powComplex, real part

Percentage Accurate: 40.6% → 81.9%

Time: 29.2s

Alternatives: 14

Speedup: 4.1×

• 35.7% 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: 40.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := y.re \cdot t_1\\ \mathbf{if}\;y.re \leq 280000000:\\ \;\;\;\;e^{t_2 - {\left(\sqrt[3]{t_0}\right)}^{3}} \cdot \cos \left({\left(\sqrt[3]{t_1 \cdot y.im}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t_2 - t_0} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (log (hypot x.re x.im)))
        (t_2 (* y.re t_1)))
   (if (<= y.re 280000000.0)
     (* (exp (- t_2 (pow (cbrt t_0) 3.0))) (cos (pow (cbrt (* t_1 y.im)) 3.0)))
     (* (exp (- t_2 t_0)) (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 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = y_46_re * t_1;
	double tmp;
	if (y_46_re <= 280000000.0) {
		tmp = exp((t_2 - pow(cbrt(t_0), 3.0))) * cos(pow(cbrt((t_1 * y_46_im)), 3.0));
	} else {
		tmp = exp((t_2 - t_0)) * 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 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.log(Math.hypot(x_46_re, x_46_im));
	double t_2 = y_46_re * t_1;
	double tmp;
	if (y_46_re <= 280000000.0) {
		tmp = Math.exp((t_2 - Math.pow(Math.cbrt(t_0), 3.0))) * Math.cos(Math.pow(Math.cbrt((t_1 * y_46_im)), 3.0));
	} else {
		tmp = Math.exp((t_2 - t_0)) * 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 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = log(hypot(x_46_re, x_46_im))
	t_2 = Float64(y_46_re * t_1)
	tmp = 0.0
	if (y_46_re <= 280000000.0)
		tmp = Float64(exp(Float64(t_2 - (cbrt(t_0) ^ 3.0))) * cos((cbrt(Float64(t_1 * y_46_im)) ^ 3.0)));
	else
		tmp = Float64(exp(Float64(t_2 - t_0)) * cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, 280000000.0], N[(N[Exp[N[(t$95$2 - N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Power[N[Power[N[(t$95$1 * y$46$im), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(t$95$2 - t$95$0), $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 y.re < 2.8e8

   1. Initial program 51.0%

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

   3. Simplified 88.8%

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

      1. add-cube-cbrt 88.8%

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

      2. pow3 88.8%

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

      3. *-commutative 88.8%

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

   5. Applied egg-rr 88.8%

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

   6. Taylor expanded in y.re around 0 50.4%

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

      1. unpow2 50.4%

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

      2. unpow2 50.4%

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

      3. hypot-def 87.1%

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

   8. Simplified 87.1%

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

      1. add-cube-cbrt 89.9%

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

      2. pow3 90.5%

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

      3. hypot-udef 52.8%

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

      4. +-commutative 52.8%

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

      5. hypot-udef 90.5%

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

      6. add-sqr-sqrt 89.9%

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

      7. add-sqr-sqrt 90.5%

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

   10. Applied egg-rr 90.5%

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

   if 2.8e8 < y.re

   1. Initial program 27.2%

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

   3. Simplified 66.7%

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

      1. add-cube-cbrt 66.7%

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

      2. pow3 66.7%

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

      3. *-commutative 66.7%

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

   5. Applied egg-rr 66.7%

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

   6. Taylor expanded in y.re around 0 35.8%

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

      1. unpow2 35.8%

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

      2. unpow2 35.8%

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

      3. hypot-def 84.0%

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

   8. Simplified 84.0%

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

      1. rem-cube-cbrt 84.0%

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

      2. *-commutative 84.0%

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

   10. Applied egg-rr 84.0%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 280000000:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - {\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}} \cdot \cos \left({\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 2: 81.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - {\left(\sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{9}} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (cos (* y.im (log (hypot x.im x.re))))
  (exp
   (-
    (* y.re (log (hypot x.re x.im)))
    (pow (cbrt (cbrt (* y.im (atan2 x.im x.re)))) 9.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - pow(cbrt(cbrt((y_46_im * atan2(x_46_im, x_46_re)))), 9.0)));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp(((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))) - Math.pow(Math.cbrt(Math.cbrt((y_46_im * Math.atan2(x_46_im, x_46_re)))), 9.0)));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - (cbrt(cbrt(Float64(y_46_im * atan(x_46_im, x_46_re)))) ^ 9.0))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Power[N[Power[N[Power[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 9.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.5%

