powComplex, real part

Percentage Accurate: 40.5% → 78.9%

Time: 17.5s

Alternatives: 13

Speedup: 6.9×

• 35.1% 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: 40.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+50}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.2:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.1e+50)
   (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
   (if (<= y.re 7.2)
     (/
      (cos (* y.im (log (hypot x.im x.re))))
      (/ (exp (* y.im (atan2 x.im x.re))) (pow (hypot x.re x.im) y.re)))
     (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e+50) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 7.2) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp((y_46_im * atan2(x_46_im, x_46_re))) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e+50) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 7.2) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / (Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re))) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.1e+50:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 7.2:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / (math.exp((y_46_im * math.atan2(x_46_im, x_46_re))) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.1e+50)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 7.2)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / Float64(exp(Float64(y_46_im * atan(x_46_im, x_46_re))) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.1e+50)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 7.2)
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / (exp((y_46_im * atan2(x_46_im, x_46_re))) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = ((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.1e+50], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.1e50

   1. Initial program 41.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 88.3%

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

   5. Simplified 88.3%

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

   if -2.1e50 < y.re < 7.20000000000000018

   1. Initial program 41.7%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 82.8%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 83.5%

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

   7. Simplified 83.5%

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

   if 7.20000000000000018 < y.re

   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. Step-by-step derivation

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 55.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 64.6%

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

   7. Simplified 64.6%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 69.2%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. frac-2neg N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right)\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

       7. sqrt-pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

       13. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(0 - \color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im\right)\right)\right)\right) \]

       18. atan2-lowering-atan2.f64 58.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right)\right) \]

   12. Applied egg-rr 58.5%

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

   13. Taylor expanded in y.im around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{1}\right)\right) \]
   14. Step-by-step derivation

   15. Simplified 77.0%

       \[\leadsto \frac{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}}{0 - \color{blue}{1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+50}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 7.2:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 10.6:\\ \;\;\;\;\frac{\cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 10.6)
   (/
    (cos (+ (* (log (hypot x.re x.im)) y.im) (* y.re (atan2 x.im x.re))))
    (/ (exp (* y.im (atan2 x.im x.re))) (pow (hypot x.re x.im) y.re)))
   (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))))

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 <= 10.6) {
		tmp = cos(((log(hypot(x_46_re, x_46_im)) * y_46_im) + (y_46_re * atan2(x_46_im, x_46_re)))) / (exp((y_46_im * atan2(x_46_im, x_46_re))) / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 10.6) {
		tmp = Math.cos(((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_im) + (y_46_re * Math.atan2(x_46_im, x_46_re)))) / (Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re))) / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= 10.6:
		tmp = math.cos(((math.log(math.hypot(x_46_re, x_46_im)) * y_46_im) + (y_46_re * math.atan2(x_46_im, x_46_re)))) / (math.exp((y_46_im * math.atan2(x_46_im, x_46_re))) / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 10.6)
		tmp = Float64(cos(Float64(Float64(log(hypot(x_46_re, x_46_im)) * y_46_im) + Float64(y_46_re * atan(x_46_im, x_46_re)))) / Float64(exp(Float64(y_46_im * atan(x_46_im, x_46_re))) / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= 10.6)
		tmp = cos(((log(hypot(x_46_re, x_46_im)) * y_46_im) + (y_46_re * atan2(x_46_im, x_46_re)))) / (exp((y_46_im * atan2(x_46_im, x_46_re))) / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = ((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 10.6], N[(N[Cos[N[(N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 10.5999999999999996

   1. Initial program 41.6%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 83.2%

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

   if 10.5999999999999996 < y.re

   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. Step-by-step derivation

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 55.4%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 64.6%

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

   7. Simplified 64.6%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 69.2%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. frac-2neg N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right)\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

       7. sqrt-pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

       13. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(0 - \color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im\right)\right)\right)\right) \]

       18. atan2-lowering-atan2.f64 58.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right)\right) \]

   12. Applied egg-rr 58.5%

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

   13. Taylor expanded in y.im around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{1}\right)\right) \]
   14. Step-by-step derivation

