powComplex, real part

Percentage Accurate: 40.4% → 77.0%

Time: 15.4s

Alternatives: 11

Speedup: 5.7×

• 36.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 77.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 320000000000:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (fma x.re x.re (* x.im x.im)) (* 0.5 y.re)))))
   (if (<= y.re -1.8e-7)
     t_0
     (if (<= y.re 320000000000.0)
       (*
        (cos (* (atan2 x.im x.re) y.re))
        (exp (* (atan2 x.im x.re) (- y.im))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(fma(x_46_re, x_46_re, (x_46_im * x_46_im)), (0.5 * y_46_re));
	double tmp;
	if (y_46_re <= -1.8e-7) {
		tmp = t_0;
	} else if (y_46_re <= 320000000000.0) {
		tmp = cos((atan2(x_46_im, x_46_re) * y_46_re)) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)) ^ Float64(0.5 * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.8e-7)
		tmp = t_0;
	elseif (y_46_re <= 320000000000.0)
		tmp = Float64(cos(Float64(atan(x_46_im, x_46_re) * y_46_re)) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	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[(1.0 * N[Power[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.8e-7], t$95$0, If[LessEqual[y$46$re, 320000000000.0], N[(N[Cos[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.79999999999999997e-7 or 3.2e11 < y.re

   1. Initial program 31.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 75.7%

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

   10. Applied rewrites 75.7%

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

   if -1.79999999999999997e-7 < y.re < 3.2e11

   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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 51.7

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in y.im around inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 78.9

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

   8. Applied rewrites 78.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 320000000000:\\ \;\;\;\;\cos \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 350000000000:\\ \;\;\;\;1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (fma x.re x.re (* x.im x.im)) (* 0.5 y.re)))))
   (if (<= y.re -1.8e-7)
     t_0
     (if (<= y.re 350000000000.0)
       (* 1.0 (exp (* (atan2 x.im x.re) (- y.im))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(fma(x_46_re, x_46_re, (x_46_im * x_46_im)), (0.5 * y_46_re));
	double tmp;
	if (y_46_re <= -1.8e-7) {
		tmp = t_0;
	} else if (y_46_re <= 350000000000.0) {
		tmp = 1.0 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)) ^ Float64(0.5 * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -1.8e-7)
		tmp = t_0;
	elseif (y_46_re <= 350000000000.0)
		tmp = Float64(1.0 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	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[(1.0 * N[Power[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.8e-7], t$95$0, If[LessEqual[y$46$re, 350000000000.0], N[(1.0 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.79999999999999997e-7 or 3.5e11 < y.re

   1. Initial program 31.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 75.7%

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

   10. Applied rewrites 75.7%

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

   if -1.79999999999999997e-7 < y.re < 3.5e11

   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.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. distribute-lft-neg-in N/A

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

      5. neg-mul-1 N/A

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

      6. lower-*.f64 N/A

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

      7. neg-mul-1 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in y.im around 0

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

   8. Applied rewrites 78.9%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.8 \cdot 10^{-7}:\\ \;\;\;\;1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 350000000000:\\ \;\;\;\;1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 61.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot {t\_0}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{\left(0.25 \cdot y.re\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (fma x.re x.re (* x.im x.im))))
   (if (<= y.re -2.1e-6)
     (* 1.0 (pow t_0 (* 0.5 y.re)))
     (if (<= y.re 3.1e-5) 1.0 (* (pow (* t_0 t_0) (* 0.25 y.re)) 1.0)))))

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_re, x_46_re, (x_46_im * x_46_im));
	double tmp;
	if (y_46_re <= -2.1e-6) {
		tmp = 1.0 * pow(t_0, (0.5 * y_46_re));
	} else if (y_46_re <= 3.1e-5) {
		tmp = 1.0;
	} else {
		tmp = pow((t_0 * t_0), (0.25 * y_46_re)) * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))
	tmp = 0.0
	if (y_46_re <= -2.1e-6)
		tmp = Float64(1.0 * (t_0 ^ Float64(0.5 * y_46_re)));
	elseif (y_46_re <= 3.1e-5)
		tmp = 1.0;
	else
		tmp = Float64((Float64(t_0 * t_0) ^ Float64(0.25 * y_46_re)) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e-6], N[(1.0 * N[Power[t$95$0, N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.1e-5], 1.0, N[(N[Power[N[(t$95$0 * t$95$0), $MachinePrecision], N[(0.25 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.0999999999999998e-6

