powComplex, real part

Percentage Accurate: 39.3% → 79.4%

Time: 9.4s

Alternatives: 12

Speedup: 5.7×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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: 39.3% 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: 79.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{\mathsf{fma}\left(t\_0, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}\\ \mathbf{if}\;x.re \leq -2.6 \cdot 10^{-137}:\\ \;\;\;\;t\_1 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos \left(\mathsf{fma}\left(t\_0, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im)))
        (t_1 (exp (fma t_0 y.re (* (- (atan2 x.im x.re)) y.im)))))
   (if (<= x.re -2.6e-137)
     (* t_1 (cos (* y.re (atan2 x.im x.re))))
     (* t_1 (cos (fma 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(hypot(x_46_re, x_46_im));
	double t_1 = exp(fma(t_0, y_46_re, (-atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (x_46_re <= -2.6e-137) {
		tmp = t_1 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_1 * cos(fma(t_0, y_46_im, (atan2(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))
	t_1 = exp(fma(t_0, y_46_re, Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_im)))
	tmp = 0.0
	if (x_46_re <= -2.6e-137)
		tmp = Float64(t_1 * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(t_1 * cos(fma(t_0, y_46_im, Float64(atan(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]}, Block[{t$95$1 = 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]}, If[LessEqual[x$46$re, -2.6e-137], N[(t$95$1 * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 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]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -2.6e-137

   1. Initial program 39.3%

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-cos.f64 N/A

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 81.2

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

   7. Applied rewrites 81.2%

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

   if -2.6e-137 < x.re

   1. Initial program 43.4%

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-cos.f64 N/A

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

   4. Applied rewrites 86.8%

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

Alternative 2: 78.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= 1.0) {
		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(t_0);
	} else {
		tmp = cos(fma(y_46_im, log(x_46_re), t_0)) * exp(((y_46_re * log(x_46_re)) - (y_46_im * 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 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= 1.0)
		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(t_0));
	else
		tmp = Float64(cos(fma(y_46_im, log(x_46_re), t_0)) * exp(Float64(Float64(y_46_re * log(x_46_re)) - Float64(y_46_im * 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, 1.0], 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[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 1

   1. Initial program 47.4%

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-cos.f64 N/A

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if 1 < x.re

   1. Initial program 25.8%

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

   3. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 81.3

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

   5. Applied rewrites 81.3%

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

Alternative 3: 78.9% 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 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. Add Preprocessing
3. Step-by-step derivation

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. fp-cancel-sub-sign-inv N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-neg.f64 N/A

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

   9. lower-atan2.f64 N/A

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

   10. lower-cos.f64 N/A

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

4. Applied rewrites 83.2%

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

5. Taylor expanded in y.re around inf

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

   1. lower-*.f64 N/A

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

   2. lower-atan2.f64 81.2

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

7. Applied rewrites 81.2%

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

Alternative 4: 76.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -0.00205 \lor \neg \left(y.re \leq 4.2 \cdot 10^{+25}\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* y.re (atan2 x.im x.re)))))
   (if (or (<= y.re -0.00205) (not (<= y.re 4.2e+25)))
     (*
      (exp
       (-
        (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
        (* (atan2 x.im x.re) y.im)))
      t_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 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if ((y_46_re <= -0.00205) || !(y_46_re <= 4.2e+25)) {
		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))) * t_0;
	} else {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * 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 = cos((y_46re * atan2(x_46im, x_46re)))
    if ((y_46re <= (-0.00205d0)) .or. (.not. (y_46re <= 4.2d+25))) then
        tmp = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * t_0
    else
        tmp = exp((-y_46im * atan2(x_46im, x_46re))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if ((y_46_re <= -0.00205) || !(y_46_re <= 4.2e+25)) {
		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))) * t_0;
	} else {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if (y_46_re <= -0.00205) or not (y_46_re <= 4.2e+25):
		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))) * t_0
	else:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if ((y_46_re <= -0.00205) || !(y_46_re <= 4.2e+25))
		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))) * t_0);
	else
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if ((y_46_re <= -0.00205) || ~((y_46_re <= 4.2e+25)))
		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))) * t_0;
	else
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y$46$re, -0.00205], N[Not[LessEqual[y$46$re, 4.2e+25]], $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] * t$95$0), $MachinePrecision], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.00205000000000000017 or 4.1999999999999998e25 < y.re

