powComplex, real part

Percentage Accurate: 39.8% → 77.4%

Time: 16.4s

Alternatives: 9

Speedup: 6.0×

• 36.0% 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: 39.8% 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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2.8:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.4e-8)
   (*
    (exp
     (-
      (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
      (* (atan2 x.im x.re) y.im)))
    (cos (* y.re (atan2 x.im x.re))))
   (if (<= y.re 2.8)
     (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 <= -1.4e-8) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 2.8) {
		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 <= -1.4e-8)
		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))))) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 2.8)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-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, -1.4e-8], 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] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $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, 2.8], 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 < -1.4e-8

   1. Initial program 47.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. atan2-lowering-atan2.f64 85.4

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

   5. Simplified 85.4%

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

   if -1.4e-8 < y.re < 2.7999999999999998

   1. Initial program 34.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.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. atan2-lowering-atan2.f64 49.7

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

   5. Simplified 49.7%

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

   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. exp-lowering-exp.f64 N/A

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

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

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

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

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

      4. atan2-lowering-atan2.f64 86.3

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

   8. Simplified 86.3%

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

   if 2.7999999999999998 < y.re

   1. Initial program 30.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 65.2

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

   5. Simplified 65.2%

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

   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 79.5%

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

      1. *-lft-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 79.5

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

   10. Applied egg-rr 79.5%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2.8:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.7% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{\left(y.re \cdot 0.25\right)}\\ \mathbf{elif}\;y.re \leq 6.8:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma x.im x.im (* x.re x.re))))
   (if (<= y.re -1.4e-8)
     (pow (* t_0 t_0) (* y.re 0.25))
     (if (<= y.re 6.8)
       (exp (* (atan2 x.im x.re) (- y.im)))
       (pow t_0 (* y.re 0.5))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_im, x_46_im, (x_46_re * x_46_re));
	double tmp;
	if (y_46_re <= -1.4e-8) {
		tmp = pow((t_0 * t_0), (y_46_re * 0.25));
	} else if (y_46_re <= 6.8) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = pow(t_0, (y_46_re * 0.5));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_re <= -1.4e-8)
		tmp = Float64(t_0 * t_0) ^ Float64(y_46_re * 0.25);
	elseif (y_46_re <= 6.8)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = t_0 ^ Float64(y_46_re * 0.5);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.4e-8], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], N[(y$46$re * 0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 6.8], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[Power[t$95$0, N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.4e-8

   1. Initial program 47.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 81.1

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

   5. Simplified 81.1%

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

   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 81.7%

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

      1. *-lft-identity N/A

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

      2. pow1/2 N/A

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

      3. +-commutative N/A

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

      4. pow-unpow N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

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

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

      18. metadata-eval 82.5

          \[\leadsto {\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right) \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot \color{blue}{0.25}\right)} \]

   10. Applied egg-rr 82.5%

       \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right) \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}} \]

   if -1.4e-8 < y.re < 6.79999999999999982

   1. Initial program 34.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.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. atan2-lowering-atan2.f64 49.7

