powComplex, real part

Percentage Accurate: 40.1% → 80.6%

Time: 13.8s

Alternatives: 8

Speedup: 4.0×

• 35.5% 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: 40.1% 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: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := e^{t_2 \cdot y.re - t_0}\\ \mathbf{if}\;x.im \leq 5.4 \cdot 10^{-258}:\\ \;\;\;\;t_3 \cdot \cos \left(\mathsf{fma}\left(t_2, y.im, t_1\right)\right)\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{-230}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right) - t_0}\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{-5}:\\ \;\;\;\;t_3 \cdot \cos \left(t_1 + y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (exp (- (* t_2 y.re) t_0))))
   (if (<= x.im 5.4e-258)
     (* t_3 (cos (fma t_2 y.im t_1)))
     (if (<= x.im 2.6e-230)
       (exp (- (* y.re (log (fma 0.5 (/ (* x.re x.re) x.im) x.im))) t_0))
       (if (<= x.im 4e-5)
         (* t_3 (cos (+ t_1 (* y.im (log x.im)))))
         (exp (- (* y.re (log x.im)) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double tmp;
	if (x_46_im <= 5.4e-258) {
		tmp = t_3 * cos(fma(t_2, y_46_im, t_1));
	} else if (x_46_im <= 2.6e-230) {
		tmp = exp(((y_46_re * log(fma(0.5, ((x_46_re * x_46_re) / x_46_im), x_46_im))) - t_0));
	} else if (x_46_im <= 4e-5) {
		tmp = t_3 * cos((t_1 + (y_46_im * log(x_46_im))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	tmp = 0.0
	if (x_46_im <= 5.4e-258)
		tmp = Float64(t_3 * cos(fma(t_2, y_46_im, t_1)));
	elseif (x_46_im <= 2.6e-230)
		tmp = exp(Float64(Float64(y_46_re * log(fma(0.5, Float64(Float64(x_46_re * x_46_re) / x_46_im), x_46_im))) - t_0));
	elseif (x_46_im <= 4e-5)
		tmp = Float64(t_3 * cos(Float64(t_1 + Float64(y_46_im * log(x_46_im)))));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, 5.4e-258], N[(t$95$3 * N[Cos[N[(t$95$2 * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.6e-230], N[Exp[N[(N[(y$46$re * N[Log[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] + x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], If[LessEqual[x$46$im, 4e-5], N[(t$95$3 * N[Cos[N[(t$95$1 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < 5.39999999999999991e-258

   1. Initial program 42.6%

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

   3. Simplified 85.2%

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

   if 5.39999999999999991e-258 < x.im < 2.6000000000000001e-230

   1. Initial program 0.0%

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

   2. Taylor expanded in y.im around 0 37.5%

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

   3. Taylor expanded in y.re around 0 87.5%

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

   4. Taylor expanded in x.re around 0 87.7%

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

      1. fma-def 87.7%

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

      2. unpow2 87.7%

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

   6. Simplified 87.7%

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

   if 2.6000000000000001e-230 < x.im < 4.00000000000000033e-5

   1. Initial program 50.2%

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

   3. Simplified 80.5%

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

   4. Taylor expanded in x.re around 0 85.9%

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

   if 4.00000000000000033e-5 < x.im

   1. Initial program 28.3%

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

   2. Taylor expanded in y.im around 0 57.3%

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

   3. Taylor expanded in y.re around 0 62.3%

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

   4. Taylor expanded in x.re around 0 88.3%

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

      1. *-commutative 88.3%

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

   6. Simplified 88.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 5.4 \cdot 10^{-258}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{-230}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4 \cdot 10^{-5}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 2: 79.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_1 := \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 27000000:\\ \;\;\;\;t_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
           (* (atan2 x.im x.re) y.im))))
        (t_1 (cos (* y.im (log (hypot x.im x.re))))))
   (if (<= y.re -1.25e-18)
     t_0
     (if (<= y.re 27000000.0)
       (* t_1 (exp (* (atan2 x.im x.re) (- y.im))))
       (* t_0 t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_1 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.25e-18) {
		tmp = t_0;
	} else if (y_46_re <= 27000000.0) {
		tmp = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0 * t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double t_1 = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (y_46_re <= -1.25e-18) {
		tmp = t_0;
	} else if (y_46_re <= 27000000.0) {
		tmp = t_1 * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0 * t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	t_1 = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if y_46_re <= -1.25e-18:
		tmp = t_0
	elif y_46_re <= 27000000.0:
		tmp = t_1 * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = t_0 * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_1 = cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.25e-18)
		tmp = t_0;
	elseif (y_46_re <= 27000000.0)
		tmp = Float64(t_1 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64(t_0 * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	t_1 = cos((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (y_46_re <= -1.25e-18)
		tmp = t_0;
	elseif (y_46_re <= 27000000.0)
		tmp = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = t_0 * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, 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, -1.25e-18], t$95$0, If[LessEqual[y$46$re, 27000000.0], N[(t$95$1 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.25000000000000009e-18