   1. Initial program 32.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 79.2%

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

   10. Applied rewrites 79.2%

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

   if -2.0999999999999998e-6 < y.re < 3.10000000000000014e-5

   1. Initial program 45.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 37.1

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

   5. Applied rewrites 37.1%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 48.5%

      \[\leadsto 1 \]

   if 3.10000000000000014e-5 < y.re

   1. Initial program 31.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 70.8%

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

   10. Applied rewrites 70.8%

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

   12. Applied rewrites 72.1%

       \[\leadsto {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(y.re \cdot 0.25\right)} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-5}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right) \cdot \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.25 \cdot y.re\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (fma x.re x.re (* x.im x.im)) (* 0.5 y.re)))))
   (if (<= y.re -2.1e-6) t_0 (if (<= y.re 4e-14) 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 = 1.0 * pow(fma(x_46_re, x_46_re, (x_46_im * x_46_im)), (0.5 * y_46_re));
	double tmp;
	if (y_46_re <= -2.1e-6) {
		tmp = t_0;
	} else if (y_46_re <= 4e-14) {
		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 = Float64(1.0 * (fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)) ^ Float64(0.5 * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2.1e-6)
		tmp = t_0;
	elseif (y_46_re <= 4e-14)
		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[(1.0 * N[Power[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e-6], t$95$0, If[LessEqual[y$46$re, 4e-14], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.0999999999999998e-6 or 4e-14 < y.re

   1. Initial program 32.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 66.1

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

   5. Applied rewrites 66.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 74.8%

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

   10. Applied rewrites 74.8%

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

   if -2.0999999999999998e-6 < y.re < 4e-14

   1. Initial program 45.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 36.5

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

   5. Applied rewrites 36.5%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-6}:\\ \;\;\;\;1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-14}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(0.5 \cdot y.re\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 54.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.35 \cdot 10^{-119}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.im \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;{\left(x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -2.35e-119)
   (* (pow (- x.im) y.re) 1.0)
   (if (<= x.im 2.55e-9)
     (* (pow (* x.re x.re) (* 0.5 y.re)) 1.0)
     (* (pow (* x.im x.im) (* 0.5 y.re)) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.35e-119) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 2.55e-9) {
		tmp = pow((x_46_re * x_46_re), (0.5 * y_46_re)) * 1.0;
	} else {
		tmp = pow((x_46_im * x_46_im), (0.5 * y_46_re)) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-2.35d-119)) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else if (x_46im <= 2.55d-9) then
        tmp = ((x_46re * x_46re) ** (0.5d0 * y_46re)) * 1.0d0
    else
        tmp = ((x_46im * x_46im) ** (0.5d0 * y_46re)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.35e-119) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 2.55e-9) {
		tmp = Math.pow((x_46_re * x_46_re), (0.5 * y_46_re)) * 1.0;
	} else {
		tmp = Math.pow((x_46_im * x_46_im), (0.5 * y_46_re)) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -2.35e-119:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	elif x_46_im <= 2.55e-9:
		tmp = math.pow((x_46_re * x_46_re), (0.5 * y_46_re)) * 1.0
	else:
		tmp = math.pow((x_46_im * x_46_im), (0.5 * y_46_re)) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -2.35e-119)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	elseif (x_46_im <= 2.55e-9)
		tmp = Float64((Float64(x_46_re * x_46_re) ^ Float64(0.5 * y_46_re)) * 1.0);
	else
		tmp = Float64((Float64(x_46_im * x_46_im) ^ Float64(0.5 * y_46_re)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -2.35e-119)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	elseif (x_46_im <= 2.55e-9)
		tmp = ((x_46_re * x_46_re) ^ (0.5 * y_46_re)) * 1.0;
	else
		tmp = ((x_46_im * x_46_im) ^ (0.5 * y_46_re)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -2.35e-119], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$im, 2.55e-9], N[(N[Power[N[(x$46$re * x$46$re), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[N[(x$46$im * x$46$im), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.35000000000000001e-119