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

   5. Simplified 49.7%

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

   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. exp-lowering-exp.f64 N/A

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

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

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

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

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

      4. atan2-lowering-atan2.f64 86.3

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

   8. Simplified 86.3%

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

   if 6.79999999999999982 < y.re

   1. Initial program 30.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 65.2

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

   5. Simplified 65.2%

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

   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 79.5%

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

      1. *-lft-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 79.5

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

   10. Applied egg-rr 79.5%

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

4. Final simplification 83.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right) \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}\\ \mathbf{elif}\;y.re \leq 6.8:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{-10}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{\left(y.re \cdot 0.25\right)}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{t\_0}^{\left(y.re \cdot 0.5\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma x.im x.im (* x.re x.re))))
   (if (<= y.re -1.8e-10)
     (pow (* t_0 t_0) (* y.re 0.25))
     (if (<= y.re 1.1e-59) 1.0 (pow t_0 (* y.re 0.5))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = fma(x_46_im, x_46_im, (x_46_re * x_46_re));
	double tmp;
	if (y_46_re <= -1.8e-10) {
		tmp = pow((t_0 * t_0), (y_46_re * 0.25));
	} else if (y_46_re <= 1.1e-59) {
		tmp = 1.0;
	} else {
		tmp = pow(t_0, (y_46_re * 0.5));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))
	tmp = 0.0
	if (y_46_re <= -1.8e-10)
		tmp = Float64(t_0 * t_0) ^ Float64(y_46_re * 0.25);
	elseif (y_46_re <= 1.1e-59)
		tmp = 1.0;
	else
		tmp = t_0 ^ Float64(y_46_re * 0.5);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.8e-10], N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], N[(y$46$re * 0.25), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.1e-59], 1.0, N[Power[t$95$0, N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.8e-10

   1. Initial program 47.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 81.1

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

   5. Simplified 81.1%

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

   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 81.7%

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

      1. *-lft-identity N/A

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

      2. pow1/2 N/A

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

      3. +-commutative N/A

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

      4. pow-unpow N/A

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

      5. sqr-pow N/A

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

      6. pow-prod-down N/A

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

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

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

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

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

      9. +-commutative N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

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

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

      12. +-commutative N/A

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

      13. accelerator-lowering-fma.f64 N/A

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

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

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

      15. *-commutative N/A

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

      16. associate-/l* N/A

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

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

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

      18. metadata-eval 82.5

          \[\leadsto {\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right) \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot \color{blue}{0.25}\right)} \]

   10. Applied egg-rr 82.5%

       \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right) \cdot \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.25\right)}} \]

   if -1.8e-10 < y.re < 1.0999999999999999e-59

   1. Initial program 34.8%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 37.0

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

   5. Simplified 37.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 55.1%

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

   if 1.0999999999999999e-59 < y.re

   1. Initial program 30.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 60.6

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

   5. Simplified 60.6%

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

   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 72.9%

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

      1. *-lft-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 72.9

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

   10. Applied egg-rr 72.9%

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

Alternative 4: 61.4% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\left(y.re \cdot 0.5\right)}\\ \mathbf{if}\;y.re \leq -5.2 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-59}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (fma x.im x.im (* x.re x.re)) (* y.re 0.5))))
   (if (<= y.re -5.2e-10) t_0 (if (<= y.re 1.55e-59) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), (y_46_re * 0.5));
	double tmp;
	if (y_46_re <= -5.2e-10) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e-59) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ Float64(y_46_re * 0.5)
	tmp = 0.0
	if (y_46_re <= -5.2e-10)
		tmp = t_0;
	elseif (y_46_re <= 1.55e-59)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[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]}, If[LessEqual[y$46$re, -5.2e-10], t$95$0, If[LessEqual[y$46$re, 1.55e-59], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.19999999999999962e-10 or 1.55e-59 < y.re