   15. Simplified 77.0%

       \[\leadsto \frac{0 - {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}}{0 - \color{blue}{1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 10.6:\\ \;\;\;\;\frac{\cos \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq 7.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.im (atan2 x.im x.re)))))
   (if (<= y.im 7.5e+47)
     (/ 1.0 (/ t_0 (pow (hypot x.re x.im) y.re)))
     (/ (cos (* y.im (log (hypot x.im x.re)))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= 7.5e+47) {
		tmp = 1.0 / (t_0 / pow(hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= 7.5e+47) {
		tmp = 1.0 / (t_0 / Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	} else {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= 7.5e+47:
		tmp = 1.0 / (t_0 / math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	else:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= 7.5e+47)
		tmp = Float64(1.0 / Float64(t_0 / (hypot(x_46_re, x_46_im) ^ y_46_re)));
	else
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= 7.5e+47)
		tmp = 1.0 / (t_0 / (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, 7.5e+47], N[(1.0 / N[(t$95$0 / N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $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] / t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 7.4999999999999999e47

   1. Initial program 43.3%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 76.9%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 78.9%

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

   7. Simplified 78.9%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 80.8%

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

   if 7.4999999999999999e47 < y.im

   1. Initial program 38.5%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 73.1%

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

   5. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. hypot-define N/A

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 80.9%

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

   7. Simplified 80.9%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 7.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+50}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.re x.im) y.re)))
   (if (<= y.re -2.9e+50)
     (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
     (if (<= y.re 1.2e-5)
       (/ 1.0 (/ (exp (* y.im (atan2 x.im x.re))) t_0))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -2.9e+50) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 1.2e-5) {
		tmp = 1.0 / (exp((y_46_im * atan2(x_46_im, x_46_re))) / t_0);
	} 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.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	double tmp;
	if (y_46_re <= -2.9e+50) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 1.2e-5) {
		tmp = 1.0 / (Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re))) / t_0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	tmp = 0
	if y_46_re <= -2.9e+50:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 1.2e-5:
		tmp = 1.0 / (math.exp((y_46_im * math.atan2(x_46_im, x_46_re))) / t_0)
	else:
		tmp = 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
	tmp = 0.0
	if (y_46_re <= -2.9e+50)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 1.2e-5)
		tmp = Float64(1.0 / Float64(exp(Float64(y_46_im * atan(x_46_im, x_46_re))) / t_0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_re, x_46_im) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -2.9e+50)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 1.2e-5)
		tmp = 1.0 / (exp((y_46_im * atan2(x_46_im, x_46_re))) / t_0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -2.9e+50], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.2e-5], N[(1.0 / N[(N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.9e50

   1. Initial program 41.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 88.3%

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

   5. Simplified 88.3%

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

   if -2.9e50 < y.re < 1.2e-5

   1. Initial program 41.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 82.5%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 82.5%

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

   if 1.2e-5 < y.re

   1. Initial program 44.1%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 57.3%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 68.8%

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

   11. Taylor expanded in y.im around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 76.3%

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

   13. Simplified 76.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+50}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{\frac{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+31}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.8e+31)
   (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
   (if (<= y.re 9e-7)
     (/ 1.0 (exp (* y.im (atan2 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 tmp;
	if (y_46_re <= -5.8e+31) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 9e-7) {
		tmp = 1.0 / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 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 tmp;
	if (y_46_re <= -5.8e+31) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 9e-7) {
		tmp = 1.0 / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = 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):
	tmp = 0
	if y_46_re <= -5.8e+31:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 9e-7:
		tmp = 1.0 / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = 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)
	tmp = 0.0
	if (y_46_re <= -5.8e+31)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 9e-7)
		tmp = Float64(1.0 / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = 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)
	tmp = 0.0;
	if (y_46_re <= -5.8e+31)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 9e-7)
		tmp = 1.0 / exp((y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = 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_] := If[LessEqual[y$46$re, -5.8e+31], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9e-7], N[(1.0 / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.8000000000000001e31

   1. Initial program 40.4%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 88.6%

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

   5. Simplified 88.6%

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

   if -5.8000000000000001e31 < y.re < 8.99999999999999959e-7

   1. Initial program 42.2%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 82.3%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 83.1%