   1. Initial program 36.1%

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

   2. Taylor expanded in y.im around 0 77.8%

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

   3. Taylor expanded in y.re around 0 82.0%

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

   if -1.25000000000000009e-18 < y.re < 2.7e7

   1. Initial program 44.4%

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

   2. Taylor expanded in y.re around 0 44.4%

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

      1. unpow2 44.4%

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

      2. unpow2 44.4%

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

      3. hypot-def 49.1%

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

   4. Simplified 49.1%

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

   5. Taylor expanded in y.re around 0 89.5%

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

      1. neg-mul-1 88.0%

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

      2. distribute-rgt-neg-in 88.0%

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

   7. Simplified 89.5%

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

   if 2.7e7 < y.re

   1. Initial program 34.3%

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

   2. Taylor expanded in y.re around 0 35.7%

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

      1. unpow2 35.7%

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

      2. unpow2 35.7%

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

      3. hypot-def 71.5%

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

   4. Simplified 71.5%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-18}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 27000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]

Alternative 3: 79.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-18} \lor \neg \left(y.re \leq 58000000000000\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.25e-18) (not (<= y.re 58000000000000.0)))
   (exp
    (-
     (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
     (* (atan2 x.im x.re) y.im)))
   (*
    (cos (* y.im (log (hypot x.im x.re))))
    (exp (* (atan2 x.im x.re) (- y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.25e-18) || !(y_46_re <= 58000000000000.0)) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.25e-18) || !(y_46_re <= 58000000000000.0)) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.25e-18) or not (y_46_re <= 58000000000000.0):
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	else:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.25e-18) || !(y_46_re <= 58000000000000.0))
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.25e-18) || ~((y_46_re <= 58000000000000.0)))
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.25e-18], N[Not[LessEqual[y$46$re, 58000000000000.0]], $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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $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[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.25000000000000009e-18 or 5.8e13 < y.re

   1. Initial program 34.3%

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

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

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

   3. Taylor expanded in y.re around 0 75.1%

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

   if -1.25000000000000009e-18 < y.re < 5.8e13

   1. Initial program 45.3%

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

   2. Taylor expanded in y.re around 0 45.3%

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

      1. unpow2 45.3%

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

      2. unpow2 45.3%

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

      3. hypot-def 49.9%

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

   4. Simplified 49.9%

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

   5. Taylor expanded in y.re around 0 88.8%

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

      1. neg-mul-1 86.5%

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

      2. distribute-rgt-neg-in 86.5%

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

   7. Simplified 88.8%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{-18} \lor \neg \left(y.re \leq 58000000000000\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 4: 79.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \lor \neg \left(y.re \leq 27000000\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.4) (not (<= y.re 27000000.0)))
   (exp
    (-
     (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
     (* (atan2 x.im x.re) y.im)))
   (exp (* (atan2 x.im x.re) (- y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.4) || !(y_46_re <= 27000000.0)) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.4d0)) .or. (.not. (y_46re <= 27000000.0d0))) then
        tmp = exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - (atan2(x_46im, x_46re) * y_46im)))
    else
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.4) || !(y_46_re <= 27000000.0)) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.4) or not (y_46_re <= 27000000.0):
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	else:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.4) || !(y_46_re <= 27000000.0))
		tmp = exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.4) || ~((y_46_re <= 27000000.0)))
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.4], N[Not[LessEqual[y$46$re, 27000000.0]], $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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.39999999999999991 or 2.7e7 < y.re

