powComplex, real part

Percentage Accurate: 41.4% → 77.4%

Time: 16.6s

Alternatives: 11

Speedup: 6.0×

• 35.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.4% 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: 77.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq -850000000:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-20}:\\ \;\;\;\;e^{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= y.re -850000000.0)
     (*
      (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
      (cos (* y.re (atan2 x.im x.re))))
     (if (<= y.re 2e-20)
       (exp (- t_0))
       (pow (pow (fma x.im x.im (* x.re x.re)) (* y.re 0.25)) 2.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= -850000000.0) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 2e-20) {
		tmp = exp(-t_0);
	} else {
		tmp = pow(pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), (y_46_re * 0.25)), 2.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (y_46_re <= -850000000.0)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 2e-20)
		tmp = exp(Float64(-t_0));
	else
		tmp = (fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ Float64(y_46_re * 0.25)) ^ 2.0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -850000000.0], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2e-20], N[Exp[(-t$95$0)], $MachinePrecision], N[Power[N[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(y$46$re * 0.25), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.5e8

   1. Initial program 41.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 82.3

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

   5. Simplified 82.3%

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

   if -8.5e8 < y.re < 1.99999999999999989e-20

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 83.0

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

   8. Simplified 83.0%

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

   if 1.99999999999999989e-20 < y.re

   1. Initial program 29.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 53.6

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

   5. Simplified 53.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. sqrt-pow2 N/A

         \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

         \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}}^{\left(\frac{y.re}{2}\right)} \]

      9. sqr-pow N/A

         \[\leadsto 1 \cdot \color{blue}{\left({\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)} \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\right)} \]

      10. pow-prod-down N/A

          \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}} \]

      11. div-inv N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\color{blue}{\left(\frac{y.re}{2} \cdot \frac{1}{2}\right)}} \]

      12. div-inv N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\color{blue}{\left(y.re \cdot \frac{1}{2}\right)} \cdot \frac{1}{2}\right)} \]

      13. metadata-eval N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\left(y.re \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{1}{2}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\color{blue}{\left(y.re \cdot \frac{1}{2}\right)} \cdot \frac{1}{2}\right)} \]

      15. metadata-eval N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\left(y.re \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      16. *-commutative N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}} \]

      17. lift-*.f64 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}} \]

      18. pow-prod-down N/A

          \[\leadsto 1 \cdot \color{blue}{\left({\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)} \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}\right)} \]

      19. lift-pow.f64 N/A

          \[\leadsto 1 \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}} \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto 1 \cdot \left({\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)} \cdot \color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}}\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -850000000:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-20}:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 75.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -850000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-20}:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -850000000.0)
   (pow (fma 0.5 (/ (* x.re x.re) x.im) x.im) y.re)
   (if (<= y.re 2e-20)
     (exp (- (* (atan2 x.im x.re) y.im)))
     (pow (pow (fma x.im x.im (* x.re x.re)) (* y.re 0.25)) 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 <= -850000000.0) {
		tmp = pow(fma(0.5, ((x_46_re * x_46_re) / x_46_im), x_46_im), y_46_re);
	} else if (y_46_re <= 2e-20) {
		tmp = exp(-(atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = pow(pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), (y_46_re * 0.25)), 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 <= -850000000.0)
		tmp = fma(0.5, Float64(Float64(x_46_re * x_46_re) / x_46_im), x_46_im) ^ y_46_re;
	elseif (y_46_re <= 2e-20)
		tmp = exp(Float64(-Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = (fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ Float64(y_46_re * 0.25)) ^ 2.0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -850000000.0], N[Power[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 2e-20], N[Exp[(-N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision])], $MachinePrecision], N[Power[N[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(y$46$re * 0.25), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.5e8

   1. Initial program 41.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 74.2

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

   5. Simplified 74.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 76.9%

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

   9. Taylor expanded in x.re around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

          \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \frac{{x.re}^{2}}{x.im}, x.im\right)\right)}}^{y.re} \]

