powComplex, real part

Percentage Accurate: 41.2% → 80.4%

Time: 10.0s

Alternatives: 16

Speedup: 5.7×

• 35.3% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.2% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq 7.5 \cdot 10^{+125}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(\mathsf{fma}\left(t\_0, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= y.re 7.5e+125)
     (*
      (exp (fma t_0 y.re (* (- (atan2 x.im x.re)) y.im)))
      (cos (fma t_0 y.im (* (atan2 x.im x.re) y.re))))
     (* 1.0 (pow (hypot 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(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= 7.5e+125) {
		tmp = exp(fma(t_0, y_46_re, (-atan2(x_46_im, x_46_re) * y_46_im))) * cos(fma(t_0, y_46_im, (atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= 7.5e+125)
		tmp = Float64(exp(fma(t_0, y_46_re, Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_im))) * cos(fma(t_0, y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))));
	else
		tmp = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, 7.5e+125], N[(N[Exp[N[(t$95$0 * y$46$re + N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t$95$0 * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < 7.5000000000000006e125

   1. Initial program 42.1%

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

   3. Applied rewrites 82.4%

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

   if 7.5000000000000006e125 < y.re

   1. Initial program 36.1%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.2

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

   4. Applied rewrites 59.2%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 68.5%

      \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_1 := e^{t\_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(t\_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

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)))))
        (t_1
         (*
          (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
          (cos (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))
   (if (<= t_1 INFINITY)
     t_1
     (*
      (exp (fma (log (hypot x.re x.im)) y.re (* (- (atan2 x.im x.re)) y.im)))
      (cos (* y.re (atan2 x.im x.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))));
	double t_1 = 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)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, (-atan2(x_46_im, x_46_re) * y_46_im))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

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))))
	t_1 = 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))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_im))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
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]}, Block[{t$95$1 = 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]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[Exp[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re + N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 80.3%

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

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (cos.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 0.0%

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

   3. Applied rewrites 79.6%

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

   4. Taylor expanded in y.re around inf

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

      1. lift-atan2.f64 N/A

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

      2. lift-*.f64 79.2

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

   6. Applied rewrites 79.2%

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

Alternative 3: 79.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (exp (fma (log (hypot x.re x.im)) y.re (* (- (atan2 x.im x.re)) y.im)))
  (cos (* y.re (atan2 x.im x.re)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 41.2%

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

3. Applied rewrites 80.0%

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

4. Taylor expanded in y.re around inf

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

   1. lift-atan2.f64 N/A

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

   2. lift-*.f64 79.6

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

6. Applied rewrites 79.6%

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

Alternative 4: 76.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+182}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -0.65:\\ \;\;\;\;t\_0 \cdot \sin t\_1\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-16}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \cos t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im))))
        (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -1.3e+182)
     (* 1.0 (pow (hypot x.im x.re) y.re))
     (if (<= y.re -0.65)
       (* t_0 (sin t_1))
       (if (<= y.re 6.2e-16)
         (*
          (cos (* y.im (log (hypot x.im x.re))))
          (exp (* (- y.im) (atan2 x.im x.re))))
         (* t_0 (cos t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -1.3e+182) {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -0.65) {
		tmp = t_0 * sin(t_1);
	} else if (y_46_re <= 6.2e-16) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0 * cos(t_1);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -1.3e+182) {
		tmp = 1.0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= -0.65) {
		tmp = t_0 * Math.sin(t_1);
	} else if (y_46_re <= 6.2e-16) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0 * Math.cos(t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_re <= -1.3e+182:
		tmp = 1.0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= -0.65:
		tmp = t_0 * math.sin(t_1)
	elif y_46_re <= 6.2e-16:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0 * math.cos(t_1)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -1.3e+182)
		tmp = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= -0.65)
		tmp = Float64(t_0 * sin(t_1));
	elseif (y_46_re <= 6.2e-16)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(t_0 * cos(t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.3e+182)
		tmp = 1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= -0.65)
		tmp = t_0 * sin(t_1);
	elseif (y_46_re <= 6.2e-16)
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0 * cos(t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.3e+182], N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -0.65], N[(t$95$0 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-16], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.3e182

