Float64(cos(fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(hypot(x_46_re, x_46_im)) * y_46_im))) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * 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[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3.6e-68], t$95$0, If[LessEqual[y$46$re, 5.8e+27], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.60000000000000007e-68

   1. Initial program 38.0%

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

   2. Taylor expanded in y.im around 0 84.9%

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

   3. Taylor expanded in y.re around 0 86.2%

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

   if -3.60000000000000007e-68 < y.re < 5.8000000000000002e27

   1. Initial program 32.9%

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

      1. exp-diff 30.6%

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

      2. exp-to-pow 30.6%

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

      3. hypot-def 30.6%

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

      4. *-commutative 30.6%

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

      5. exp-prod 30.2%

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

      6. +-commutative 30.2%

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

      7. *-commutative 30.2%

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

      8. fma-def 30.2%

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

      9. hypot-def 79.2%

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

   3. Simplified 79.2%

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

   4. Taylor expanded in y.re around 0 82.1%

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

      1. rec-exp 82.1%

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

      2. distribute-rgt-neg-in 82.1%

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

   6. Simplified 82.1%

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

   if 5.8000000000000002e27 < 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. Taylor expanded in y.im around 0 69.2%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{-68}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 4: 75.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{-5}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\left(1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -9e-5)
     t_0
     (if (<= y.re 5.8e+27)
       (*
        (+ 1.0 (* y.re (log (hypot x.im x.re))))
        (pow (exp (- y.im)) (atan2 x.im x.re)))
       (* (cos (* y.re (atan2 x.im x.re))) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -9e-5) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e+27) {
		tmp = (1.0 + (y_46_re * log(hypot(x_46_im, x_46_re)))) * pow(exp(-y_46_im), atan2(x_46_im, x_46_re));
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -9e-5) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e+27) {
		tmp = (1.0 + (y_46_re * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.pow(Math.exp(-y_46_im), Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	tmp = 0
	if y_46_re <= -9e-5:
		tmp = t_0
	elif y_46_re <= 5.8e+27:
		tmp = (1.0 + (y_46_re * math.log(math.hypot(x_46_im, x_46_re)))) * math.pow(math.exp(-y_46_im), math.atan2(x_46_im, x_46_re))
	else:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -9e-5)
		tmp = t_0;
	elseif (y_46_re <= 5.8e+27)
		tmp = Float64(Float64(1.0 + Float64(y_46_re * log(hypot(x_46_im, x_46_re)))) * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	tmp = 0.0;
	if (y_46_re <= -9e-5)
		tmp = t_0;
	elseif (y_46_re <= 5.8e+27)
		tmp = (1.0 + (y_46_re * log(hypot(x_46_im, x_46_re)))) * (exp(-y_46_im) ^ atan2(x_46_im, x_46_re));
	else
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -9e-5], t$95$0, If[LessEqual[y$46$re, 5.8e+27], N[(N[(1.0 + N[(y$46$re * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.00000000000000057e-5

   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. Taylor expanded in y.im around 0 86.8%

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

   3. Taylor expanded in y.re around 0 88.3%

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

   if -9.00000000000000057e-5 < y.re < 5.8000000000000002e27

   1. Initial program 34.5%

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

   2. Taylor expanded in y.im around 0 42.8%

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

   3. Taylor expanded in y.re around 0 44.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 44.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