       3. lower-/.f64 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{x.re}^{2}}{x.im}}, x.im\right)\right)}^{y.re} \]

       4. unpow2 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.re \cdot x.re}}{x.im}, x.im\right)\right)}^{y.re} \]

       5. lower-*.f64 78.3

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

   11. Simplified 78.3%

       \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}}^{y.re} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-fma.f64 N/A

          \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}}^{y.re} \]

       4. lift-pow.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re}} \]

       5. *-lft-identity 78.3

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

   13. Applied egg-rr 78.3%

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

   if -8.5e8 < y.re < 1.99999999999999989e-20

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 83.0

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

   8. Simplified 83.0%

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

   if 1.99999999999999989e-20 < y.re

   1. Initial program 29.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 53.6

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

   5. Simplified 53.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. sqrt-pow2 N/A

         \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

         \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}}^{\left(\frac{y.re}{2}\right)} \]

      9. sqr-pow N/A

         \[\leadsto 1 \cdot \color{blue}{\left({\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)} \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}\right)} \]

      10. pow-prod-down N/A

          \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{\frac{y.re}{2}}{2}\right)}} \]

      11. div-inv N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\color{blue}{\left(\frac{y.re}{2} \cdot \frac{1}{2}\right)}} \]

      12. div-inv N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\color{blue}{\left(y.re \cdot \frac{1}{2}\right)} \cdot \frac{1}{2}\right)} \]

      13. metadata-eval N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\left(y.re \cdot \color{blue}{\frac{1}{2}}\right) \cdot \frac{1}{2}\right)} \]

      14. lift-*.f64 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\color{blue}{\left(y.re \cdot \frac{1}{2}\right)} \cdot \frac{1}{2}\right)} \]

      15. metadata-eval N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\left(y.re \cdot \frac{1}{2}\right) \cdot \color{blue}{\frac{1}{2}}\right)} \]

      16. *-commutative N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}} \]

      17. lift-*.f64 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\color{blue}{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}} \]

      18. pow-prod-down N/A

          \[\leadsto 1 \cdot \color{blue}{\left({\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)} \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}\right)} \]

      19. lift-pow.f64 N/A

          \[\leadsto 1 \cdot \left(\color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}} \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}\right) \]

      20. lift-pow.f64 N/A

          \[\leadsto 1 \cdot \left({\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)} \cdot \color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{1}{2} \cdot \left(y.re \cdot \frac{1}{2}\right)\right)}}\right) \]

   10. Applied egg-rr 68.7%

       \[\leadsto \color{blue}{{\left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}\right)}^{2}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -850000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-20}:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -880000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-20}:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -880000000.0)
   (pow (fma 0.5 (/ (* x.re x.re) x.im) x.im) y.re)
   (if (<= y.re 2e-20)
     (exp (- (* (atan2 x.im x.re) y.im)))
     (pow (fma x.im x.im (* x.re x.re)) (* y.re 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 <= -880000000.0) {
		tmp = pow(fma(0.5, ((x_46_re * x_46_re) / x_46_im), x_46_im), y_46_re);
	} else if (y_46_re <= 2e-20) {
		tmp = exp(-(atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), (y_46_re * 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 <= -880000000.0)
		tmp = fma(0.5, Float64(Float64(x_46_re * x_46_re) / x_46_im), x_46_im) ^ y_46_re;
	elseif (y_46_re <= 2e-20)
		tmp = exp(Float64(-Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ Float64(y_46_re * 0.5);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -880000000.0], N[Power[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] + x$46$im), $MachinePrecision], y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 2e-20], N[Exp[(-N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision])], $MachinePrecision], N[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.8e8

   1. Initial program 41.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 74.2

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

   5. Simplified 74.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 76.9%

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

   9. Taylor expanded in x.re around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

          \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \frac{{x.re}^{2}}{x.im}, x.im\right)\right)}}^{y.re} \]

       3. lower-/.f64 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{x.re}^{2}}{x.im}}, x.im\right)\right)}^{y.re} \]