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

3. Simplified 81.8%

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

   1. add-cube-cbrt 81.8%

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

   2. pow3 81.8%

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

   3. *-commutative 81.8%

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

5. Applied egg-rr 81.8%

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

6. Taylor expanded in y.re around 0 45.8%

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

   1. unpow2 45.8%

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

   2. unpow2 45.8%

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

   3. hypot-def 86.1%

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

8. Simplified 86.1%

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

   1. add-cube-cbrt 86.1%

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

   2. pow3 86.1%

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

10. Applied egg-rr 86.1%

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

    1. pow-pow 86.1%

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

    2. pow-to-exp 51.7%

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

    3. metadata-eval 51.7%

       \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - e^{\log \left(\sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot \color{blue}{9}}} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

12. Applied egg-rr 51.7%

    \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{e^{\log \left(\sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \cdot 9}}} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]
13. Step-by-step derivation

    1. exp-to-pow 86.1%

       \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{{\left(\sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{9}}} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

14. Simplified 86.1%

    \[\leadsto e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \color{blue}{{\left(\sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{9}}} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

15. Final simplification 86.1%

    \[\leadsto \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - {\left(\sqrt[3]{\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)}^{9}} \]

Alternative 3: 81.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - {\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (exp
   (-
    (* y.re (log (hypot x.re x.im)))
    (pow (cbrt (* y.im (atan2 x.im x.re))) 3.0)))
  (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) {
	return exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - pow(cbrt((y_46_im * atan2(x_46_im, x_46_re))), 3.0))) * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.exp(((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))) - Math.pow(Math.cbrt((y_46_im * Math.atan2(x_46_im, x_46_re))), 3.0))) * Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - (cbrt(Float64(y_46_im * atan(x_46_im, x_46_re))) ^ 3.0))) * cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[Power[N[Power[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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. Initial program 43.5%

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

3. Simplified 81.8%

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

   1. add-cube-cbrt 81.8%

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

   2. pow3 81.8%

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

   3. *-commutative 81.8%

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

5. Applied egg-rr 81.8%

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

6. Taylor expanded in y.re around 0 45.8%

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

   1. unpow2 45.8%

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

   2. unpow2 45.8%

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

   3. hypot-def 86.1%

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

8. Simplified 86.1%

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

9. Final simplification 86.1%

   \[\leadsto e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - {\left(\sqrt[3]{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]

Alternative 4: 78.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_3 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_1}\\ \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t_3\\ \mathbf{elif}\;y.re \leq 4.8:\\ \;\;\;\;t_2 \cdot \frac{t_0}{e^{t_1}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+131}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re))
        (t_1 (* y.im (atan2 x.im x.re)))
        (t_2 (cos (* y.im (log (hypot x.im x.re)))))
        (t_3
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_1))))
   (if (<= y.re -1.76e-22)
     (* (cos (* y.re (atan2 x.im x.re))) t_3)
     (if (<= y.re 4.8)
       (* t_2 (/ t_0 (exp t_1)))
       (if (<= y.re 5e+131) t_3 (* t_2 t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_3 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	double tmp;
	if (y_46_re <= -1.76e-22) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * t_3;
	} else if (y_46_re <= 4.8) {
		tmp = t_2 * (t_0 / exp(t_1));
	} else if (y_46_re <= 5e+131) {
		tmp = t_3;
	} else {
		tmp = t_2 * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double t_3 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	double tmp;
	if (y_46_re <= -1.76e-22) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * t_3;
	} else if (y_46_re <= 4.8) {
		tmp = t_2 * (t_0 / Math.exp(t_1));
	} else if (y_46_re <= 5e+131) {
		tmp = t_3;
	} else {
		tmp = t_2 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	t_1 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_2 = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	t_3 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1))
	tmp = 0
	if y_46_re <= -1.76e-22:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * t_3
	elif y_46_re <= 4.8:
		tmp = t_2 * (t_0 / math.exp(t_1))
	elif y_46_re <= 5e+131:
		tmp = t_3
	else:
		tmp = t_2 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	t_3 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_1))
	tmp = 0.0
	if (y_46_re <= -1.76e-22)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * t_3);
	elseif (y_46_re <= 4.8)
		tmp = Float64(t_2 * Float64(t_0 / exp(t_1)));
	elseif (y_46_re <= 5e+131)
		tmp = t_3;
	else
		tmp = Float64(t_2 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re;
	t_1 = y_46_im * atan2(x_46_im, x_46_re);
	t_2 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	t_3 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	tmp = 0.0;
	if (y_46_re <= -1.76e-22)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * t_3;
	elseif (y_46_re <= 4.8)
		tmp = t_2 * (t_0 / exp(t_1));
	elseif (y_46_re <= 5e+131)
		tmp = t_3;
	else
		tmp = t_2 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.76e-22], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[y$46$re, 4.8], N[(t$95$2 * N[(t$95$0 / N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5e+131], t$95$3, N[(t$95$2 * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.76e-22