   5. Simplified 44.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

   6. Taylor expanded in y.re around 0 36.6%

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

      1. *-commutative 36.6%

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

      2. distribute-rgt1-in 43.5%

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

   8. Simplified 80.9%

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

   if 5.8000000000000002e27 < 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. Taylor expanded in y.im around 0 69.2%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{-5}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+27}:\\ \;\;\;\;\left(1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 5: 75.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.048:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 10^{+28}:\\ \;\;\;\;\left(1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -0.048)
   (exp
    (-
     (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
     (* (atan2 x.im x.re) y.im)))
   (if (<= y.re 1e+28)
     (*
      (+ 1.0 (* y.re (log (hypot x.im x.re))))
      (pow (exp (- y.im)) (atan2 x.im x.re)))
     (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.048) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 1e+28) {
		tmp = (1.0 + (y_46_re * log(hypot(x_46_im, x_46_re)))) * pow(exp(-y_46_im), atan2(x_46_im, x_46_re));
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.048) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 1e+28) {
		tmp = (1.0 + (y_46_re * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.pow(Math.exp(-y_46_im), Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -0.048:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_re <= 1e+28:
		tmp = (1.0 + (y_46_re * math.log(math.hypot(x_46_im, x_46_re)))) * math.pow(math.exp(-y_46_im), math.atan2(x_46_im, x_46_re))
	else:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -0.048)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 1e+28)
		tmp = Float64(Float64(1.0 + Float64(y_46_re * log(hypot(x_46_im, x_46_re)))) * (exp(Float64(-y_46_im)) ^ atan(x_46_im, x_46_re)));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -0.048)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 1e+28)
		tmp = (1.0 + (y_46_re * log(hypot(x_46_im, x_46_re)))) * (exp(-y_46_im) ^ atan2(x_46_im, x_46_re));
	else
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -0.048], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1e+28], N[(N[(1.0 + N[(y$46$re * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[(-y$46$im)], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.048000000000000001

   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. Taylor expanded in y.im around 0 86.8%

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

   3. Taylor expanded in y.re around 0 88.3%

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

   if -0.048000000000000001 < y.re < 9.99999999999999958e27

   1. Initial program 34.5%

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

   2. Taylor expanded in y.im around 0 42.8%

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

   3. Taylor expanded in y.re around 0 44.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 44.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

   5. Simplified 44.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

   6. Taylor expanded in y.re around 0 36.6%

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

      1. *-commutative 36.6%

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

      2. distribute-rgt1-in 43.5%

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

   8. Simplified 80.9%

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

   if 9.99999999999999958e27 < 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. Taylor expanded in x.im around inf 19.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 21.5%

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

   4. Simplified 21.5%

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

   5. Taylor expanded in y.im around 0 64.5%

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

      1. *-commutative 64.5%

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

      2. unpow2 64.5%

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

      3. unpow2 64.5%

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

      4. hypot-def 64.5%

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

   7. Simplified 64.5%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.048:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 10^{+28}:\\ \;\;\;\;\left(1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(e^{-y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 6: 74.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.5e-12)
   (exp
    (-
     (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
     (* (atan2 x.im x.re) y.im)))
   (if (<= y.re 1.05e+29)
     (exp (* (atan2 x.im x.re) (- y.im)))
     (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.5e-12) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 1.05e+29) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.5e-12) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 1.05e+29) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.5e-12:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_re <= 1.05e+29:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.5e-12)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 1.05e+29)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.5e-12)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	elseif (y_46_re <= 1.05e+29)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.5e-12], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.05e+29], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.49999999999999985e-12

   1. Initial program 36.6%

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

   2. Taylor expanded in y.im around 0 86.0%

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

   3. Taylor expanded in y.re around 0 87.4%

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

   if -2.49999999999999985e-12 < y.re < 1.0500000000000001e29

   1. Initial program 33.8%

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

   2. Taylor expanded in y.im around 0 42.2%

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

   3. Taylor expanded in y.re around 0 43.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 43.6%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

   5. Simplified 43.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

   6. Taylor expanded in y.re around 0 80.7%

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

      1. distribute-rgt-neg-in 80.7%

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

   8. Simplified 80.7%

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

   if 1.0500000000000001e29 < 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. Taylor expanded in x.im around inf 19.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 21.5%

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

   4. Simplified 21.5%

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

   5. Taylor expanded in y.im around 0 64.5%

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

      1. *-commutative 64.5%

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

      2. unpow2 64.5%

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

      3. unpow2 64.5%

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

      4. hypot-def 64.5%

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

   7. Simplified 64.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+29}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 7: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{-16} \lor \neg \left(y.re \leq 1.45 \cdot 10^{+28}\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -4e-16) (not (<= y.re 1.45e+28)))
   (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
   (exp (* (atan2 x.im x.re) (- y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4e-16) || !(y_46_re <= 1.45e+28)) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -4e-16) || !(y_46_re <= 1.45e+28)) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -4e-16) or not (y_46_re <= 1.45e+28):
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -4e-16) || !(y_46_re <= 1.45e+28))
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -4e-16) || ~((y_46_re <= 1.45e+28)))
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -4e-16], N[Not[LessEqual[y$46$re, 1.45e+28]], $MachinePrecision]], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.9999999999999999e-16 or 1.4500000000000001e28 < y.re