   1. Initial program 38.3%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 70.5

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

   5. Simplified 70.5%

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

   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 77.2%

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

      1. *-lft-identity N/A

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

      2. sqrt-pow2 N/A

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

      3. +-commutative N/A

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

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

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

      5. +-commutative N/A

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

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

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

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

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

      8. div-inv N/A

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

      9. metadata-eval N/A

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

      10. *-lowering-*.f64 77.2

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

   10. Applied egg-rr 77.2%

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

   if -5.19999999999999962e-10 < y.re < 1.55e-59

   1. Initial program 34.8%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 37.0

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

   5. Simplified 37.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 55.1%

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

Alternative 5: 58.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -2.7e+29)
   (pow (- x.re) y.re)
   (if (<= x.re 1.02e-8) (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 <= -2.7e+29) {
		tmp = pow(-x_46_re, y_46_re);
	} else if (x_46_re <= 1.02e-8) {
		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 <= (-2.7d+29)) then
        tmp = -x_46re ** y_46re
    else if (x_46re <= 1.02d-8) 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 <= -2.7e+29) {
		tmp = Math.pow(-x_46_re, y_46_re);
	} else if (x_46_re <= 1.02e-8) {
		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 <= -2.7e+29:
		tmp = math.pow(-x_46_re, y_46_re)
	elif x_46_re <= 1.02e-8:
		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 <= -2.7e+29)
		tmp = Float64(-x_46_re) ^ y_46_re;
	elseif (x_46_re <= 1.02e-8)
		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 <= -2.7e+29)
		tmp = -x_46_re ^ y_46_re;
	elseif (x_46_re <= 1.02e-8)
		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, -2.7e+29], N[Power[(-x$46$re), y$46$re], $MachinePrecision], If[LessEqual[x$46$re, 1.02e-8], 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 < -2.7e29

   1. Initial program 25.4%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 47.5

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

   5. Simplified 47.5%

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

   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 50.6%

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

   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. neg-lowering-neg.f64 63.2

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

   11. Simplified 63.2%

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

   if -2.7e29 < x.re < 1.02000000000000003e-8

   1. Initial program 44.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.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 59.4%

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

   9. Taylor expanded in x.re around 0

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

       1. pow-lowering-pow.f64 48.4

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

   11. Simplified 48.4%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   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. pow-lowering-pow.f64 N/A

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

       4. *-lowering-*.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. *-lowering-*.f64 58.7

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

   13. Applied egg-rr 58.7%

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

   if 1.02000000000000003e-8 < x.re

   1. Initial program 33.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 66.5

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

   5. Simplified 66.5%

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

   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 66.5%

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

   9. Taylor expanded in x.im around 0

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

       1. pow-lowering-pow.f64 74.5

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

   11. Simplified 74.5%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.7 \cdot 10^{+29}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 1.02 \cdot 10^{-8}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)}\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.0% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -7.2 \cdot 10^{-198}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 10^{-41}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -7.2e-198)
   (pow (- x.im) y.re)
   (if (<= x.im 1e-41) (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 <= -7.2e-198) {
		tmp = pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 1e-41) {
		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 <= (-7.2d-198)) then
        tmp = -x_46im ** y_46re
    else if (x_46im <= 1d-41) 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 <= -7.2e-198) {
		tmp = Math.pow(-x_46_im, y_46_re);
	} else if (x_46_im <= 1e-41) {
		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 <= -7.2e-198:
		tmp = math.pow(-x_46_im, y_46_re)
	elif x_46_im <= 1e-41:
		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 <= -7.2e-198)
		tmp = Float64(-x_46_im) ^ y_46_re;
	elseif (x_46_im <= 1e-41)
		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 <= -7.2e-198)
		tmp = -x_46_im ^ y_46_re;
	elseif (x_46_im <= 1e-41)
		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, -7.2e-198], N[Power[(-x$46$im), y$46$re], $MachinePrecision], If[LessEqual[x$46$im, 1e-41], 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 < -7.19999999999999996e-198

   1. Initial program 39.3%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 53.7

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

   5. Simplified 53.7%

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

   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 59.4%

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

   9. Taylor expanded in x.re around 0

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

       1. pow-lowering-pow.f64 33.2

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

   11. Simplified 33.2%

       \[\leadsto \color{blue}{{x.im}^{y.re}} \]
   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. sqr-neg N/A