   1. Initial program 45.0%

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

   2. Taylor expanded in y.im around 0 95.1%

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

   if -1.76e-22 < y.re < 4.79999999999999982

   1. Initial program 55.2%

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

      1. exp-diff 55.2%

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

      2. exp-to-pow 55.2%

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

      3. hypot-def 55.2%

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

      4. *-commutative 55.2%

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

      5. exp-prod 55.0%

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

      6. +-commutative 55.0%

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

      7. *-commutative 55.0%

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

      8. fma-def 55.0%

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

      9. +-commutative 55.0%

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

   3. Simplified 88.2%

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

   4. Taylor expanded in y.re around 0 55.0%

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

      1. unpow2 55.0%

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

      2. unpow2 55.0%

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

      3. hypot-def 88.2%

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

   6. Simplified 88.2%

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

   7. Taylor expanded in y.im around inf 88.6%

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

   if 4.79999999999999982 < y.re < 4.99999999999999995e131

   1. Initial program 42.5%

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

   2. Taylor expanded in y.im around 0 62.6%

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

   3. Taylor expanded in y.re around 0 77.6%

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

   if 4.99999999999999995e131 < y.re

   1. Initial program 13.3%

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

      1. exp-diff 11.1%

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

      2. exp-to-pow 11.1%

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

      3. hypot-def 11.1%

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

      4. *-commutative 11.1%

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

      5. exp-prod 11.1%

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

      6. +-commutative 11.1%

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

      7. *-commutative 11.1%

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

      8. fma-def 11.1%

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

      9. +-commutative 11.1%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in y.re around 0 22.2%

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

      1. unpow2 22.2%

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

      2. unpow2 22.2%

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

      3. hypot-def 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in y.im around 0 80.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 4.8:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+131}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 5: 81.7% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (exp (- (* y.re (log (hypot x.re x.im))) (* 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) {
	return exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - (y_46_im * atan2(x_46_im, x_46_re)))) * cos((y_46_im * log(hypot(x_46_im, x_46_re))));
}

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $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. Initial program 43.5%

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

3. Simplified 81.8%

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

   1. add-cube-cbrt 81.8%

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

   2. pow3 81.8%

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

   3. *-commutative 81.8%

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

5. Applied egg-rr 81.8%

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

6. Taylor expanded in y.re around 0 45.8%

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

   1. unpow2 45.8%

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

   2. unpow2 45.8%

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

   3. hypot-def 86.1%

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

8. Simplified 86.1%

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

   1. rem-cube-cbrt 86.1%

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

   2. *-commutative 86.1%

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

10. Applied egg-rr 86.1%

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

11. Final simplification 86.1%

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

Alternative 6: 78.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot t_0\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 10^{+133}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* y.im (atan2 x.im x.re))))))
   (if (<= y.re -1.76e-22)
     (* (cos (* y.re (atan2 x.im x.re))) t_0)
     (if (<= y.re 2e-6)
       (exp (* y.im (- (atan2 x.im x.re))))
       (if (<= y.re 1e+133)
         t_0
         (*
          (cos (* y.im (log (hypot x.im x.re))))
          (pow (hypot x.re x.im) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.76e-22) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * t_0;
	} else if (y_46_re <= 2e-6) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1e+133) {
		tmp = t_0;
	} else {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * pow(hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.76e-22) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * t_0;
	} else if (y_46_re <= 2e-6) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1e+133) {
		tmp = t_0;
	} else {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -1.76e-22:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * t_0
	elif y_46_re <= 2e-6:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 1e+133:
		tmp = t_0
	else:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(y_46_im * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.76e-22)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * t_0);
	elseif (y_46_re <= 2e-6)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 1e+133)
		tmp = t_0;
	else
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -1.76e-22)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * t_0;
	elseif (y_46_re <= 2e-6)
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 1e+133)
		tmp = t_0;
	else
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.76e-22], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 2e-6], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1e+133], t$95$0, N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.76e-22

   1. Initial program 45.0%

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

   2. Taylor expanded in y.im around 0 95.1%

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

   if -1.76e-22 < y.re < 1.99999999999999991e-6

   1. Initial program 54.4%

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

   2. Taylor expanded in y.im around 0 58.0%

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

   3. Taylor expanded in y.re around 0 58.0%

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

   4. Taylor expanded in y.re around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

   6. Simplified 85.6%

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

   if 1.99999999999999991e-6 < y.re < 1e133

   1. Initial program 45.2%

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

   2. Taylor expanded in y.im around 0 62.0%

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

   3. Taylor expanded in y.re around 0 76.3%

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

   if 1e133 < y.re

   1. Initial program 13.3%

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

      1. exp-diff 11.1%

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

      2. exp-to-pow 11.1%

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

      3. hypot-def 11.1%

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

      4. *-commutative 11.1%

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

      5. exp-prod 11.1%

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

      6. +-commutative 11.1%

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

      7. *-commutative 11.1%

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

      8. fma-def 11.1%

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

      9. +-commutative 11.1%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in y.re around 0 22.2%