       4. unpow2 N/A

          \[\leadsto 1 \cdot {\left(\mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.re \cdot x.re}}{x.im}, x.im\right)\right)}^{y.re} \]

       5. lower-*.f64 78.3

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

   11. Simplified 78.3%

       \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}}^{y.re} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-fma.f64 N/A

          \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}}^{y.re} \]

       4. lift-pow.f64 N/A

          \[\leadsto 1 \cdot \color{blue}{{\left(\mathsf{fma}\left(\frac{1}{2}, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re}} \]

       5. *-lft-identity 78.3

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

   13. Applied egg-rr 78.3%

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

   if -8.8e8 < y.re < 1.99999999999999989e-20

   1. Initial program 43.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 58.2

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

   5. Simplified 58.2%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 83.0

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

   8. Simplified 83.0%

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

   if 1.99999999999999989e-20 < y.re

   1. Initial program 29.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 53.6

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

   5. Simplified 53.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 68.7%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. lift-pow.f64 N/A

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

      7. *-lft-identity 68.7

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

      8. lift-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-pow2 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

          \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}}^{\left(\frac{y.re}{2}\right)} \]

      16. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

   10. Applied egg-rr 68.7%

       \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -880000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(0.5, \frac{x.re \cdot x.re}{x.im}, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-20}:\\ \;\;\;\;e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{-50}:\\ \;\;\;\;{\left(x.re \cdot x.re\right)}^{\left(y.re \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -3.5e-6)
   (pow (- x.im) y.re)
   (if (<= x.im 2.4e-50)
     (pow (* x.re x.re) (* y.re 0.5))
     (pow (* x.im x.im) (* y.re 0.5)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -3.5e-6) {
		tmp = pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 2.4e-50) {
		tmp = pow((x_46_re * x_46_re), (y_46_re * 0.5));
	} else {
		tmp = pow((x_46_im * x_46_im), (y_46_re * 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 (x_46im <= (-3.5d-6)) then
        tmp = -x_46im ** y_46re
    else if (x_46im <= 2.4d-50) then
        tmp = (x_46re * x_46re) ** (y_46re * 0.5d0)
    else
        tmp = (x_46im * x_46im) ** (y_46re * 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 (x_46_im <= -3.5e-6) {
		tmp = Math.pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 2.4e-50) {
		tmp = Math.pow((x_46_re * x_46_re), (y_46_re * 0.5));
	} else {
		tmp = Math.pow((x_46_im * x_46_im), (y_46_re * 0.5));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -3.5e-6:
		tmp = math.pow(-x_46_im, y_46_re)
	elif x_46_im <= 2.4e-50:
		tmp = math.pow((x_46_re * x_46_re), (y_46_re * 0.5))
	else:
		tmp = math.pow((x_46_im * x_46_im), (y_46_re * 0.5))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -3.5e-6)
		tmp = Float64(-x_46_im) ^ y_46_re;
	elseif (x_46_im <= 2.4e-50)
		tmp = Float64(x_46_re * x_46_re) ^ Float64(y_46_re * 0.5);
	else
		tmp = Float64(x_46_im * x_46_im) ^ Float64(y_46_re * 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 (x_46_im <= -3.5e-6)
		tmp = -x_46_im ^ y_46_re;
	elseif (x_46_im <= 2.4e-50)
		tmp = (x_46_re * x_46_re) ^ (y_46_re * 0.5);
	else
		tmp = (x_46_im * x_46_im) ^ (y_46_re * 0.5);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -3.5e-6], N[Power[(-x$46$im), y$46$re], $MachinePrecision], If[LessEqual[x$46$im, 2.4e-50], N[Power[N[(x$46$re * x$46$re), $MachinePrecision], N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision], N[Power[N[(x$46$im * x$46$im), $MachinePrecision], N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.49999999999999995e-6

   1. Initial program 28.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   5. Simplified 50.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 60.7%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 67.8