   1. Initial program 42.1%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 83.8

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

   4. Applied rewrites 83.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 83.8%

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

   if -1.3e182 < y.re < -0.650000000000000022

   1. Initial program 45.2%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-atan2.f64 N/A

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

      11. sin-+PI/2-rev N/A

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

      12. lower-sin.f64 N/A

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

   3. Applied rewrites 82.2%

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

   4. Taylor expanded in y.re around inf

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

      1. lift-atan2.f64 N/A

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

      2. lift-*.f64 79.9

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

   6. Applied rewrites 79.9%

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

   if -0.650000000000000022 < y.re < 6.2000000000000002e-16

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 N/A

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

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

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

      10. lower-exp.f64 N/A

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

   4. Applied rewrites 49.6%

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

      1. lift-log.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. sqrt-pow2 N/A

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

      8. log-pow-rev N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-hypot.f64 81.2

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

   6. Applied rewrites 81.2%

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

   if 6.2000000000000002e-16 < y.re

   1. Initial program 37.4%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 63.1

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

   4. Applied rewrites 63.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 76.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.95:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{-16}:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.95)
   (* 1.0 (pow (hypot x.im x.re) y.re))
   (if (<= y.re 6.2e-16)
     (*
      (cos (* y.im (log (hypot x.im x.re))))
      (exp (* (- y.im) (atan2 x.im x.re))))
     (*
      (exp
       (-
        (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
        (* (atan2 x.im x.re) y.im)))
      (cos (* y.re (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.95) {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 6.2e-16) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.95) {
		tmp = 1.0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 6.2e-16) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.95:
		tmp = 1.0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 6.2e-16:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.95)
		tmp = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 6.2e-16)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.95)
		tmp = 1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_re <= 6.2e-16)
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.95], N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e-16], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.94999999999999996

   1. Initial program 43.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 80.4%

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

   if -1.94999999999999996 < y.re < 6.2000000000000002e-16

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 N/A

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

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

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

      10. lower-exp.f64 N/A

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

   4. Applied rewrites 49.5%

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

      1. lift-log.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. sqrt-pow2 N/A

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

      8. log-pow-rev N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-hypot.f64 81.1

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

   6. Applied rewrites 81.1%

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

   if 6.2000000000000002e-16 < y.re

   1. Initial program 37.4%

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

   2. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 63.1

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

   4. Applied rewrites 63.1%

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

Alternative 6: 77.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.95:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1040000000:\\ \;\;\;\;\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -1.95)
     t_0
     (if (<= y.re 1040000000.0)
       (*
        (cos (* y.im (log (hypot x.im x.re))))
        (exp (* (- y.im) (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.95) {
		tmp = t_0;
	} else if (y_46_re <= 1040000000.0) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.95) {
		tmp = t_0;
	} else if (y_46_re <= 1040000000.0) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -1.95:
		tmp = t_0
	elif y_46_re <= 1040000000.0:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	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 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.95)
		tmp = t_0;
	elseif (y_46_re <= 1040000000.0)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.95)
		tmp = t_0;
	elseif (y_46_re <= 1040000000.0)
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.95], t$95$0, If[LessEqual[y$46$re, 1040000000.0], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.94999999999999996 or 1.04e9 < y.re

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 74.3%

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

   if -1.94999999999999996 < y.re < 1.04e9

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 N/A

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

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

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

      10. lower-exp.f64 N/A

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

   4. Applied rewrites 48.0%

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

      1. lift-log.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-hypot.f64 N/A

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

      4. sqrt-pow2 N/A

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

      5. pow2 N/A

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

      6. pow2 N/A

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

      7. sqrt-pow2 N/A

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

      8. log-pow-rev N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-hypot.f64 79.8