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

      1. unpow2 22.2%

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

      2. unpow2 22.2%

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

      3. hypot-def 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in y.im around 0 80.0%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 10^{+133}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 7: 77.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.re \leq -215000000000:\\ \;\;\;\;t_2 \cdot \frac{t_0}{t_1 + 1}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+131}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re))
        (t_1 (* y.im (atan2 x.im x.re)))
        (t_2 (cos (* y.im (log (hypot x.im x.re))))))
   (if (<= y.re -215000000000.0)
     (* t_2 (/ t_0 (+ t_1 1.0)))
     (if (<= y.re 1.3e-6)
       (exp (* y.im (- (atan2 x.im x.re))))
       (if (<= y.re 4e+131)
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_1))
         (* t_2 t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -215000000000.0) {
		tmp = t_2 * (t_0 / (t_1 + 1.0));
	} else if (y_46_re <= 1.3e-6) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 4e+131) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	} else {
		tmp = t_2 * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -215000000000.0) {
		tmp = t_2 * (t_0 / (t_1 + 1.0));
	} else if (y_46_re <= 1.3e-6) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 4e+131) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	} else {
		tmp = t_2 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	t_1 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_2 = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -215000000000.0:
		tmp = t_2 * (t_0 / (t_1 + 1.0))
	elif y_46_re <= 1.3e-6:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 4e+131:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1))
	else:
		tmp = t_2 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -215000000000.0)
		tmp = Float64(t_2 * Float64(t_0 / Float64(t_1 + 1.0)));
	elseif (y_46_re <= 1.3e-6)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 4e+131)
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_1));
	else
		tmp = Float64(t_2 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re;
	t_1 = y_46_im * atan2(x_46_im, x_46_re);
	t_2 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -215000000000.0)
		tmp = t_2 * (t_0 / (t_1 + 1.0));
	elseif (y_46_re <= 1.3e-6)
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 4e+131)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1));
	else
		tmp = t_2 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -215000000000.0], N[(t$95$2 * N[(t$95$0 / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.3e-6], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 4e+131], N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision], N[(t$95$2 * t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.15e11

   1. Initial program 49.1%

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

      1. exp-diff 37.7%

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

      2. exp-to-pow 37.7%

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

      3. hypot-def 37.7%

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

      4. *-commutative 37.7%

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

      5. exp-prod 37.7%

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

      6. +-commutative 37.7%

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

      7. *-commutative 37.7%

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

      8. fma-def 37.7%

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

      9. +-commutative 37.7%

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

   3. Simplified 77.4%

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

   4. Taylor expanded in y.re around 0 35.8%

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

      1. unpow2 35.8%

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

      2. unpow2 35.8%

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

      3. hypot-def 75.5%

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

   6. Simplified 75.5%

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

   7. Taylor expanded in y.im around 0 86.9%

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

   if -2.15e11 < y.re < 1.30000000000000005e-6

   1. Initial program 51.9%

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

   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 58.8%

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

   4. Taylor expanded in y.re around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. *-commutative 84.8%

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

      3. distribute-rgt-neg-in 84.8%

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

   6. Simplified 84.8%

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

   if 1.30000000000000005e-6 < y.re < 3.9999999999999996e131

   1. Initial program 45.2%

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

   2. Taylor expanded in y.im around 0 62.0%

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

   3. Taylor expanded in y.re around 0 76.3%

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

   if 3.9999999999999996e131 < y.re

   1. Initial program 13.3%

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

      1. exp-diff 11.1%

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

      2. exp-to-pow 11.1%

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

      3. hypot-def 11.1%

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

      4. *-commutative 11.1%

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

      5. exp-prod 11.1%

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

      6. +-commutative 11.1%

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

      7. *-commutative 11.1%

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

      8. fma-def 11.1%

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

      9. +-commutative 11.1%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in y.re around 0 22.2%