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

   7. Simplified 83.1%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 82.3%

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

   11. Taylor expanded in y.re around 0

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

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

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

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

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

       3. atan2-lowering-atan2.f64 81.3%

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

   13. Simplified 81.3%

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

   if 8.99999999999999959e-7 < y.re

   1. Initial program 44.1%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 57.3%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 66.2%

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

   7. Simplified 66.2%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 68.8%

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

   11. Taylor expanded in y.im around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 76.3%

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

   13. Simplified 76.3%

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

Alternative 6: 74.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{if}\;y.im \leq -2.45 \cdot 10^{+61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 55000:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ 1.0 (exp (* y.im (atan2 x.im x.re))))))
   (if (<= y.im -2.45e+61)
     t_0
     (if (<= y.im 55000.0) (pow (hypot x.re x.im) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.45e+61) {
		tmp = t_0;
	} else if (y_46_im <= 55000.0) {
		tmp = pow(hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.45e+61) {
		tmp = t_0;
	} else if (y_46_im <= 55000.0) {
		tmp = Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -2.45e+61:
		tmp = t_0
	elif y_46_im <= 55000.0:
		tmp = math.pow(math.hypot(x_46_re, x_46_im), y_46_re)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / exp(Float64(y_46_im * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_im <= -2.45e+61)
		tmp = t_0;
	elseif (y_46_im <= 55000.0)
		tmp = hypot(x_46_re, x_46_im) ^ y_46_re;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / exp((y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -2.45e+61)
		tmp = t_0;
	elseif (y_46_im <= 55000.0)
		tmp = hypot(x_46_re, x_46_im) ^ y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.45e+61], t$95$0, If[LessEqual[y$46$im, 55000.0], N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.45000000000000013e61 or 55000 < y.im

   1. Initial program 34.9%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 65.9%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 63.4%

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

   7. Simplified 63.4%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 63.5%

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

   11. Taylor expanded in y.re around 0

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

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

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

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

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

       3. atan2-lowering-atan2.f64 68.7%

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

   13. Simplified 68.7%

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

   if -2.45000000000000013e61 < y.im < 55000

   1. Initial program 48.7%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 84.9%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 89.2%

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

   7. Simplified 89.2%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 89.7%

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

   11. Taylor expanded in y.im around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 90.2%

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

   13. Simplified 90.2%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 65.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-5}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))))
   (if (<= y.im -4.6e+76)
     t_0
     (if (<= y.im 8e-5) (pow (hypot x.re x.im) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	double tmp;
	if (y_46_im <= -4.6e+76) {
		tmp = t_0;
	} else if (y_46_im <= 8e-5) {
		tmp = pow(hypot(x_46_re, x_46_im), y_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.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	double tmp;
	if (y_46_im <= -4.6e+76) {
		tmp = t_0;
	} else if (y_46_im <= 8e-5) {
		tmp = Math.pow(Math.hypot(x_46_re, x_46_im), y_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.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	tmp = 0
	if y_46_im <= -4.6e+76:
		tmp = t_0
	elif y_46_im <= 8e-5:
		tmp = math.pow(math.hypot(x_46_re, x_46_im), y_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(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0)
	tmp = 0.0
	if (y_46_im <= -4.6e+76)
		tmp = t_0;
	elseif (y_46_im <= 8e-5)
		tmp = hypot(x_46_re, x_46_im) ^ y_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 = ((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0);
	tmp = 0.0;
	if (y_46_im <= -4.6e+76)
		tmp = t_0;
	elseif (y_46_im <= 8e-5)
		tmp = hypot(x_46_re, x_46_im) ^ y_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[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -4.6e+76], t$95$0, If[LessEqual[y$46$im, 8e-5], N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.60000000000000002e76 or 8.00000000000000065e-5 < y.im

   1. Initial program 33.5%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 66.5%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 63.8%

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

   7. Simplified 63.8%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 63.1%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. frac-2neg N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right)\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

       7. sqrt-pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

       13. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(0 - \color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im\right)\right)\right)\right) \]