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

   11. Simplified 67.8%

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

   if -3.49999999999999995e-6 < x.im < 2.40000000000000002e-50

   1. Initial program 47.8%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.4

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

   5. Simplified 60.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. lift-pow.f64 N/A

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

      7. *-lft-identity 63.8

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

      8. lift-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-pow2 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

          \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}}^{\left(\frac{y.re}{2}\right)} \]

      16. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

   10. Applied egg-rr 63.8%

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

   11. Taylor expanded in x.im around 0

       \[\leadsto {\color{blue}{\left({x.re}^{2}\right)}}^{\left(y.re \cdot \frac{1}{2}\right)} \]
   12. Step-by-step derivation

       1. unpow2 N/A

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

       2. lower-*.f64 62.0

          \[\leadsto {\color{blue}{\left(x.re \cdot x.re\right)}}^{\left(y.re \cdot 0.5\right)} \]

   13. Simplified 62.0%

       \[\leadsto {\color{blue}{\left(x.re \cdot x.re\right)}}^{\left(y.re \cdot 0.5\right)} \]

   if 2.40000000000000002e-50 < x.im

   1. Initial program 35.2%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.0

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

   5. Simplified 52.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 56.0%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 55.1

          \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   11. Simplified 55.1%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   12. Step-by-step derivation

       1. sqr-pow N/A

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

       2. pow-prod-down N/A

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

       3. lift-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. div-inv N/A

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

       6. metadata-eval N/A

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

       7. lift-*.f64 56.3

          \[\leadsto {\left(x.im \cdot x.im\right)}^{\color{blue}{\left(y.re \cdot 0.5\right)}} \]

   13. Applied egg-rr 56.3%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{-50}:\\ \;\;\;\;{\left(x.re \cdot x.re\right)}^{\left(y.re \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.6 \cdot 10^{-40}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 0.02:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -1.6e-40)
   (pow (- x.re) y.re)
   (if (<= x.re 0.02) (pow (* x.im x.im) (* y.re 0.5)) (pow x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1.6e-40) {
		tmp = pow(-x_46_re, y_46_re);
	} else if (x_46_re <= 0.02) {
		tmp = pow((x_46_im * x_46_im), (y_46_re * 0.5));
	} else {
		tmp = pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-1.6d-40)) then
        tmp = -x_46re ** y_46re
    else if (x_46re <= 0.02d0) then
        tmp = (x_46im * x_46im) ** (y_46re * 0.5d0)
    else
        tmp = x_46re ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1.6e-40) {
		tmp = Math.pow(-x_46_re, y_46_re);
	} else if (x_46_re <= 0.02) {
		tmp = Math.pow((x_46_im * x_46_im), (y_46_re * 0.5));
	} else {
		tmp = Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -1.6e-40:
		tmp = math.pow(-x_46_re, y_46_re)
	elif x_46_re <= 0.02:
		tmp = math.pow((x_46_im * x_46_im), (y_46_re * 0.5))
	else:
		tmp = math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.6e-40)
		tmp = Float64(-x_46_re) ^ y_46_re;
	elseif (x_46_re <= 0.02)
		tmp = Float64(x_46_im * x_46_im) ^ Float64(y_46_re * 0.5);
	else
		tmp = x_46_re ^ y_46_re;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.6e-40)
		tmp = -x_46_re ^ y_46_re;
	elseif (x_46_re <= 0.02)
		tmp = (x_46_im * x_46_im) ^ (y_46_re * 0.5);
	else
		tmp = x_46_re ^ y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -1.6e-40], N[Power[(-x$46$re), y$46$re], $MachinePrecision], If[LessEqual[x$46$re, 0.02], N[Power[N[(x$46$im * x$46$im), $MachinePrecision], N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision], N[Power[x$46$re, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.60000000000000001e-40