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

      1. unpow2 22.2%

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

      2. unpow2 22.2%

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

      3. hypot-def 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in y.im around 0 80.0%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -215000000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} + 1}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-6}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+131}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 8: 76.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ t_1 := \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+207}:\\ \;\;\;\;\frac{t_1}{y.im} \cdot \frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{t_0}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+132}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re))
        (t_1 (cos (* y.im (log (hypot x.im x.re))))))
   (if (<= y.re -1.7e+207)
     (* (/ t_1 y.im) (/ (pow (hypot x.im x.re) y.re) (atan2 x.im x.re)))
     (if (<= y.re 1.4e-5)
       (/ t_0 (pow (exp y.im) (atan2 x.im x.re)))
       (if (<= y.re 3.4e+132)
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* y.im (atan2 x.im x.re))))
         (* t_1 t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.7e+207) {
		tmp = (t_1 / y_46_im) * (pow(hypot(x_46_im, x_46_re), y_46_re) / atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 1.4e-5) {
		tmp = t_0 / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 3.4e+132) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double t_1 = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.7e+207) {
		tmp = (t_1 / y_46_im) * (Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re) / Math.atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 1.4e-5) {
		tmp = t_0 / Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re));
	} else if (y_46_re <= 3.4e+132) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	t_1 = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -1.7e+207:
		tmp = (t_1 / y_46_im) * (math.pow(math.hypot(x_46_im, x_46_re), y_46_re) / math.atan2(x_46_im, x_46_re))
	elif y_46_re <= 1.4e-5:
		tmp = t_0 / math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re))
	elif y_46_re <= 3.4e+132:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	else:
		tmp = t_1 * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re
	t_1 = cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.7e+207)
		tmp = Float64(Float64(t_1 / y_46_im) * Float64((hypot(x_46_im, x_46_re) ^ y_46_re) / atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.4e-5)
		tmp = Float64(t_0 / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	elseif (y_46_re <= 3.4e+132)
		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(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re;
	t_1 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -1.7e+207)
		tmp = (t_1 / y_46_im) * ((hypot(x_46_im, x_46_re) ^ y_46_re) / atan2(x_46_im, x_46_re));
	elseif (y_46_re <= 1.4e-5)
		tmp = t_0 / (exp(y_46_im) ^ atan2(x_46_im, x_46_re));
	elseif (y_46_re <= 3.4e+132)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	else
		tmp = t_1 * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.7e+207], N[(N[(t$95$1 / y$46$im), $MachinePrecision] * N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.4e-5], N[(t$95$0 / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.4e+132], 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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.6999999999999999e207

   1. Initial program 42.9%

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

      1. exp-diff 28.6%

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

      2. exp-to-pow 28.6%

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

      3. hypot-def 28.6%

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

      4. *-commutative 28.6%

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

      5. exp-prod 28.6%

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

      6. +-commutative 28.6%

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

      7. *-commutative 28.6%

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

      8. fma-def 28.6%

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

      9. +-commutative 28.6%

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

   3. Simplified 61.9%

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

   4. Taylor expanded in y.re around 0 28.6%

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

      1. unpow2 28.6%

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

      2. unpow2 28.6%

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

      3. hypot-def 61.9%

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

   6. Simplified 61.9%

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

   7. Taylor expanded in y.im around 0 90.5%

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

   8. Taylor expanded in y.im around inf 90.5%

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

      1. times-frac 90.5%

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

      2. unpow2 90.5%

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

      3. unpow2 90.5%

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

      4. hypot-def 90.5%

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

   10. Simplified 90.5%

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

   if -1.6999999999999999e207 < y.re < 1.39999999999999998e-5

   1. Initial program 52.2%

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

      1. exp-diff 50.2%

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

      2. exp-to-pow 50.2%

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

      3. hypot-def 50.2%

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

      4. *-commutative 50.2%

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

      5. exp-prod 50.0%

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

      6. +-commutative 50.0%

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

      7. *-commutative 50.0%

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

      8. fma-def 50.0%

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

      9. +-commutative 50.0%

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

   3. Simplified 87.1%

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

   4. Taylor expanded in y.re around 0 49.4%

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

      1. unpow2 49.4%

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

      2. unpow2 49.4%

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

      3. hypot-def 86.4%

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

   6. Simplified 86.4%

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

   7. Taylor expanded in x.re around -inf 44.3%

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

      1. mul-1-neg 44.3%

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

      2. cos-neg 44.3%

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

   9. Simplified 44.3%

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

   10. Taylor expanded in y.im around 0 84.5%

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

   if 1.39999999999999998e-5 < y.re < 3.40000000000000025e132

   1. Initial program 45.2%

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

   2. Taylor expanded in y.im around 0 62.0%

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

   3. Taylor expanded in y.re around 0 76.3%

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

   if 3.40000000000000025e132 < y.re

   1. Initial program 13.3%

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

      1. exp-diff 11.1%

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

      2. exp-to-pow 11.1%

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

      3. hypot-def 11.1%

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

      4. *-commutative 11.1%

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

      5. exp-prod 11.1%

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

      6. +-commutative 11.1%

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

      7. *-commutative 11.1%

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

      8. fma-def 11.1%

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

      9. +-commutative 11.1%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in y.re around 0 22.2%