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

   6. Applied rewrites 79.8%

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

Alternative 7: 77.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1040000000:\\ \;\;\;\;{\left(e^{-1}\right)}^{\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -3.0)
     t_0
     (if (<= y.re 1040000000.0)
       (*
        (pow (exp -1.0) (* y.im (atan2 x.im x.re)))
        (cos (* y.re (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -3.0) {
		tmp = t_0;
	} else if (y_46_re <= 1040000000.0) {
		tmp = pow(exp(-1.0), (y_46_im * atan2(x_46_im, x_46_re))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -3.0) {
		tmp = t_0;
	} else if (y_46_re <= 1040000000.0) {
		tmp = Math.pow(Math.exp(-1.0), (y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -3.0:
		tmp = t_0
	elif y_46_re <= 1040000000.0:
		tmp = math.pow(math.exp(-1.0), (y_46_im * math.atan2(x_46_im, x_46_re))) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.0)
		tmp = t_0;
	elseif (y_46_re <= 1040000000.0)
		tmp = Float64((exp(-1.0) ^ Float64(y_46_im * atan(x_46_im, x_46_re))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -3.0)
		tmp = t_0;
	elseif (y_46_re <= 1040000000.0)
		tmp = (exp(-1.0) ^ (y_46_im * atan2(x_46_im, x_46_re))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.0], t$95$0, If[LessEqual[y$46$re, 1040000000.0], N[(N[Power[N[Exp[-1.0], $MachinePrecision], N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3 or 1.04e9 < y.re

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 74.3%

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

   if -3 < y.re < 1.04e9

   1. Initial program 41.9%

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

   3. Applied rewrites 82.0%

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

   4. Taylor expanded in y.re around inf

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

      1. lift-atan2.f64 N/A

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

      2. lift-*.f64 81.7

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

   6. Applied rewrites 81.7%

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

   7. Taylor expanded in y.re around 0

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

      1. exp-prod N/A

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

      2. lower-pow.f64 N/A

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

      3. lower-exp.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-atan2.f64 79.6

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

   9. Applied rewrites 79.6%

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

Alternative 8: 77.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.4:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1040000000:\\ \;\;\;\;1 \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -1.4)
     t_0
     (if (<= y.re 1040000000.0)
       (* 1.0 (exp (* (- y.im) (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.4) {
		tmp = t_0;
	} else if (y_46_re <= 1040000000.0) {
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.4) {
		tmp = t_0;
	} else if (y_46_re <= 1040000000.0) {
		tmp = 1.0 * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -1.4:
		tmp = t_0
	elif y_46_re <= 1040000000.0:
		tmp = 1.0 * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.4)
		tmp = t_0;
	elseif (y_46_re <= 1040000000.0)
		tmp = Float64(1.0 * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.4)
		tmp = t_0;
	elseif (y_46_re <= 1040000000.0)
		tmp = 1.0 * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.4], t$95$0, If[LessEqual[y$46$re, 1040000000.0], N[(1.0 * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.3999999999999999 or 1.04e9 < y.re

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 74.3%

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

   if -1.3999999999999999 < y.re < 1.04e9

   1. Initial program 41.9%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 N/A

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

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

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

      10. lower-exp.f64 N/A

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

   4. Applied rewrites 48.0%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 79.6%

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

Alternative 9: 63.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(x.re + 0.5 \cdot \frac{x.im \cdot x.im}{x.re}\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -1.46 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{+45}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (+ x.re (* 0.5 (/ (* x.im x.im) x.re))) y.re))))
   (if (<= y.im -1.46e+159)
     t_0
     (if (<= y.im 2.25e+45) (* 1.0 (pow (hypot x.im x.re) y.re)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))), y_46_re);
	double tmp;
	if (y_46_im <= -1.46e+159) {
		tmp = t_0;
	} else if (y_46_im <= 2.25e+45) {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))), y_46_re);
	double tmp;
	if (y_46_im <= -1.46e+159) {
		tmp = t_0;
	} else if (y_46_im <= 2.25e+45) {
		tmp = 1.0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))), y_46_re)
	tmp = 0
	if y_46_im <= -1.46e+159:
		tmp = t_0
	elif y_46_im <= 2.25e+45:
		tmp = 1.0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (Float64(x_46_re + Float64(0.5 * Float64(Float64(x_46_im * x_46_im) / x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_im <= -1.46e+159)
		tmp = t_0;
	elseif (y_46_im <= 2.25e+45)
		tmp = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * ((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))) ^ y_46_re);
	tmp = 0.0;
	if (y_46_im <= -1.46e+159)
		tmp = t_0;
	elseif (y_46_im <= 2.25e+45)
		tmp = 1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[(x$46$re + N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.46e+159], t$95$0, If[LessEqual[y$46$im, 2.25e+45], N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.46e159 or 2.2499999999999999e45 < y.im