   1. Initial program 30.8%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.8

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

   5. Simplified 52.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 55.4%

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

   9. Taylor expanded in x.re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}}^{y.re} \]

       2. lower-neg.f64 56.2

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

   11. Simplified 56.2%

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

   if -1.60000000000000001e-40 < x.re < 0.0200000000000000004

   1. Initial program 44.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 55.0

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

   5. Simplified 55.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 63.6%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 49.7

          \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   11. Simplified 49.7%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   12. Step-by-step derivation

       1. sqr-pow N/A

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

       2. pow-prod-down N/A

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

       3. lift-*.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. div-inv N/A

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

       6. metadata-eval N/A

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

       7. lift-*.f64 62.1

          \[\leadsto {\left(x.im \cdot x.im\right)}^{\color{blue}{\left(y.re \cdot 0.5\right)}} \]

   13. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{{\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}} \]

   if 0.0200000000000000004 < x.re

   1. Initial program 37.9%

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

   3. Taylor expanded in y.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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 58.7

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

   5. Simplified 58.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 61.8%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 65.1

          \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   11. Simplified 65.1%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.6 \cdot 10^{-40}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 0.02:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.2 \cdot 10^{+182}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -1.2e+182)
   (pow (- x.im) y.re)
   (pow (fma x.im x.im (* x.re x.re)) (* y.re 0.5))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1.2e+182) {
		tmp = pow(-x_46_im, y_46_re);
	} else {
		tmp = pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), (y_46_re * 0.5));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.2e+182)
		tmp = Float64(-x_46_im) ^ y_46_re;
	else
		tmp = fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ Float64(y_46_re * 0.5);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -1.2e+182], N[Power[(-x$46$im), y$46$re], $MachinePrecision], N[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.20000000000000005e182

   1. Initial program 0.0%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 29.4

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

   5. Simplified 29.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 38.8%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 63.2

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

   11. Simplified 63.2%

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

   if -1.20000000000000005e182 < x.im

   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 59.0

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

   5. Simplified 59.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 63.8%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. sqr-pow N/A

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

      5. sqr-pow N/A

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

      6. lift-pow.f64 N/A

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

      7. *-lft-identity 63.8

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

      8. lift-pow.f64 N/A

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

      9. lift-sqrt.f64 N/A

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

      10. sqrt-pow2 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. lift-fma.f64 N/A

          \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}}^{\left(\frac{y.re}{2}\right)} \]

      16. lower-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(\frac{y.re}{2}\right)}} \]

   10. Applied egg-rr 63.8%

       \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.2 \cdot 10^{+182}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq -4.15 \cdot 10^{-162}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{elif}\;x.im \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -2.9e-9)
   (pow (- x.im) y.re)
   (if (<= x.im -4.15e-162)
     (pow x.re y.re)
     (if (<= x.im 2.7e-33) (pow (- x.re) y.re) (pow x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.9e-9) {
		tmp = pow(-x_46_im, y_46_re);
	} else if (x_46_im <= -4.15e-162) {
		tmp = pow(x_46_re, y_46_re);
	} else if (x_46_im <= 2.7e-33) {
		tmp = pow(-x_46_re, y_46_re);
	} else {
		tmp = pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-2.9d-9)) then
        tmp = -x_46im ** y_46re
    else if (x_46im <= (-4.15d-162)) then
        tmp = x_46re ** y_46re
    else if (x_46im <= 2.7d-33) then
        tmp = -x_46re ** y_46re
    else
        tmp = x_46im ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.9e-9) {
		tmp = Math.pow(-x_46_im, y_46_re);
	} else if (x_46_im <= -4.15e-162) {
		tmp = Math.pow(x_46_re, y_46_re);
	} else if (x_46_im <= 2.7e-33) {
		tmp = Math.pow(-x_46_re, y_46_re);
	} else {
		tmp = Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -2.9e-9:
		tmp = math.pow(-x_46_im, y_46_re)
	elif x_46_im <= -4.15e-162:
		tmp = math.pow(x_46_re, y_46_re)
	elif x_46_im <= 2.7e-33:
		tmp = math.pow(-x_46_re, y_46_re)
	else:
		tmp = math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -2.9e-9)
		tmp = Float64(-x_46_im) ^ y_46_re;
	elseif (x_46_im <= -4.15e-162)
		tmp = x_46_re ^ y_46_re;
	elseif (x_46_im <= 2.7e-33)
		tmp = Float64(-x_46_re) ^ y_46_re;
	else
		tmp = 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 (x_46_im <= -2.9e-9)
		tmp = -x_46_im ^ y_46_re;
	elseif (x_46_im <= -4.15e-162)
		tmp = x_46_re ^ y_46_re;
	elseif (x_46_im <= 2.7e-33)
		tmp = -x_46_re ^ y_46_re;
	else
		tmp = 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[x$46$im, -2.9e-9], N[Power[(-x$46$im), y$46$re], $MachinePrecision], If[LessEqual[x$46$im, -4.15e-162], N[Power[x$46$re, y$46$re], $MachinePrecision], If[LessEqual[x$46$im, 2.7e-33], N[Power[(-x$46$re), y$46$re], $MachinePrecision], N[Power[x$46$im, y$46$re], $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -2.89999999999999991e-9