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

      1. unpow2 22.2%

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

      2. unpow2 22.2%

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

      3. hypot-def 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in y.im around 0 80.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+207}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{y.im} \cdot \frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-5}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+132}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 9: 78.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22}:\\ \;\;\;\;e^{t_1 - \left|t_0\right|}\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;e^{t_1 - t_0}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))))
   (if (<= y.re -1.76e-22)
     (exp (- t_1 (fabs t_0)))
     (if (<= y.re 1.06e-5)
       (exp (* y.im (- (atan2 x.im x.re))))
       (if (<= y.re 1.5e+132)
         (exp (- t_1 t_0))
         (*
          (cos (* y.im (log (hypot x.im x.re))))
          (pow (hypot x.re x.im) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double tmp;
	if (y_46_re <= -1.76e-22) {
		tmp = exp((t_1 - fabs(t_0)));
	} else if (y_46_re <= 1.06e-5) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1.5e+132) {
		tmp = exp((t_1 - t_0));
	} else {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * pow(hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double tmp;
	if (y_46_re <= -1.76e-22) {
		tmp = Math.exp((t_1 - Math.abs(t_0)));
	} else if (y_46_re <= 1.06e-5) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1.5e+132) {
		tmp = Math.exp((t_1 - t_0));
	} else {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	tmp = 0
	if y_46_re <= -1.76e-22:
		tmp = math.exp((t_1 - math.fabs(t_0)))
	elif y_46_re <= 1.06e-5:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 1.5e+132:
		tmp = math.exp((t_1 - t_0))
	else:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))))
	tmp = 0.0
	if (y_46_re <= -1.76e-22)
		tmp = exp(Float64(t_1 - abs(t_0)));
	elseif (y_46_re <= 1.06e-5)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 1.5e+132)
		tmp = exp(Float64(t_1 - t_0));
	else
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = 0.0;
	if (y_46_re <= -1.76e-22)
		tmp = exp((t_1 - abs(t_0)));
	elseif (y_46_re <= 1.06e-5)
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.5e+132)
		tmp = exp((t_1 - t_0));
	else
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$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]}, If[LessEqual[y$46$re, -1.76e-22], N[Exp[N[(t$95$1 - N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.06e-5], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.5e+132], N[Exp[N[(t$95$1 - t$95$0), $MachinePrecision]], $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.76e-22

   1. Initial program 45.0%

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

   2. Taylor expanded in y.im around 0 95.1%

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

   3. Taylor expanded in y.re around 0 83.4%

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

      1. add-sqr-sqrt 55.0%

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

      2. sqrt-unprod 83.5%

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

      3. pow2 83.5%

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

      4. *-commutative 83.5%

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

   5. Applied egg-rr 83.5%

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

      1. unpow2 83.5%

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

      2. rem-sqrt-square 83.5%

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

   7. Simplified 83.5%

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

   if -1.76e-22 < y.re < 1.06e-5

   1. Initial program 54.4%

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

   2. Taylor expanded in y.im around 0 58.0%

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

   3. Taylor expanded in y.re around 0 58.0%

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

   4. Taylor expanded in y.re around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

   6. Simplified 85.6%

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

   if 1.06e-5 < y.re < 1.4999999999999999e132

   1. Initial program 45.2%

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

   2. Taylor expanded in y.im around 0 62.0%

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

   3. Taylor expanded in y.re around 0 76.3%

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

   if 1.4999999999999999e132 < y.re

   1. Initial program 13.3%

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

      1. exp-diff 11.1%

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

      2. exp-to-pow 11.1%

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

      3. hypot-def 11.1%

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

      4. *-commutative 11.1%

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

      5. exp-prod 11.1%

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

      6. +-commutative 11.1%

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

      7. *-commutative 11.1%

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

      8. fma-def 11.1%

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

      9. +-commutative 11.1%

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

   3. Simplified 46.7%

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

   4. Taylor expanded in y.re around 0 22.2%

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

      1. unpow2 22.2%

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

      2. unpow2 22.2%

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

      3. hypot-def 64.4%

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

   6. Simplified 64.4%

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

   7. Taylor expanded in y.im around 0 80.0%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \left|y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|}\\ \mathbf{elif}\;y.re \leq 1.06 \cdot 10^{-5}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+132}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 10: 77.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -215000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+132}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (cos (* y.im (log (hypot x.im x.re))))
          (pow (hypot x.re x.im) y.re))))
   (if (<= y.re -215000000000.0)
     t_0
     (if (<= y.re 8.5e-11)
       (exp (* y.im (- (atan2 x.im x.re))))
       (if (<= y.re 5.2e+132)
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* y.im (atan2 x.im x.re))))
         t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -215000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 8.5e-11) {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 5.2e+132) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -215000000000.0) {
		tmp = t_0;
	} else if (y_46_re <= 8.5e-11) {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 5.2e+132) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -215000000000.0:
		tmp = t_0
	elif y_46_re <= 8.5e-11:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 5.2e+132:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -215000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 8.5e-11)
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 5.2e+132)
		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(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -215000000000.0)
		tmp = t_0;
	elseif (y_46_re <= 8.5e-11)
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 5.2e+132)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -215000000000.0], t$95$0, If[LessEqual[y$46$re, 8.5e-11], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 5.2e+132], 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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.15e11 or 5.2e132 < y.re