   1. Initial program 36.4%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 29.6

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

   4. Applied rewrites 29.6%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 30.2

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 30.2%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 29.6%

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

   11. Taylor expanded in x.im around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 33.3

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

   13. Applied rewrites 33.3%

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

   if -1.46e159 < y.im < 2.2499999999999999e45

   1. Initial program 43.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 76.0

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

   4. Applied rewrites 76.0%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 79.3%

      \[\leadsto 1 \cdot {\color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}}^{y.re} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 59.6% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;1 + -1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(x.re + 0.5 \cdot \frac{x.im \cdot x.im}{x.re}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.1e-16)
   (* 1.0 (pow (+ x.im (* 0.5 (/ (* x.re x.re) x.im))) y.re))
   (if (<= y.re 1.8e-7)
     (+ 1.0 (* -1.0 (* y.im (atan2 x.im x.re))))
     (* 1.0 (pow (+ x.re (* 0.5 (/ (* x.im 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 tmp;
	if (y_46_re <= -2.1e-16) {
		tmp = 1.0 * pow((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))), y_46_re);
	} else if (y_46_re <= 1.8e-7) {
		tmp = 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * pow((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))), y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.1d-16)) then
        tmp = 1.0d0 * ((x_46im + (0.5d0 * ((x_46re * x_46re) / x_46im))) ** y_46re)
    else if (y_46re <= 1.8d-7) then
        tmp = 1.0d0 + ((-1.0d0) * (y_46im * atan2(x_46im, x_46re)))
    else
        tmp = 1.0d0 * ((x_46re + (0.5d0 * ((x_46im * x_46im) / x_46re))) ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e-16) {
		tmp = 1.0 * Math.pow((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))), y_46_re);
	} else if (y_46_re <= 1.8e-7) {
		tmp = 1.0 + (-1.0 * (y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * Math.pow((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.1e-16:
		tmp = 1.0 * math.pow((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))), y_46_re)
	elif y_46_re <= 1.8e-7:
		tmp = 1.0 + (-1.0 * (y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = 1.0 * math.pow((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.1e-16)
		tmp = Float64(1.0 * (Float64(x_46_im + Float64(0.5 * Float64(Float64(x_46_re * x_46_re) / x_46_im))) ^ y_46_re));
	elseif (y_46_re <= 1.8e-7)
		tmp = Float64(1.0 + Float64(-1.0 * Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * (Float64(x_46_re + Float64(0.5 * Float64(Float64(x_46_im * x_46_im) / x_46_re))) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.1e-16)
		tmp = 1.0 * ((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))) ^ y_46_re);
	elseif (y_46_re <= 1.8e-7)
		tmp = 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = 1.0 * ((x_46_re + (0.5 * ((x_46_im * x_46_im) / x_46_re))) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.1e-16], N[(1.0 * N[Power[N[(x$46$im + N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.8e-7], N[(1.0 + N[(-1.0 * N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[N[(x$46$re + N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.1000000000000001e-16

   1. Initial program 43.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 79.1

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 73.8

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 73.8%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 73.8%

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

   if -2.1000000000000001e-16 < y.re < 1.79999999999999997e-7

   1. Initial program 42.0%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 N/A

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

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

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

      10. lower-exp.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 50.9

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

   7. Applied rewrites 50.9%

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

   if 1.79999999999999997e-7 < y.re

   1. Initial program 37.2%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 51.0

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 51.0%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 59.4%