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

       4. pow-prod-down N/A

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

       5. sqr-pow N/A

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

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

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

       7. neg-lowering-neg.f64 61.5

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

   13. Applied egg-rr 61.5%

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

   if -7.19999999999999996e-198 < x.im < 1.00000000000000001e-41

   1. Initial program 46.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 64.4

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

   5. Simplified 64.4%

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

   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 64.4%

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

   9. Taylor expanded in x.im around 0

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

       1. pow-lowering-pow.f64 63.8

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

   11. Simplified 63.8%

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

   if 1.00000000000000001e-41 < x.im

   1. Initial program 22.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 48.2

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

   5. Simplified 48.2%

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

   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 52.6%

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

   9. Taylor expanded in x.re around 0

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

       1. pow-lowering-pow.f64 64.1

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

   11. Simplified 64.1%

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

Alternative 7: 52.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.2 \cdot 10^{-10}:\\ \;\;\;\;{x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 260:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.2e-10)
   (pow x.re y.re)
   (if (<= y.re 260.0) 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 <= -8.2e-10) {
		tmp = pow(x_46_re, y_46_re);
	} else if (y_46_re <= 260.0) {
		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 <= (-8.2d-10)) then
        tmp = x_46re ** y_46re
    else if (y_46re <= 260.0d0) 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 <= -8.2e-10) {
		tmp = Math.pow(x_46_re, y_46_re);
	} else if (y_46_re <= 260.0) {
		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 <= -8.2e-10:
		tmp = math.pow(x_46_re, y_46_re)
	elif y_46_re <= 260.0:
		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 <= -8.2e-10)
		tmp = x_46_re ^ y_46_re;
	elseif (y_46_re <= 260.0)
		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 <= -8.2e-10)
		tmp = x_46_re ^ y_46_re;
	elseif (y_46_re <= 260.0)
		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, -8.2e-10], N[Power[x$46$re, y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 260.0], 1.0, N[Power[x$46$im, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.1999999999999996e-10

   1. Initial program 47.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 81.1

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

   5. Simplified 81.1%

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

   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 81.7%

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

   9. Taylor expanded in x.im around 0

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

       1. pow-lowering-pow.f64 62.1

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

   11. Simplified 62.1%

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

   if -8.1999999999999996e-10 < y.re < 260

   1. Initial program 34.2%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 37.0

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

   5. Simplified 37.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 52.6%

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

   if 260 < y.re

   1. Initial program 30.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 64.6

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

   5. Simplified 64.6%

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

   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 79.1%

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

   9. Taylor expanded in x.re around 0

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

       1. pow-lowering-pow.f64 66.6

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

   11. Simplified 66.6%

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

Alternative 8: 52.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.2:\\ \;\;\;\;{x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 4500:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.2) (pow x.im y.re) (if (<= y.re 4500.0) 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 <= -1.2) {
		tmp = pow(x_46_im, y_46_re);
	} else if (y_46_re <= 4500.0) {
		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 <= (-1.2d0)) then
        tmp = x_46im ** y_46re
    else if (y_46re <= 4500.0d0) 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 <= -1.2) {
		tmp = Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= 4500.0) {
		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 <= -1.2:
		tmp = math.pow(x_46_im, y_46_re)
	elif y_46_re <= 4500.0:
		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 <= -1.2)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 4500.0)
		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 <= -1.2)
		tmp = x_46_im ^ y_46_re;
	elseif (y_46_re <= 4500.0)
		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, -1.2], N[Power[x$46$im, y$46$re], $MachinePrecision], If[LessEqual[y$46$re, 4500.0], 1.0, N[Power[x$46$im, y$46$re], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.19999999999999996 or 4500 < y.re

   1. Initial program 37.8%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 73.4

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

   5. Simplified 73.4%

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

   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 81.2%

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

   9. Taylor expanded in x.re around 0

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

       1. pow-lowering-pow.f64 60.3

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

   11. Simplified 60.3%

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

   if -1.19999999999999996 < y.re < 4500

   1. Initial program 35.7%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 37.8

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

   5. Simplified 37.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Simplified 52.0%

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

Alternative 9: 25.4% 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 36.7%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. atan2-lowering-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. pow-lowering-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. sqrt-lowering-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. accelerator-lowering-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. *-lowering-*.f64 55.4

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

5. Simplified 55.4%

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

6. Taylor expanded in y.re around 0

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

8. Simplified 27.6%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024204 
(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)))))