   1. Initial program 32.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in x.im around -inf

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

   11. Applied rewrites 63.2%

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

   if -2.35000000000000001e-119 < x.im < 2.55000000000000009e-9

   1. Initial program 48.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 56.9%

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

   10. Applied rewrites 56.9%

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

   11. Taylor expanded in x.im around 0

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

   13. Applied rewrites 58.8%

       \[\leadsto {\left(x.re \cdot x.re\right)}^{\left(y.re \cdot 0.5\right)} \cdot 1 \]

   if 2.55000000000000009e-9 < x.im

   1. Initial program 29.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 60.3%

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

   10. Applied rewrites 60.3%

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

   11. Taylor expanded in x.im around inf

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

   13. Applied rewrites 65.0%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -2.35 \cdot 10^{-119}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.im \leq 2.55 \cdot 10^{-9}:\\ \;\;\;\;{\left(x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -185000:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 1.75 \cdot 10^{+27}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -185000.0)
   (* (pow (- x.re) y.re) 1.0)
   (if (<= x.re 1.75e+27)
     (* (pow (* x.im x.im) (* 0.5 y.re)) 1.0)
     (* (pow x.re y.re) 1.0))))

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 <= -185000.0) {
		tmp = pow(-x_46_re, y_46_re) * 1.0;
	} else if (x_46_re <= 1.75e+27) {
		tmp = pow((x_46_im * x_46_im), (0.5 * y_46_re)) * 1.0;
	} else {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-185000.0d0)) then
        tmp = (-x_46re ** y_46re) * 1.0d0
    else if (x_46re <= 1.75d+27) then
        tmp = ((x_46im * x_46im) ** (0.5d0 * y_46re)) * 1.0d0
    else
        tmp = (x_46re ** y_46re) * 1.0d0
    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 <= -185000.0) {
		tmp = Math.pow(-x_46_re, y_46_re) * 1.0;
	} else if (x_46_re <= 1.75e+27) {
		tmp = Math.pow((x_46_im * x_46_im), (0.5 * y_46_re)) * 1.0;
	} else {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -185000.0:
		tmp = math.pow(-x_46_re, y_46_re) * 1.0
	elif x_46_re <= 1.75e+27:
		tmp = math.pow((x_46_im * x_46_im), (0.5 * y_46_re)) * 1.0
	else:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -185000.0)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * 1.0);
	elseif (x_46_re <= 1.75e+27)
		tmp = Float64((Float64(x_46_im * x_46_im) ^ Float64(0.5 * y_46_re)) * 1.0);
	else
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -185000.0)
		tmp = (-x_46_re ^ y_46_re) * 1.0;
	elseif (x_46_re <= 1.75e+27)
		tmp = ((x_46_im * x_46_im) ^ (0.5 * y_46_re)) * 1.0;
	else
		tmp = (x_46_re ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -185000.0], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 1.75e+27], N[(N[Power[N[(x$46$im * x$46$im), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -185000

   1. Initial program 23.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 52.2

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

   5. Applied rewrites 52.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 56.9%

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

   9. Taylor expanded in x.re around -inf

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

   11. Applied rewrites 63.5%

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

   if -185000 < x.re < 1.7500000000000001e27

   1. Initial program 50.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 57.9%

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

   10. Applied rewrites 57.9%

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

   11. Taylor expanded in x.im around inf

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

   13. Applied rewrites 60.1%

       \[\leadsto {\left(x.im \cdot x.im\right)}^{\left(y.re \cdot 0.5\right)} \cdot 1 \]

   if 1.7500000000000001e27 < x.re

   1. Initial program 26.7%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 55.7

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

   5. Applied rewrites 55.7%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 55.7%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto {x.re}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 60.9%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -185000:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.re \leq 1.75 \cdot 10^{+27}:\\ \;\;\;\;{\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.0% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4.4 \cdot 10^{-79}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.im \leq 1.7 \cdot 10^{-14}:\\ \;\;\;\;{\left(-x.re\right)}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -4.4e-79)
   (* (pow (- x.im) y.re) 1.0)
   (if (<= x.im 1.7e-14) (* (pow (- x.re) y.re) 1.0) (* (pow x.im y.re) 1.0))))