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

   11. Taylor expanded in x.im around 0

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

       1. lower-+.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 61.5

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

   13. Applied rewrites 61.5%

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

Alternative 11: 57.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-16}:\\ \;\;\;\;1 \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;1 + -1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.1e-16)
   (* 1.0 (pow (+ x.im (* 0.5 (/ (* x.re x.re) x.im))) y.re))
   (if (<= y.re 3.7e-7)
     (+ 1.0 (* -1.0 (* y.im (atan2 x.im x.re))))
     (* 1.0 (pow x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e-16) {
		tmp = 1.0 * pow((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))), y_46_re);
	} else if (y_46_re <= 3.7e-7) {
		tmp = 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.1d-16)) then
        tmp = 1.0d0 * ((x_46im + (0.5d0 * ((x_46re * x_46re) / x_46im))) ** y_46re)
    else if (y_46re <= 3.7d-7) then
        tmp = 1.0d0 + ((-1.0d0) * (y_46im * atan2(x_46im, x_46re)))
    else
        tmp = 1.0d0 * (x_46re ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.1e-16) {
		tmp = 1.0 * Math.pow((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))), y_46_re);
	} else if (y_46_re <= 3.7e-7) {
		tmp = 1.0 + (-1.0 * (y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.1e-16:
		tmp = 1.0 * math.pow((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))), y_46_re)
	elif y_46_re <= 3.7e-7:
		tmp = 1.0 + (-1.0 * (y_46_im * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = 1.0 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.1e-16)
		tmp = Float64(1.0 * (Float64(x_46_im + Float64(0.5 * Float64(Float64(x_46_re * x_46_re) / x_46_im))) ^ y_46_re));
	elseif (y_46_re <= 3.7e-7)
		tmp = Float64(1.0 + Float64(-1.0 * Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.1e-16)
		tmp = 1.0 * ((x_46_im + (0.5 * ((x_46_re * x_46_re) / x_46_im))) ^ y_46_re);
	elseif (y_46_re <= 3.7e-7)
		tmp = 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_46_re)));
	else
		tmp = 1.0 * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.1e-16], N[(1.0 * N[Power[N[(x$46$im + N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.7e-7], N[(1.0 + N[(-1.0 * N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.1000000000000001e-16

   1. Initial program 43.8%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 79.1

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 73.8

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 73.8%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 73.8%

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

   if -2.1000000000000001e-16 < y.re < 3.70000000000000004e-7

   1. Initial program 42.0%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 N/A

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

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

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

      10. lower-exp.f64 N/A

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 50.9

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

   7. Applied rewrites 50.9%

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

   if 3.70000000000000004e-7 < y.re

   1. Initial program 37.2%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 51.0

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 51.0%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 59.4%

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

   11. Taylor expanded in x.re around inf

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

   13. Applied rewrites 53.5%

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

Alternative 12: 54.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -7.2 \cdot 10^{-21}:\\ \;\;\;\;1 \cdot {\left(-1 \cdot x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 9.2 \cdot 10^{-271}:\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -7.2e-21)
   (* 1.0 (pow (* -1.0 x.re) y.re))
   (if (<= x.re 9.2e-271) (* 1.0 (pow x.im y.re)) (* 1.0 (pow x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -7.2e-21) {
		tmp = 1.0 * pow((-1.0 * x_46_re), y_46_re);
	} else if (x_46_re <= 9.2e-271) {
		tmp = 1.0 * pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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 <= (-7.2d-21)) then
        tmp = 1.0d0 * (((-1.0d0) * x_46re) ** y_46re)
    else if (x_46re <= 9.2d-271) then
        tmp = 1.0d0 * (x_46im ** y_46re)
    else
        tmp = 1.0d0 * (x_46re ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -7.2e-21) {
		tmp = 1.0 * Math.pow((-1.0 * x_46_re), y_46_re);
	} else if (x_46_re <= 9.2e-271) {
		tmp = 1.0 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -7.2e-21:
		tmp = 1.0 * math.pow((-1.0 * x_46_re), y_46_re)
	elif x_46_re <= 9.2e-271:
		tmp = 1.0 * math.pow(x_46_im, y_46_re)
	else:
		tmp = 1.0 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -7.2e-21)
		tmp = Float64(1.0 * (Float64(-1.0 * x_46_re) ^ y_46_re));
	elseif (x_46_re <= 9.2e-271)
		tmp = Float64(1.0 * (x_46_im ^ y_46_re));
	else
		tmp = Float64(1.0 * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -7.2e-21)
		tmp = 1.0 * ((-1.0 * x_46_re) ^ y_46_re);
	elseif (x_46_re <= 9.2e-271)
		tmp = 1.0 * (x_46_im ^ y_46_re);
	else
		tmp = 1.0 * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -7.2e-21], N[(1.0 * N[Power[N[(-1.0 * x$46$re), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 9.2e-271], N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -7.19999999999999979e-21