   1. Initial program 36.5%

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

   2. Taylor expanded in x.im around inf 20.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 20.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 20.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 20.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 20.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 21.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 21.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 23.6%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 23.6%

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

   4. Simplified 23.6%

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

   5. Taylor expanded in y.im around 0 75.8%

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

      1. *-commutative 75.8%

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

      2. unpow2 75.8%

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

      3. unpow2 75.8%

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

      4. hypot-def 77.5%

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

   7. Simplified 77.5%

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

   if -3.9999999999999999e-16 < y.re < 1.4500000000000001e28

   1. Initial program 33.6%

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

   2. Taylor expanded in y.im around 0 42.1%

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

   3. Taylor expanded in y.re around 0 43.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 43.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

   5. Simplified 43.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

   6. Taylor expanded in y.re around 0 80.8%

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

      1. distribute-rgt-neg-in 80.8%

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

   8. Simplified 80.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{-16} \lor \neg \left(y.re \leq 1.45 \cdot 10^{+28}\right):\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 8: 72.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+29}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\frac{x.re \cdot x.re}{x.im} \cdot -0.5 - x.im\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.5e-12)
   (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re)
   (if (<= y.re 3.8e+29)
     (exp (* (atan2 x.im x.re) (- y.im)))
     (*
      (cos (* y.re (atan2 x.im x.re)))
      (pow (- (* (/ (* x.re x.re) x.im) -0.5) 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 <= -2.5e-12) {
		tmp = pow(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else if (y_46_re <= 3.8e+29) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(((((x_46_re * x_46_re) / x_46_im) * -0.5) - x_46_im), y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.5d-12)) then
        tmp = sqrt(((x_46re * x_46re) + (x_46im * x_46im))) ** y_46re
    else if (y_46re <= 3.8d+29) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else
        tmp = cos((y_46re * atan2(x_46im, x_46re))) * (((((x_46re * x_46re) / x_46im) * (-0.5d0)) - x_46im) ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.5e-12) {
		tmp = Math.pow(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else if (y_46_re <= 3.8e+29) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(((((x_46_re * x_46_re) / x_46_im) * -0.5) - x_46_im), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.5e-12:
		tmp = math.pow(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re)
	elif y_46_re <= 3.8e+29:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(((((x_46_re * x_46_re) / x_46_im) * -0.5) - 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 <= -2.5e-12)
		tmp = sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) ^ y_46_re;
	elseif (y_46_re <= 3.8e+29)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (Float64(Float64(Float64(Float64(x_46_re * x_46_re) / x_46_im) * -0.5) - x_46_im) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.5e-12)
		tmp = sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))) ^ y_46_re;
	elseif (y_46_re <= 3.8e+29)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (((((x_46_re * x_46_re) / x_46_im) * -0.5) - x_46_im) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.5e-12], N[Power[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 3.8e+29], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] * -0.5), $MachinePrecision] - x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.49999999999999985e-12

   1. Initial program 36.6%

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

   2. Taylor expanded in x.im around inf 19.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 24.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 24.0%

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

   4. Simplified 24.0%

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

   5. Taylor expanded in y.im around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. +-commutative 83.2%

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

      3. unpow2 83.2%

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

      4. unpow2 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in y.re around 0 84.6%

      \[\leadsto \color{blue}{1} \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re} \]

   if -2.49999999999999985e-12 < y.re < 3.79999999999999971e29

   1. Initial program 33.8%

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

   2. Taylor expanded in y.im around 0 42.2%

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

   3. Taylor expanded in y.re around 0 43.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 43.6%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

   5. Simplified 43.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

   6. Taylor expanded in y.re around 0 80.7%

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

      1. distribute-rgt-neg-in 80.7%

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

   8. Simplified 80.7%

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

   if 3.79999999999999971e29 < 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. Taylor expanded in x.im around inf 19.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 19.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 21.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 21.5%