       18. atan2-lowering-atan2.f64 51.3%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right)\right) \]

   12. Applied egg-rr 51.3%

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

   13. Taylor expanded in y.im around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{1}\right)\right) \]
   14. Step-by-step derivation

   15. Simplified 44.3%

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

   if -4.60000000000000002e76 < y.im < 8.00000000000000065e-5

   1. Initial program 49.4%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 83.9%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 88.1%

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

   7. Simplified 88.1%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 89.3%

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

   11. Taylor expanded in y.im around 0

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

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

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. unpow2 N/A

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

       5. hypot-define N/A

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

       6. hypot-lowering-hypot.f64 87.0%

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

   13. Simplified 87.0%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.6 \cdot 10^{+76}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-5}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.1% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-136}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-102}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (+ (* x.re x.re) (* x.im x.im)) (/ y.re 2.0))))
   (if (<= y.re -7.5e-136)
     t_0
     (if (<= y.re 2.2e-102) (- 1.0 (* y.im (atan2 x.im x.re))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	double tmp;
	if (y_46_re <= -7.5e-136) {
		tmp = t_0;
	} else if (y_46_re <= 2.2e-102) {
		tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * x_46re) + (x_46im * x_46im)) ** (y_46re / 2.0d0)
    if (y_46re <= (-7.5d-136)) then
        tmp = t_0
    else if (y_46re <= 2.2d-102) then
        tmp = 1.0d0 - (y_46im * atan2(x_46im, x_46re))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0));
	double tmp;
	if (y_46_re <= -7.5e-136) {
		tmp = t_0;
	} else if (y_46_re <= 2.2e-102) {
		tmp = 1.0 - (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.pow(((x_46_re * x_46_re) + (x_46_im * x_46_im)), (y_46_re / 2.0))
	tmp = 0
	if y_46_re <= -7.5e-136:
		tmp = t_0
	elif y_46_re <= 2.2e-102:
		tmp = 1.0 - (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(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)) ^ Float64(y_46_re / 2.0)
	tmp = 0.0
	if (y_46_re <= -7.5e-136)
		tmp = t_0;
	elseif (y_46_re <= 2.2e-102)
		tmp = Float64(1.0 - 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 = ((x_46_re * x_46_re) + (x_46_im * x_46_im)) ^ (y_46_re / 2.0);
	tmp = 0.0;
	if (y_46_re <= -7.5e-136)
		tmp = t_0;
	elseif (y_46_re <= 2.2e-102)
		tmp = 1.0 - (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[Power[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e-136], t$95$0, If[LessEqual[y$46$re, 2.2e-102], N[(1.0 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.5000000000000003e-136 or 2.20000000000000013e-102 < y.re

   1. Initial program 40.7%

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

      1. exp-diff N/A

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

      2. associate-*l/ N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. associate-/r/ N/A

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

      6. exp-diff N/A

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

   3. Simplified 73.6%

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

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(\color{blue}{\cos \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}, \mathsf{/.f64}\left(\mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right), \mathsf{pow.f64}\left(\mathsf{hypot.f64}\left(x.re, x.im\right), y.re\right)\right)\right) \]
   6. Step-by-step derivation

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. hypot-define N/A

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

      7. hypot-lowering-hypot.f64 75.3%

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

   7. Simplified 75.3%

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

   8. Taylor expanded in y.im around 0

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

   10. Simplified 75.2%

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

       1. *-commutative N/A

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

       2. clear-num N/A

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

       3. frac-2neg N/A

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

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

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

       5. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(0 - {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\right)\right), \left(\mathsf{neg}\left(\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) \]

       7. sqrt-pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\left(x.re \cdot x.re + x.im \cdot x.im\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\left(x.re \cdot x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \left(x.im \cdot x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \left(\frac{y.re}{2}\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(\mathsf{neg}\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\right)\right) \]

       13. neg-sub0 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \left(0 - \color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{\left(e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) \]

       16. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)\right)\right) \]

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

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im\right)\right)\right)\right) \]