   1. Initial program 32.7%

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

      1. exp-diff 25.5%

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

      2. exp-to-pow 25.5%

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

      3. hypot-def 25.5%

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

      4. *-commutative 25.5%

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

      5. exp-prod 25.5%

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

      6. +-commutative 25.5%

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

      7. *-commutative 25.5%

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

      8. fma-def 25.5%

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

      9. +-commutative 25.5%

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

   3. Simplified 63.3%

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

   4. Taylor expanded in y.re around 0 29.6%

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

      1. unpow2 29.6%

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

      2. unpow2 29.6%

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

      3. hypot-def 70.4%

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

   6. Simplified 70.4%

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

   7. Taylor expanded in y.im around 0 82.7%

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

   if -2.15e11 < y.re < 8.50000000000000037e-11

   1. Initial program 51.9%

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

   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 58.8%

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

   4. Taylor expanded in y.re around 0 84.8%

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

      1. mul-1-neg 84.8%

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

      2. *-commutative 84.8%

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

      3. distribute-rgt-neg-in 84.8%

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

   6. Simplified 84.8%

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

   if 8.50000000000000037e-11 < y.re < 5.2e132

   1. Initial program 45.2%

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

   2. Taylor expanded in y.im around 0 62.0%

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

   3. Taylor expanded in y.re around 0 76.3%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -215000000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-11}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+132}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 11: 79.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22} \lor \neg \left(y.re \leq 1.15 \cdot 10^{-5}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.76e-22) (not (<= y.re 1.15e-5)))
   (exp
    (-
     (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
     (* y.im (atan2 x.im x.re))))
   (exp (* y.im (- (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.76e-22) || !(y_46_re <= 1.15e-5)) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.76d-22)) .or. (.not. (y_46re <= 1.15d-5))) then
        tmp = exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - (y_46im * atan2(x_46im, x_46re))))
    else
        tmp = exp((y_46im * -atan2(x_46im, x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.76e-22) || !(y_46_re <= 1.15e-5)) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.76e-22) or not (y_46_re <= 1.15e-5):
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * math.atan2(x_46_im, x_46_re))))
	else:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.76e-22) || !(y_46_re <= 1.15e-5))
		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(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.76e-22) || ~((y_46_re <= 1.15e-5)))
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	else
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.76e-22], N[Not[LessEqual[y$46$re, 1.15e-5]], $MachinePrecision]], 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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.76e-22 or 1.15e-5 < y.re

   1. Initial program 35.4%

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

   2. Taylor expanded in y.im around 0 72.8%

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

   3. Taylor expanded in y.re around 0 75.6%

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

   if -1.76e-22 < y.re < 1.15e-5

   1. Initial program 54.4%

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

   2. Taylor expanded in y.im around 0 58.0%

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

   3. Taylor expanded in y.re around 0 58.0%

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

   4. Taylor expanded in y.re around 0 85.6%

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

      1. mul-1-neg 85.6%

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

      2. *-commutative 85.6%

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

      3. distribute-rgt-neg-in 85.6%

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

   6. Simplified 85.6%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.76 \cdot 10^{-22} \lor \neg \left(y.re \leq 1.15 \cdot 10^{-5}\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \]

Alternative 12: 77.1% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-16} \lor \neg \left(y.re \leq 0.00015\right):\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.5e-16) (not (<= y.re 0.00015)))
   (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re)
   (exp (* y.im (- (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.5e-16) || !(y_46_re <= 0.00015)) {
		tmp = pow(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.5d-16)) .or. (.not. (y_46re <= 0.00015d0))) then
        tmp = sqrt(((x_46re * x_46re) + (x_46im * x_46im))) ** y_46re
    else
        tmp = exp((y_46im * -atan2(x_46im, x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.5e-16) || !(y_46_re <= 0.00015)) {
		tmp = Math.pow(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.5e-16) or not (y_46_re <= 0.00015):
		tmp = math.pow(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re)
	else:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.5e-16) || !(y_46_re <= 0.00015))
		tmp = sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) ^ y_46_re;
	else
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.5e-16) || ~((y_46_re <= 0.00015)))
		tmp = sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))) ^ y_46_re;
	else
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.5e-16], N[Not[LessEqual[y$46$re, 0.00015]], $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[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.50000000000000017e-16 or 1.49999999999999987e-4 < y.re