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

   4. Simplified 21.5%

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

   5. Taylor expanded in y.im around 0 64.5%

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

      1. *-commutative 64.5%

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

      2. +-commutative 64.5%

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

      3. unpow2 64.5%

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

      4. unpow2 64.5%

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

   7. Simplified 64.5%

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

   8. Taylor expanded in x.im around -inf 59.8%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(-1 \cdot x.im + -0.5 \cdot \frac{{x.re}^{2}}{x.im}\right)}}^{y.re} \]
   9. Step-by-step derivation

      1. +-commutative 59.8%

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(-0.5 \cdot \frac{{x.re}^{2}}{x.im} + -1 \cdot x.im\right)}}^{y.re} \]

      2. mul-1-neg 59.8%

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(-0.5 \cdot \frac{{x.re}^{2}}{x.im} + \color{blue}{\left(-x.im\right)}\right)}^{y.re} \]

      3. unsub-neg 59.8%

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(-0.5 \cdot \frac{{x.re}^{2}}{x.im} - x.im\right)}}^{y.re} \]

      4. *-commutative 59.8%

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\color{blue}{\frac{{x.re}^{2}}{x.im} \cdot -0.5} - x.im\right)}^{y.re} \]

      5. unpow2 59.8%

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

   10. Simplified 59.8%

       \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\color{blue}{\left(\frac{x.re \cdot x.re}{x.im} \cdot -0.5 - x.im\right)}}^{y.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+29}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\frac{x.re \cdot x.re}{x.im} \cdot -0.5 - x.im\right)}^{y.re}\\ \end{array} \]

Alternative 9: 71.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.5e-12)
   (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re)
   (if (<= y.re 5.5e+25)
     (exp (* (atan2 x.im x.re) (- y.im)))
     (* (cos (* y.re (atan2 x.im x.re))) (pow x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.5e-12) {
		tmp = pow(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else if (y_46_re <= 5.5e+25) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.5d-12)) then
        tmp = sqrt(((x_46re * x_46re) + (x_46im * x_46im))) ** y_46re
    else if (y_46re <= 5.5d+25) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else
        tmp = cos((y_46re * atan2(x_46im, x_46re))) * (x_46im ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.5e-12) {
		tmp = Math.pow(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else if (y_46_re <= 5.5e+25) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.5e-12:
		tmp = math.pow(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re)
	elif y_46_re <= 5.5e+25:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.5e-12)
		tmp = sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) ^ y_46_re;
	elseif (y_46_re <= 5.5e+25)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_im ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.5e-12)
		tmp = sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))) ^ y_46_re;
	elseif (y_46_re <= 5.5e+25)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.5e-12], N[Power[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 5.5e+25], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.49999999999999985e-12

   1. Initial program 36.6%

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

   2. Taylor expanded in x.im around inf 19.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 19.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 24.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 24.0%

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

   4. Simplified 24.0%

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

   5. Taylor expanded in y.im around 0 83.2%

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

      1. *-commutative 83.2%

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

      2. +-commutative 83.2%

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

      3. unpow2 83.2%

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

      4. unpow2 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in y.re around 0 84.6%

      \[\leadsto \color{blue}{1} \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re} \]

   if -2.49999999999999985e-12 < y.re < 5.50000000000000018e25

   1. Initial program 33.8%

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

   2. Taylor expanded in y.im around 0 42.4%

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

   3. Taylor expanded in y.re around 0 43.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 43.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

   5. Simplified 43.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

   6. Taylor expanded in y.re around 0 81.0%

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

      1. distribute-rgt-neg-in 81.0%

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

   8. Simplified 81.0%

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

   if 5.50000000000000018e25 < y.re

   1. Initial program 35.6%

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

   2. Taylor expanded in x.im around inf 17.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 17.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 17.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 17.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 17.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 20.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 20.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 20.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 20.1%

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

   4. Simplified 20.1%

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

   5. Taylor expanded in y.im around 0 60.2%

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

      1. *-commutative 60.2%

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

      2. +-commutative 60.2%

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

      3. unpow2 60.2%

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

      4. unpow2 60.2%

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

   7. Simplified 60.2%

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

   8. Taylor expanded in x.re around 0 58.2%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12}:\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+25}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \]