   1. Initial program 34.8%

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

   2. Taylor expanded in y.im around 0 71.1%

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

   3. Taylor expanded in y.re around 0 75.4%

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

   if -3.39999999999999991 < y.re < 2.7e7

   1. Initial program 44.6%

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

   2. Taylor expanded in y.im around 0 48.5%

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

   3. Taylor expanded in y.re around 0 48.5%

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

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

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

      1. neg-mul-1 87.4%

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

      2. distribute-rgt-neg-in 87.4%

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

   6. Simplified 87.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \lor \neg \left(y.re \leq 27000000\right):\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 5: 72.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -4e-310)
     (exp (- (* y.re (log (- x.im))) t_0))
     (exp (- (* y.re (log x.im)) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -4e-310) {
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46im <= (-4d-310)) then
        tmp = exp(((y_46re * log(-x_46im)) - t_0))
    else
        tmp = exp(((y_46re * log(x_46im)) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -4e-310) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -4e-310:
		tmp = math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -4e-310)
		tmp = exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -4e-310)
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -4e-310], N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.999999999999988e-310

   1. Initial program 42.6%

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

   2. Taylor expanded in y.im around 0 62.2%

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

   3. Taylor expanded in y.re around 0 61.5%

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

   4. Taylor expanded in x.im around -inf 71.5%

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

      1. mul-1-neg 71.5%

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

   6. Simplified 71.5%

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

   if -3.999999999999988e-310 < x.im

   1. Initial program 35.5%

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

   2. Taylor expanded in y.im around 0 58.9%

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

   3. Taylor expanded in y.re around 0 64.7%

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

   4. Taylor expanded in x.re around 0 77.8%

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

      1. *-commutative 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 6: 62.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -4e-310)
   (exp (* (atan2 x.im x.re) (- y.im)))
   (exp (- (* y.re (log x.im)) (* (atan2 x.im x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -4e-310) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-4d-310)) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else
        tmp = exp(((y_46re * log(x_46im)) - (atan2(x_46im, x_46re) * y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -4e-310) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -4e-310:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -4e-310)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -4e-310)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = exp(((y_46_re * log(x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -4e-310], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.999999999999988e-310

   1. Initial program 42.6%

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

   2. Taylor expanded in y.im around 0 62.2%

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

   3. Taylor expanded in y.re around 0 61.5%

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

   4. Taylor expanded in y.re around 0 53.7%

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

      1. neg-mul-1 53.7%

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

      2. distribute-rgt-neg-in 53.7%

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

   6. Simplified 53.7%

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

   if -3.999999999999988e-310 < x.im

   1. Initial program 35.5%

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

   2. Taylor expanded in y.im around 0 58.9%

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

   3. Taylor expanded in y.re around 0 64.7%

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

   4. Taylor expanded in x.re around 0 77.8%

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

      1. *-commutative 77.8%

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

   6. Simplified 77.8%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 7: 64.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq 10^{-292}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re 1e-292)
   (exp (* (atan2 x.im x.re) (- y.im)))
   (exp (- (* y.re (log x.re)) (* (atan2 x.im x.re) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1e-292) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= 1d-292) then
        tmp = exp((atan2(x_46im, x_46re) * -y_46im))
    else
        tmp = exp(((y_46re * log(x_46re)) - (atan2(x_46im, x_46re) * y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= 1e-292) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_re)) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= 1e-292:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_re)) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= 1e-292)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = exp(Float64(Float64(y_46_re * log(x_46_re)) - Float64(atan(x_46_im, x_46_re) * y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= 1e-292)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = exp(((y_46_re * log(x_46_re)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, 1e-292], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1.0000000000000001e-292

   1. Initial program 37.6%

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

   2. Taylor expanded in y.im around 0 62.2%

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

   3. Taylor expanded in y.re around 0 60.8%

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

   4. Taylor expanded in y.re around 0 57.4%

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

      1. neg-mul-1 57.4%

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

      2. distribute-rgt-neg-in 57.4%

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

   6. Simplified 57.4%

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

   if 1.0000000000000001e-292 < x.re

   1. Initial program 41.4%

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

   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 65.7%

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

   4. Taylor expanded in x.re around inf 75.3%

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

      1. mul-1-neg 75.3%

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

      2. distribute-lft-neg-in 75.3%

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

      3. log-rec 75.3%

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

      4. remove-double-neg 75.3%

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

   6. Simplified 75.3%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 10^{-292}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 8: 53.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.3%

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

2. Taylor expanded in y.im around 0 60.7%

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

3. Taylor expanded in y.re around 0 63.0%

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

4. Taylor expanded in y.re around 0 53.7%

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

   1. neg-mul-1 53.7%

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

   2. distribute-rgt-neg-in 53.7%

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

6. Simplified 53.7%

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

7. Final simplification 53.7%

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

Reproduce

herbie shell --seed 2023238 
(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)))))