       18. atan2-lowering-atan2.f64 64.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \mathsf{exp.f64}\left(\mathsf{*.f64}\left(\mathsf{atan2.f64}\left(x.im, x.re\right), y.im\right)\right)\right)\right) \]

   12. Applied egg-rr 64.5%

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

   13. Taylor expanded in y.im around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{pow.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.re, x.re\right), \mathsf{*.f64}\left(x.im, x.im\right)\right), \mathsf{/.f64}\left(y.re, 2\right)\right)\right), \mathsf{\_.f64}\left(0, \color{blue}{1}\right)\right) \]
   14. Step-by-step derivation

   15. Simplified 70.5%

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

   if -7.5000000000000003e-136 < y.re < 2.20000000000000013e-102

   1. Initial program 45.8%

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

   3. Taylor expanded in x.im around 0

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

      1. exp-diff N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. associate-/l* N/A

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

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

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

   5. Simplified 45.3%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. atan2-lowering-atan2.f64 45.3%

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

   8. Simplified 45.3%

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

   9. Taylor expanded in y.im around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. atan2-lowering-atan2.f64 45.7%

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

   11. Simplified 45.7%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-136}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-102}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\left(\frac{y.re}{2}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 31.4% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 2.15 \cdot 10^{-102}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 2.15e-102)
   (- 1.0 (* y.im (atan2 x.im x.re)))
   (+ 1.0 (* y.re (* (log (+ (* x.re x.re) (* x.im x.im))) 0.5)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 2.15e-102) {
		tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
	} else {
		tmp = 1.0 + (y_46_re * (log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= 2.15d-102) then
        tmp = 1.0d0 - (y_46im * atan2(x_46im, x_46re))
    else
        tmp = 1.0d0 + (y_46re * (log(((x_46re * x_46re) + (x_46im * x_46im))) * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 2.15e-102) {
		tmp = 1.0 - (y_46_im * Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = 1.0 + (y_46_re * (Math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= 2.15e-102:
		tmp = 1.0 - (y_46_im * math.atan2(x_46_im, x_46_re))
	else:
		tmp = 1.0 + (y_46_re * (math.log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 2.15e-102)
		tmp = Float64(1.0 - Float64(y_46_im * atan(x_46_im, x_46_re)));
	else
		tmp = Float64(1.0 + Float64(y_46_re * Float64(log(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= 2.15e-102)
		tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
	else
		tmp = 1.0 + (y_46_re * (log(((x_46_re * x_46_re) + (x_46_im * x_46_im))) * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 2.15e-102], N[(1.0 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y$46$re * N[(N[Log[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 2.1499999999999999e-102

   1. Initial program 41.3%

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

   3. Taylor expanded in x.im around 0

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

      1. exp-diff N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. associate-/l* N/A

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

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

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

   5. Simplified 41.7%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. atan2-lowering-atan2.f64 35.4%

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

   8. Simplified 35.4%

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

   9. Taylor expanded in y.im around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. atan2-lowering-atan2.f64 29.5%

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

   11. Simplified 29.5%

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

   if 2.1499999999999999e-102 < y.re

   1. Initial program 44.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 54.1%

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

   5. Simplified 54.1%

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

   6. Taylor expanded in y.re around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 33.8%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 33.8%

      \[\leadsto \color{blue}{1 + y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)} \]
   9. Step-by-step derivation

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), \color{blue}{1}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(y.re, \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)\right), 1\right) \]