Alternative 10: 75.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12} \lor \neg \left(y.re \leq 5.8 \cdot 10^{+27}\right):\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.5e-12) (not (<= y.re 5.8e+27)))
   (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re)
   (exp (* (atan2 x.im x.re) (- y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.5e-12) || !(y_46_re <= 5.8e+27)) {
		tmp = pow(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.5d-12)) .or. (.not. (y_46re <= 5.8d+27))) then
        tmp = sqrt(((x_46re * x_46re) + (x_46im * x_46im))) ** y_46re
    else
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.5e-12) || !(y_46_re <= 5.8e+27)) {
		tmp = Math.pow(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.5e-12) or not (y_46_re <= 5.8e+27):
		tmp = math.pow(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re)
	else:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.5e-12) || !(y_46_re <= 5.8e+27))
		tmp = sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) ^ y_46_re;
	else
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.5e-12) || ~((y_46_re <= 5.8e+27)))
		tmp = sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))) ^ y_46_re;
	else
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.5e-12], N[Not[LessEqual[y$46$re, 5.8e+27]], $MachinePrecision]], N[Power[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.49999999999999985e-12 or 5.8000000000000002e27 < y.re

   1. Initial program 36.3%

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

   2. Taylor expanded in x.im around inf 19.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 19.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \color{blue}{\left(-\log \left(\frac{1}{x.im}\right) \cdot y.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      2. unsub-neg 19.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log \left(\frac{1}{x.im}\right) \cdot y.im\right)} + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      3. *-commutative 19.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - \color{blue}{y.im \cdot \log \left(\frac{1}{x.im}\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      4. log-rec 19.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \color{blue}{\left(-\log x.im\right)}\right) + -0.5 \cdot \frac{{x.re}^{2} \cdot \left(y.im \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)\right)}{{x.im}^{2}}\right) \]

      5. associate-*r* 20.4%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\color{blue}{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}}{{x.im}^{2}}\right) \]

      6. unpow2 20.4%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \frac{\left({x.re}^{2} \cdot y.im\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{\color{blue}{x.im \cdot x.im}}\right) \]

      7. times-frac 23.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} - y.im \cdot \left(-\log x.im\right)\right) + -0.5 \cdot \color{blue}{\left(\frac{{x.re}^{2} \cdot y.im}{x.im} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + -1 \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot y.im\right)\right)}{x.im}\right)}\right) \]

      8. unpow2 23.1%

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

   4. Simplified 23.1%

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

   5. Taylor expanded in y.im around 0 76.2%

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

      1. *-commutative 76.2%

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

      2. +-commutative 76.2%

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

      3. unpow2 76.2%

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

      4. unpow2 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in y.re around 0 73.6%

      \[\leadsto \color{blue}{1} \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re} \]

   if -2.49999999999999985e-12 < y.re < 5.8000000000000002e27

   1. Initial program 33.8%

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

   2. Taylor expanded in y.im around 0 42.2%

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

   3. Taylor expanded in y.re around 0 43.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
   4. Step-by-step derivation

      1. unpow2 43.6%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

   5. Simplified 43.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

   6. Taylor expanded in y.re around 0 80.7%

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

      1. distribute-rgt-neg-in 80.7%

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

   8. Simplified 80.7%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-12} \lor \neg \left(y.re \leq 5.8 \cdot 10^{+27}\right):\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 11: 53.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (exp (* (atan2 x.im x.re) (- y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp((atan2(x_46_im, x_46_re) * -y_46_im));
}

Fortran

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

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 34.9%

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

2. Taylor expanded in y.im around 0 58.8%

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

3. Taylor expanded in y.re around 0 45.1%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. unpow2 45.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(1 + -0.5 \cdot \left(\color{blue}{\left(y.re \cdot y.re\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) \]

5. Simplified 45.1%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(\left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)} \]

6. Taylor expanded in y.re around 0 54.0%

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

   1. distribute-rgt-neg-in 54.0%

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

8. Simplified 54.0%

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

9. Final simplification 54.0%

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

Reproduce

herbie shell --seed 2023229 
(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)))))