   1. Initial program 34.1%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 56.3

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

   4. Applied rewrites 56.3%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 41.6

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 41.6%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 45.3%

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

   11. Taylor expanded in x.re around -inf

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

       1. lower-*.f64 58.5

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

   13. Applied rewrites 58.5%

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

   if -7.19999999999999979e-21 < x.re < 9.20000000000000034e-271

   1. Initial program 50.6%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.5

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

   4. Applied rewrites 59.5%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 42.3

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 42.3%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 44.8%

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

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 47.6%

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

   if 9.20000000000000034e-271 < x.re

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 43.4

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 43.4%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 44.4%

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

   11. Taylor expanded in x.re around inf

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

   13. Applied rewrites 56.3%

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

Alternative 13: 54.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -2.15 \cdot 10^{-245}:\\ \;\;\;\;1 \cdot {\left(-1 \cdot x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;1 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -2.15e-245)
   (* 1.0 (pow (* -1.0 x.im) y.re))
   (if (<= x.im 7.2e-45) (* 1.0 (pow x.re y.re)) (* 1.0 (pow x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.15e-245) {
		tmp = 1.0 * pow((-1.0 * x_46_im), y_46_re);
	} else if (x_46_im <= 7.2e-45) {
		tmp = 1.0 * pow(x_46_re, y_46_re);
	} else {
		tmp = 1.0 * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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 <= (-2.15d-245)) then
        tmp = 1.0d0 * (((-1.0d0) * x_46im) ** y_46re)
    else if (x_46im <= 7.2d-45) then
        tmp = 1.0d0 * (x_46re ** y_46re)
    else
        tmp = 1.0d0 * (x_46im ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -2.15e-245) {
		tmp = 1.0 * Math.pow((-1.0 * x_46_im), y_46_re);
	} else if (x_46_im <= 7.2e-45) {
		tmp = 1.0 * Math.pow(x_46_re, y_46_re);
	} else {
		tmp = 1.0 * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -2.15e-245:
		tmp = 1.0 * math.pow((-1.0 * x_46_im), y_46_re)
	elif x_46_im <= 7.2e-45:
		tmp = 1.0 * math.pow(x_46_re, y_46_re)
	else:
		tmp = 1.0 * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -2.15e-245)
		tmp = Float64(1.0 * (Float64(-1.0 * x_46_im) ^ y_46_re));
	elseif (x_46_im <= 7.2e-45)
		tmp = Float64(1.0 * (x_46_re ^ y_46_re));
	else
		tmp = Float64(1.0 * (x_46_im ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -2.15e-245)
		tmp = 1.0 * ((-1.0 * x_46_im) ^ y_46_re);
	elseif (x_46_im <= 7.2e-45)
		tmp = 1.0 * (x_46_re ^ y_46_re);
	else
		tmp = 1.0 * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -2.15e-245], N[(1.0 * N[Power[N[(-1.0 * x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 7.2e-45], N[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.15000000000000001e-245

   1. Initial program 41.3%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.7

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

   4. Applied rewrites 59.7%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 33.8

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 33.8%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 36.1%

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

   11. Taylor expanded in x.im around -inf

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

       1. lower-*.f64 53.5

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

   13. Applied rewrites 53.5%

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

   if -2.15000000000000001e-245 < x.im < 7.20000000000000001e-45

   1. Initial program 47.2%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 62.5

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

   4. Applied rewrites 62.5%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 44.9

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 44.9%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 46.0%

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

   11. Taylor expanded in x.re around inf

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

   13. Applied rewrites 50.5%

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

   if 7.20000000000000001e-45 < x.im

   1. Initial program 35.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.6

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

   4. Applied rewrites 59.6%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 55.1

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 55.1%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 57.8%