      4. pow1/2 N/A

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

      5. +-commutative N/A

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

      6. pow-to-exp N/A

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

      7. rem-log-exp N/A

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

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

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

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

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

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

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

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

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

      12. *-lowering-*.f64 33.8%

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

   10. Applied egg-rr 33.8%

       \[\leadsto \color{blue}{y.re \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot 0.5\right) + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 2.15 \cdot 10^{-102}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \left(\log \left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot 0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 27.7% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \log \left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 1.8e-23)
   (- 1.0 (* y.im (atan2 x.im x.re)))
   (+ 1.0 (* y.re (log (+ x.re (/ (* (* x.im x.im) 0.5) 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 <= 1.8e-23) {
		tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
	} else {
		tmp = 1.0 + (y_46_re * log((x_46_re + (((x_46_im * x_46_im) * 0.5) / 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 <= 1.8d-23) then
        tmp = 1.0d0 - (y_46im * atan2(x_46im, x_46re))
    else
        tmp = 1.0d0 + (y_46re * log((x_46re + (((x_46im * x_46im) * 0.5d0) / 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 <= 1.8e-23) {
		tmp = 1.0 - (y_46_im * Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = 1.0 + (y_46_re * Math.log((x_46_re + (((x_46_im * x_46_im) * 0.5) / 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 <= 1.8e-23:
		tmp = 1.0 - (y_46_im * math.atan2(x_46_im, x_46_re))
	else:
		tmp = 1.0 + (y_46_re * math.log((x_46_re + (((x_46_im * x_46_im) * 0.5) / 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 <= 1.8e-23)
		tmp = Float64(1.0 - Float64(y_46_im * atan(x_46_im, x_46_re)));
	else
		tmp = Float64(1.0 + Float64(y_46_re * log(Float64(x_46_re + Float64(Float64(Float64(x_46_im * x_46_im) * 0.5) / 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 <= 1.8e-23)
		tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
	else
		tmp = 1.0 + (y_46_re * log((x_46_re + (((x_46_im * x_46_im) * 0.5) / x_46_re))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 1.8e-23], N[(1.0 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y$46$re * N[Log[N[(x$46$re + N[(N[(N[(x$46$im * x$46$im), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 1.7999999999999999e-23

   1. Initial program 41.6%

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

   3. Taylor expanded in x.im around 0

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

      1. exp-diff N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. associate-/l* N/A

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

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

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

   5. Simplified 40.9%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. atan2-lowering-atan2.f64 35.2%

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

   8. Simplified 35.2%

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

   9. Taylor expanded in y.im around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. atan2-lowering-atan2.f64 30.6%

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

   11. Simplified 30.6%

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

   if 1.7999999999999999e-23 < y.re

   1. Initial program 44.3%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 58.5%

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

   5. Simplified 58.5%

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

   6. Taylor expanded in y.re around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 30.9%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 30.9%

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

   9. Taylor expanded in x.im around 0

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

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

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

       2. associate-*r/ N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 18.5%

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

   11. Simplified 18.5%

       \[\leadsto 1 + y.re \cdot \log \color{blue}{\left(x.re + \frac{0.5 \cdot \left(x.im \cdot x.im\right)}{x.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 1.8 \cdot 10^{-23}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \log \left(x.re + \frac{\left(x.im \cdot x.im\right) \cdot 0.5}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 27.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 2.6 \cdot 10^{+82}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 2.6e+82)
   (- 1.0 (* y.im (atan2 x.im x.re)))
   (+ 1.0 (* y.re (log (+ x.im (/ (* (* x.re x.re) 0.5) x.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 2.6e+82) {
		tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
	} else {
		tmp = 1.0 + (y_46_re * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= 2.6d+82) then
        tmp = 1.0d0 - (y_46im * atan2(x_46im, x_46re))
    else
        tmp = 1.0d0 + (y_46re * log((x_46im + (((x_46re * x_46re) * 0.5d0) / x_46im))))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 2.6e+82) {
		tmp = 1.0 - (y_46_im * Math.atan2(x_46_im, x_46_re));
	} else {
		tmp = 1.0 + (y_46_re * Math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= 2.6e+82:
		tmp = 1.0 - (y_46_im * math.atan2(x_46_im, x_46_re))
	else:
		tmp = 1.0 + (y_46_re * math.log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 2.6e+82)
		tmp = Float64(1.0 - Float64(y_46_im * atan(x_46_im, x_46_re)));
	else
		tmp = Float64(1.0 + Float64(y_46_re * log(Float64(x_46_im + Float64(Float64(Float64(x_46_re * x_46_re) * 0.5) / x_46_im)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= 2.6e+82)
		tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
	else
		tmp = 1.0 + (y_46_re * log((x_46_im + (((x_46_re * x_46_re) * 0.5) / x_46_im))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 2.6e+82], N[(1.0 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(y$46$re * N[Log[N[(x$46$im + N[(N[(N[(x$46$re * x$46$re), $MachinePrecision] * 0.5), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 2.5999999999999998e82