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 <= -4.4e-79) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 1.7e-14) {
		tmp = pow(-x_46_re, y_46_re) * 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-4.4d-79)) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else if (x_46im <= 1.7d-14) then
        tmp = (-x_46re ** y_46re) * 1.0d0
    else
        tmp = (x_46im ** y_46re) * 1.0d0
    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 <= -4.4e-79) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 1.7e-14) {
		tmp = Math.pow(-x_46_re, y_46_re) * 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -4.4e-79:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	elif x_46_im <= 1.7e-14:
		tmp = math.pow(-x_46_re, y_46_re) * 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -4.4e-79)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	elseif (x_46_im <= 1.7e-14)
		tmp = Float64((Float64(-x_46_re) ^ y_46_re) * 1.0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -4.4e-79)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	elseif (x_46_im <= 1.7e-14)
		tmp = (-x_46_re ^ y_46_re) * 1.0;
	else
		tmp = (x_46_im ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -4.4e-79], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$im, 1.7e-14], N[(N[Power[(-x$46$re), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -4.3999999999999998e-79

   1. Initial program 26.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 48.3

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 53.4%

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

   9. Taylor expanded in x.im around -inf

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

   11. Applied rewrites 61.3%

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

   if -4.3999999999999998e-79 < x.im < 1.70000000000000001e-14

   1. Initial program 50.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 53.2

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

   5. Applied rewrites 53.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 58.5%

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

   9. Taylor expanded in x.re around -inf

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

   11. Applied rewrites 55.9%

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

   if 1.70000000000000001e-14 < x.im

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto {x.im}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 62.6%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 54.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8.5 \cdot 10^{-130}:\\ \;\;\;\;{\left(-x.im\right)}^{y.re} \cdot 1\\ \mathbf{elif}\;x.im \leq 6.4 \cdot 10^{-15}:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -8.5e-130)
   (* (pow (- x.im) y.re) 1.0)
   (if (<= x.im 6.4e-15) (* (pow x.re y.re) 1.0) (* (pow x.im y.re) 1.0))))

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 <= -8.5e-130) {
		tmp = pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 6.4e-15) {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-8.5d-130)) then
        tmp = (-x_46im ** y_46re) * 1.0d0
    else if (x_46im <= 6.4d-15) then
        tmp = (x_46re ** y_46re) * 1.0d0
    else
        tmp = (x_46im ** y_46re) * 1.0d0
    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 <= -8.5e-130) {
		tmp = Math.pow(-x_46_im, y_46_re) * 1.0;
	} else if (x_46_im <= 6.4e-15) {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -8.5e-130:
		tmp = math.pow(-x_46_im, y_46_re) * 1.0
	elif x_46_im <= 6.4e-15:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -8.5e-130)
		tmp = Float64((Float64(-x_46_im) ^ y_46_re) * 1.0);
	elseif (x_46_im <= 6.4e-15)
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	else
		tmp = Float64((x_46_im ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -8.5e-130)
		tmp = (-x_46_im ^ y_46_re) * 1.0;
	elseif (x_46_im <= 6.4e-15)
		tmp = (x_46_re ^ y_46_re) * 1.0;
	else
		tmp = (x_46_im ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -8.5e-130], N[(N[Power[(-x$46$im), y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$im, 6.4e-15], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -8.50000000000000033e-130

   1. Initial program 32.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 50.8

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

   5. Applied rewrites 50.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in x.im around -inf

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

   11. Applied rewrites 63.2%

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

   if -8.50000000000000033e-130 < x.im < 6.3999999999999999e-15

   1. Initial program 48.3%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 51.6

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

   5. Applied rewrites 51.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 57.4%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto {x.re}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 49.5%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]