   1. Initial program 35.0%

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

      1. exp-diff 26.6%

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

      2. exp-to-pow 26.6%

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

      3. hypot-def 26.6%

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

      4. *-commutative 26.6%

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

      5. exp-prod 25.9%

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

      6. +-commutative 25.9%

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

      7. *-commutative 25.9%

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

      8. fma-def 25.9%

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

      9. +-commutative 25.9%

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

   3. Simplified 61.5%

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

   4. Taylor expanded in y.re around 0 29.4%

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

      1. unpow2 29.4%

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

      2. unpow2 29.4%

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

      3. hypot-def 67.1%

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

   6. Simplified 67.1%

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

   7. Taylor expanded in y.im around 0 75.7%

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

   8. Taylor expanded in y.im around 0 70.8%

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

      1. +-commutative 70.8%

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

      2. unpow2 70.8%

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

      3. unpow2 70.8%

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

   10. Simplified 70.8%

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

   if -3.50000000000000017e-16 < y.re < 1.49999999999999987e-4

   1. Initial program 54.2%

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

   2. Taylor expanded in y.im around 0 58.6%

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

   3. Taylor expanded in y.re around 0 58.6%

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

   4. Taylor expanded in y.re around 0 85.2%

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

      1. mul-1-neg 85.2%

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

      2. *-commutative 85.2%

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

      3. distribute-rgt-neg-in 85.2%

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

   6. Simplified 85.2%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-16} \lor \neg \left(y.re \leq 0.00015\right):\\ \;\;\;\;{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \]

Alternative 13: 77.2% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -235000000000 \lor \neg \left(y.re \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -235000000000.0) (not (<= y.re 3.8e-5)))
   (pow (hypot x.im x.re) y.re)
   (exp (* y.im (- (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -235000000000.0) || !(y_46_re <= 3.8e-5)) {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = exp((y_46_im * -atan2(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 tmp;
	if ((y_46_re <= -235000000000.0) || !(y_46_re <= 3.8e-5)) {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -235000000000.0) or not (y_46_re <= 3.8e-5):
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -235000000000.0) || !(y_46_re <= 3.8e-5))
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	else
		tmp = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -235000000000.0) || ~((y_46_re <= 3.8e-5)))
		tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
	else
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -235000000000.0], N[Not[LessEqual[y$46$re, 3.8e-5]], $MachinePrecision]], N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision], N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.35e11 or 3.8000000000000002e-5 < y.re

   1. Initial program 36.0%

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

      1. exp-diff 27.3%

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

      2. exp-to-pow 27.3%

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

      3. hypot-def 27.3%

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

      4. *-commutative 27.3%

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

      5. exp-prod 26.6%

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

      6. +-commutative 26.6%

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

      7. *-commutative 26.6%

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

      8. fma-def 26.6%

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

      9. +-commutative 26.6%

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

   3. Simplified 61.2%

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

   4. Taylor expanded in y.re around 0 30.2%

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

      1. unpow2 30.2%

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

      2. unpow2 30.2%

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

      3. hypot-def 66.9%

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

   6. Simplified 66.9%

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

   7. Taylor expanded in y.im around 0 76.4%

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

   8. Taylor expanded in y.im around 0 70.7%

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

      1. unpow2 70.7%

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

      2. unpow2 70.7%

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

      3. hypot-def 70.7%

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

   10. Simplified 70.7%

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

   if -2.35e11 < y.re < 3.8000000000000002e-5

   1. Initial program 52.4%

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

   2. Taylor expanded in y.im around 0 59.2%

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

   3. Taylor expanded in y.re around 0 59.2%

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

   4. Taylor expanded in y.re around 0 84.9%

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

      1. mul-1-neg 84.9%

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

      2. *-commutative 84.9%

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

      3. distribute-rgt-neg-in 84.9%

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

   6. Simplified 84.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -235000000000 \lor \neg \left(y.re \leq 3.8 \cdot 10^{-5}\right):\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \]

Alternative 14: 62.6% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (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) {
	return pow(hypot(x_46_im, x_46_re), y_46_re);
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.pow(math.hypot(x_46_im, x_46_re), y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return hypot(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)
	tmp = hypot(x_46_im, x_46_re) ^ y_46_re;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]

Derivation

1. Initial program 43.5%

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

   1. exp-diff 38.8%

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

   2. exp-to-pow 38.8%

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

   3. hypot-def 38.8%

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

   4. *-commutative 38.8%

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

   5. exp-prod 38.3%

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

   6. +-commutative 38.3%

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

   7. *-commutative 38.3%

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

   8. fma-def 38.3%

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

   9. +-commutative 38.3%

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

3. Simplified 73.0%

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

4. Taylor expanded in y.re around 0 40.3%

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

   1. unpow2 40.3%

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

   2. unpow2 40.3%

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

   3. hypot-def 76.1%

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

6. Simplified 76.1%

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

7. Taylor expanded in y.im around 0 68.7%

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

8. Taylor expanded in y.im around 0 57.5%

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

   1. unpow2 57.5%

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

   2. unpow2 57.5%

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

   3. hypot-def 65.0%

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

10. Simplified 65.0%

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

11. Final simplification 65.0%

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

Reproduce

herbie shell --seed 2023283 
(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)))))