   1. Initial program 41.5%

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

   3. Taylor expanded in x.im around 0

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

      1. exp-diff N/A

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

      2. *-commutative N/A

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

      3. exp-to-pow N/A

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

      4. associate-/l* N/A

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

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

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

   5. Simplified 40.4%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

      7. atan2-lowering-atan2.f64 33.2%

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

   8. Simplified 33.2%

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

   9. Taylor expanded in y.im around 0

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

       1. neg-mul-1 N/A

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

       2. unsub-neg N/A

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

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

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

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

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

       5. atan2-lowering-atan2.f64 28.2%

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

   11. Simplified 28.2%

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

   if 2.5999999999999998e82 < y.re

   1. Initial program 45.8%

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

   3. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

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

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

      6. unpow2 N/A

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

      7. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

      8. hypot-define N/A

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

      9. hypot-lowering-hypot.f64 58.4%

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

   5. Simplified 58.4%

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

   6. Taylor expanded in y.re around 0

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \color{blue}{\left(y.re \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\left({x.im}^{2}\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      6. unpow2 N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left({x.re}^{2}\right)\right)\right)\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \left(x.re \cdot x.re\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 27.6%

         \[\leadsto \mathsf{+.f64}\left(1, \mathsf{*.f64}\left(y.re, \mathsf{log.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(x.im, x.im\right), \mathsf{*.f64}\left(x.re, x.re\right)\right)\right)\right)\right)\right) \]

   8. Simplified 27.6%

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

   9. Taylor expanded in x.re around 0

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

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

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

       2. associate-*r/ N/A

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

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

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

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

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

       5. unpow2 N/A

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

       6. *-lowering-*.f64 12.3%

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

   11. Simplified 12.3%

       \[\leadsto 1 + y.re \cdot \log \color{blue}{\left(x.im + \frac{0.5 \cdot \left(x.re \cdot x.re\right)}{x.im}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 2.6 \cdot 10^{+82}:\\ \;\;\;\;1 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;1 + y.re \cdot \log \left(x.im + \frac{\left(x.re \cdot x.re\right) \cdot 0.5}{x.im}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.8% accurate, 7.8× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0 - (y_46im * atan2(x_46im, x_46re))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 - (y_46_im * Math.atan2(x_46_im, x_46_re));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 - (y_46_im * math.atan2(x_46_im, x_46_re))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 - Float64(y_46_im * atan(x_46_im, x_46_re)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 - (y_46_im * atan2(x_46_im, x_46_re));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in x.im around 0

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

   1. exp-diff N/A

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

   2. *-commutative N/A

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

   3. exp-to-pow N/A

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

   4. associate-/l* N/A

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

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

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

5. Simplified 36.8%

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

6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

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

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

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

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

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

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

   7. atan2-lowering-atan2.f64 28.3%

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

8. Simplified 28.3%

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

9. Taylor expanded in y.im around 0

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

    1. neg-mul-1 N/A

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

    2. unsub-neg N/A

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

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

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

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

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

    5. atan2-lowering-atan2.f64 23.4%

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

11. Simplified 23.4%

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

Alternative 13: 25.7% accurate, 829.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 42.3%

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

3. Taylor expanded in y.im around 0

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

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

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

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

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

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

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

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

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \left({\left(\sqrt{{x.im}^{2} + \color{blue}{{x.re}^{2}}}\right)}^{y.re}\right)\right) \]

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

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

   6. unpow2 N/A

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

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{cos.f64}\left(\mathsf{*.f64}\left(y.re, \mathsf{atan2.f64}\left(x.im, x.re\right)\right)\right), \mathsf{pow.f64}\left(\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right), y.re\right)\right) \]

   8. hypot-define N/A

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

   9. hypot-lowering-hypot.f64 56.7%

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

5. Simplified 56.7%

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

6. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{1} \]
7. Step-by-step derivation

8. Simplified 23.2%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(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)))))