   if 6.3999999999999999e-15 < x.im

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 59.4%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto {x.im}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 62.6%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 51.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -44:\\ \;\;\;\;{x.re}^{y.re} \cdot 1\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -44.0)
   (* (pow x.re y.re) 1.0)
   (if (<= y.re 4.3e+15) 1.0 (* (pow x.im y.re) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -44.0) {
		tmp = pow(x_46_re, y_46_re) * 1.0;
	} else if (y_46_re <= 4.3e+15) {
		tmp = 1.0;
	} else {
		tmp = pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-44.0d0)) then
        tmp = (x_46re ** y_46re) * 1.0d0
    else if (y_46re <= 4.3d+15) then
        tmp = 1.0d0
    else
        tmp = (x_46im ** y_46re) * 1.0d0
    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 <= -44.0) {
		tmp = Math.pow(x_46_re, y_46_re) * 1.0;
	} else if (y_46_re <= 4.3e+15) {
		tmp = 1.0;
	} else {
		tmp = Math.pow(x_46_im, y_46_re) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -44.0:
		tmp = math.pow(x_46_re, y_46_re) * 1.0
	elif y_46_re <= 4.3e+15:
		tmp = 1.0
	else:
		tmp = math.pow(x_46_im, y_46_re) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -44.0)
		tmp = Float64((x_46_re ^ y_46_re) * 1.0);
	elseif (y_46_re <= 4.3e+15)
		tmp = 1.0;
	else
		tmp = Float64((x_46_im ^ y_46_re) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -44.0)
		tmp = (x_46_re ^ y_46_re) * 1.0;
	elseif (y_46_re <= 4.3e+15)
		tmp = 1.0;
	else
		tmp = (x_46_im ^ y_46_re) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -44.0], N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[y$46$re, 4.3e+15], 1.0, N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -44

   1. Initial program 31.1%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 75.6

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

   5. Applied rewrites 75.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 78.8%

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

   9. Taylor expanded in x.im around 0

      \[\leadsto {x.re}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 64.2%

       \[\leadsto {x.re}^{y.re} \cdot 1 \]

   if -44 < y.re < 4.3e15

   1. Initial program 45.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 46.6%

      \[\leadsto 1 \]

   if 4.3e15 < y.re

   1. Initial program 31.0%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 74.7%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto {x.im}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 56.8%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 51.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {x.im}^{y.re} \cdot 1\\ \mathbf{if}\;y.re \leq -430:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{+15}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (pow x.im y.re) 1.0)))
   (if (<= y.re -430.0) t_0 (if (<= y.re 4.3e+15) 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(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -430.0) {
		tmp = t_0;
	} else if (y_46_re <= 4.3e+15) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im ** y_46re) * 1.0d0
    if (y_46re <= (-430.0d0)) then
        tmp = t_0
    else if (y_46re <= 4.3d+15) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.pow(x_46_im, y_46_re) * 1.0;
	double tmp;
	if (y_46_re <= -430.0) {
		tmp = t_0;
	} else if (y_46_re <= 4.3e+15) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(x_46_im, y_46_re) * 1.0
	tmp = 0
	if y_46_re <= -430.0:
		tmp = t_0
	elif y_46_re <= 4.3e+15:
		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 = Float64((x_46_im ^ y_46_re) * 1.0)
	tmp = 0.0
	if (y_46_re <= -430.0)
		tmp = t_0;
	elseif (y_46_re <= 4.3e+15)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im ^ y_46_re) * 1.0;
	tmp = 0.0;
	if (y_46_re <= -430.0)
		tmp = t_0;
	elseif (y_46_re <= 4.3e+15)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -430.0], t$95$0, If[LessEqual[y$46$re, 4.3e+15], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -430 or 4.3e15 < y.re

   1. Initial program 30.5%

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

   3. Taylor expanded in 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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 76.4%

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

   9. Taylor expanded in x.re around 0

      \[\leadsto {x.im}^{y.re} \cdot 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 55.5%

       \[\leadsto {x.im}^{y.re} \cdot 1 \]

   if -430 < y.re < 4.3e15

   1. Initial program 46.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-atan2.f64 37.7

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 46.2%

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

Alternative 11: 25.5% accurate, 680.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 38.2%

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

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

   2. lower-*.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. +-commutative N/A

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

   6. unpow2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-atan2.f64 52.5

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

5. Applied rewrites 52.5%

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

6. Taylor expanded in y.re around 0

   \[\leadsto 1 \]
7. Step-by-step derivation

8. Applied rewrites 23.9%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(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)))))