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

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 59.4%

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

Alternative 14: 52.8% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.45:\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.45)
   (* 1.0 (pow x.im y.re))
   (if (<= y.re 3.7e-7) 1.0 (* 1.0 (pow x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.45) {
		tmp = 1.0 * pow(x_46_im, y_46_re);
	} else if (y_46_re <= 3.7e-7) {
		tmp = 1.0;
	} else {
		tmp = 1.0 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.45d0)) then
        tmp = 1.0d0 * (x_46im ** y_46re)
    else if (y_46re <= 3.7d-7) then
        tmp = 1.0d0
    else
        tmp = 1.0d0 * (x_46re ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.45) {
		tmp = 1.0 * Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= 3.7e-7) {
		tmp = 1.0;
	} else {
		tmp = 1.0 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.45:
		tmp = 1.0 * math.pow(x_46_im, y_46_re)
	elif y_46_re <= 3.7e-7:
		tmp = 1.0
	else:
		tmp = 1.0 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.45)
		tmp = Float64(1.0 * (x_46_im ^ y_46_re));
	elseif (y_46_re <= 3.7e-7)
		tmp = 1.0;
	else
		tmp = Float64(1.0 * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.45)
		tmp = 1.0 * (x_46_im ^ y_46_re);
	elseif (y_46_re <= 3.7e-7)
		tmp = 1.0;
	else
		tmp = 1.0 * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.45], N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.7e-7], 1.0, N[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.4500000000000002

   1. Initial program 43.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 76.6

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 76.6%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 76.6%

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

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 57.4%

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

   if -2.4500000000000002 < y.re < 3.70000000000000004e-7

   1. Initial program 42.0%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 50.1%

      \[\leadsto 1 \]

   if 3.70000000000000004e-7 < y.re

   1. Initial program 37.2%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 51.0

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 51.0%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 59.4%

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

   11. Taylor expanded in x.re around inf

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

   13. Applied rewrites 53.5%

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

Alternative 15: 53.0% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {x.re}^{y.re}\\ \mathbf{if}\;y.re \leq -0.155:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow x.re y.re))))
   (if (<= y.re -0.155) t_0 (if (<= y.re 3.7e-7) 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(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -0.155) {
		tmp = t_0;
	} else if (y_46_re <= 3.7e-7) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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 = 1.0d0 * (x_46re ** y_46re)
    if (y_46re <= (-0.155d0)) then
        tmp = t_0
    else if (y_46re <= 3.7d-7) 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 = 1.0 * Math.pow(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -0.155) {
		tmp = t_0;
	} else if (y_46_re <= 3.7e-7) {
		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 = 1.0 * math.pow(x_46_re, y_46_re)
	tmp = 0
	if y_46_re <= -0.155:
		tmp = t_0
	elif y_46_re <= 3.7e-7:
		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 * (x_46_re ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -0.155)
		tmp = t_0;
	elseif (y_46_re <= 3.7e-7)
		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 = 1.0 * (x_46_re ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -0.155)
		tmp = t_0;
	elseif (y_46_re <= 3.7e-7)
		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[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -0.155], t$95$0, If[LessEqual[y$46$re, 3.7e-7], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.154999999999999999 or 3.70000000000000004e-7 < y.re

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 69.6

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

   4. Applied rewrites 69.6%

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

   5. Taylor expanded in x.re around 0

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

      1. pow2 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 63.5

         \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   7. Applied rewrites 63.5%

      \[\leadsto \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im + 0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re} \]

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 67.7%

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

   11. Taylor expanded in x.re around inf

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

   13. Applied rewrites 55.7%

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

   if -0.154999999999999999 < y.re < 3.70000000000000004e-7

   1. Initial program 42.0%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 51.1

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

   4. Applied rewrites 51.1%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 50.3%

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

Alternative 16: 26.4% accurate, 680.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    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 41.2%

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

2. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lift-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. lower-hypot.f64 60.4

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

4. Applied rewrites 60.4%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 26.4%

   \[\leadsto 1 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025093 
(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)))))