   1. Initial program 28.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   5. Simplified 50.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 60.7%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 67.8

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

   11. Simplified 67.8%

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

   if -2.89999999999999991e-9 < x.im < -4.1499999999999999e-162

   1. Initial program 58.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 68.9%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 65.2

          \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   11. Simplified 65.2%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   if -4.1499999999999999e-162 < x.im < 2.7000000000000001e-33

   1. Initial program 45.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 59.7

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

   5. Simplified 59.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 60.6%

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

   9. Taylor expanded in x.re around -inf

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

       1. mul-1-neg N/A

          \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{neg}\left(x.re\right)\right)}}^{y.re} \]

       2. lower-neg.f64 56.7

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

   11. Simplified 56.7%

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

   if 2.7000000000000001e-33 < x.im

   1. Initial program 32.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 52.8

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

   5. Simplified 52.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 57.0%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 54.6

          \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   11. Simplified 54.6%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.9 \cdot 10^{-9}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq -4.15 \cdot 10^{-162}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{elif}\;x.im \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -3.25 \cdot 10^{-6}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 2.2 \cdot 10^{-195}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -3.25e-6)
   (pow (- x.im) y.re)
   (if (<= x.im 2.2e-195) (pow x.re y.re) (pow x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -3.25e-6) {
		tmp = pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 2.2e-195) {
		tmp = pow(x_46_re, y_46_re);
	} else {
		tmp = pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-3.25d-6)) then
        tmp = -x_46im ** y_46re
    else if (x_46im <= 2.2d-195) then
        tmp = x_46re ** y_46re
    else
        tmp = x_46im ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -3.25e-6) {
		tmp = Math.pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 2.2e-195) {
		tmp = Math.pow(x_46_re, y_46_re);
	} else {
		tmp = Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -3.25e-6:
		tmp = math.pow(-x_46_im, y_46_re)
	elif x_46_im <= 2.2e-195:
		tmp = math.pow(x_46_re, y_46_re)
	else:
		tmp = math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -3.25e-6)
		tmp = Float64(-x_46_im) ^ y_46_re;
	elseif (x_46_im <= 2.2e-195)
		tmp = x_46_re ^ y_46_re;
	else
		tmp = 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 (x_46_im <= -3.25e-6)
		tmp = -x_46_im ^ y_46_re;
	elseif (x_46_im <= 2.2e-195)
		tmp = x_46_re ^ y_46_re;
	else
		tmp = 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[x$46$im, -3.25e-6], N[Power[(-x$46$im), y$46$re], $MachinePrecision], If[LessEqual[x$46$im, 2.2e-195], N[Power[x$46$re, y$46$re], $MachinePrecision], N[Power[x$46$im, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.2499999999999998e-6

   1. Initial program 28.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 50.7

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

   5. Simplified 50.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 60.7%

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

   9. Taylor expanded in x.im around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 67.8

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

   11. Simplified 67.8%

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

   if -3.2499999999999998e-6 < x.im < 2.20000000000000005e-195

   1. Initial program 45.4%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 60.3

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

   5. Simplified 60.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 63.8%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 52.2

          \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   11. Simplified 52.2%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   if 2.20000000000000005e-195 < x.im

   1. Initial program 40.3%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 54.2

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

   5. Simplified 54.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 57.9%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 54.1

          \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   11. Simplified 54.1%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.25 \cdot 10^{-6}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 2.2 \cdot 10^{-195}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 51.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.0034:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 9.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -0.0034) (pow x.re y.re) (if (<= y.re 9.5) 1.0 (pow x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.0034) {
		tmp = pow(x_46_re, y_46_re);
	} else if (y_46_re <= 9.5) {
		tmp = 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-0.0034d0)) then
        tmp = x_46re ** y_46re
    else if (y_46re <= 9.5d0) then
        tmp = 1.0d0
    else
        tmp = x_46im ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -0.0034) {
		tmp = Math.pow(x_46_re, y_46_re);
	} else if (y_46_re <= 9.5) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -0.0034:
		tmp = math.pow(x_46_re, y_46_re)
	elif y_46_re <= 9.5:
		tmp = 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -0.0034)
		tmp = x_46_re ^ y_46_re;
	elseif (y_46_re <= 9.5)
		tmp = 1.0;
	else
		tmp = 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 <= -0.0034)
		tmp = x_46_re ^ y_46_re;
	elseif (y_46_re <= 9.5)
		tmp = 1.0;
	else
		tmp = 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, -0.0034], N[Power[x$46$re, y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 9.5], 1.0, N[Power[x$46$im, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.00339999999999999981

   1. Initial program 40.8%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 73.9

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

   5. Simplified 73.9%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 77.9%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto \color{blue}{{x.re}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 54.4

          \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   11. Simplified 54.4%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \]

   if -0.00339999999999999981 < y.re < 9.5

   1. Initial program 45.0%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 45.0

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

   5. Simplified 45.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 48.7%

      \[\leadsto \color{blue}{1} \]

   if 9.5 < y.re

   1. Initial program 25.8%

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

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 51.6

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

   5. Simplified 51.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 69.8%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 56.4

          \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   11. Simplified 56.4%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5600000000:\\ \;\;\;\;{x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 9.5:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5600000000.0)
   (pow x.im y.re)
   (if (<= y.re 9.5) 1.0 (pow x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5600000000.0) {
		tmp = pow(x_46_im, y_46_re);
	} else if (y_46_re <= 9.5) {
		tmp = 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-5600000000.0d0)) then
        tmp = x_46im ** y_46re
    else if (y_46re <= 9.5d0) then
        tmp = 1.0d0
    else
        tmp = x_46im ** y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5600000000.0) {
		tmp = Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= 9.5) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -5600000000.0:
		tmp = math.pow(x_46_im, y_46_re)
	elif y_46_re <= 9.5:
		tmp = 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5600000000.0)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 9.5)
		tmp = 1.0;
	else
		tmp = 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 <= -5600000000.0)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 9.5)
		tmp = 1.0;
	else
		tmp = 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, -5600000000.0], N[Power[x$46$im, y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 9.5], 1.0, N[Power[x$46$im, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.6e9 or 9.5 < y.re

   1. Initial program 34.1%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 63.2

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

   5. Simplified 63.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 73.3%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   10. Step-by-step derivation

       1. lower-pow.f64 52.7

          \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   11. Simplified 52.7%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]

   if -5.6e9 < y.re < 9.5

   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. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 46.0

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

   5. Simplified 46.0%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{1} \]
   7. Step-by-step derivation

   8. Simplified 47.3%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 26.3% accurate, 680.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = 1.0d0
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 38.8%

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

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 55.3

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

5. Simplified 55.3%

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

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 2024207 
(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)))))