   1. Initial program 40.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 76.0

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

   5. Applied rewrites 76.0%

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

   if -0.00205000000000000017 < y.re < 4.1999999999999998e25

   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. Add Preprocessing
   3. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-cos.f64 N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 82.4

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

   7. Applied rewrites 82.4%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 81.7

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

   10. Applied rewrites 81.7%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00205 \lor \neg \left(y.re \leq 4.2 \cdot 10^{+25}\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+25}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (cos (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -3.5e-8)
     (* t_0 (pow (hypot x.im x.re) y.re))
     (if (<= y.re 4.2e+25)
       (* (exp (* (- y.im) (atan2 x.im x.re))) t_0)
       (* t_0 (pow (sqrt (fma x.im x.im (* x.re 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 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -3.5e-8) {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 4.2e+25) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * 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 = cos(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -3.5e-8)
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 4.2e+25)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * t_0);
	else
		tmp = Float64(t_0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * 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[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -3.5e-8], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.2e+25], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.50000000000000024e-8

   1. Initial program 42.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 83.7

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

   5. Applied rewrites 83.7%

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

   if -3.50000000000000024e-8 < y.re < 4.1999999999999998e25

   1. Initial program 44.0%

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. fp-cancel-sub-sign-inv N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-atan2.f64 N/A

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

      10. lower-cos.f64 N/A

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

   4. Applied rewrites 84.5%

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

   5. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-atan2.f64 82.3

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 82.2

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

   10. Applied rewrites 82.2%

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

   if 4.1999999999999998e25 < y.re

   1. Initial program 36.7%

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

   3. Taylor expanded in y.im around 0

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

      1. 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. lower-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 63.6

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

   5. Applied rewrites 63.6%

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

      1. pow2 N/A

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

      2. pow2 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 N/A

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

      6. pow2 N/A

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

      7. lower-*.f64 63.6

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

   7. Applied rewrites 63.6%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{-8}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{+25}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 60.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+194}:\\ \;\;\;\;1 \cdot {\left(\frac{\mathsf{fma}\left(0.5, x.re \cdot x.re, x.im \cdot x.im\right)}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 2.55 \cdot 10^{+216}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log x.re\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.9e+194)
   (* 1.0 (pow (/ (fma 0.5 (* x.re x.re) (* x.im x.im)) x.im) y.re))
   (if (<= y.im 2.55e+216)
     (* (cos (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
     (* (cos (* y.im (log x.re))) (exp (* (- y.im) (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.9e+194) {
		tmp = 1.0 * pow((fma(0.5, (x_46_re * x_46_re), (x_46_im * x_46_im)) / x_46_im), y_46_re);
	} else if (y_46_im <= 2.55e+216) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = cos((y_46_im * log(x_46_re))) * exp((-y_46_im * 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_im <= -1.9e+194)
		tmp = Float64(1.0 * (Float64(fma(0.5, Float64(x_46_re * x_46_re), Float64(x_46_im * x_46_im)) / x_46_im) ^ y_46_re));
	elseif (y_46_im <= 2.55e+216)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(cos(Float64(y_46_im * log(x_46_re))) * exp(Float64(Float64(-y_46_im) * 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_] := If[LessEqual[y$46$im, -1.9e+194], N[(1.0 * N[Power[N[(N[(0.5 * N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.55e+216], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.8999999999999999e194

   1. Initial program 38.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 24.1

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

   5. Applied rewrites 24.1%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 17.3

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

   8. Applied rewrites 17.3%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 25.7%

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

   12. Taylor expanded in x.im around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lower-*.f64 39.9

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

   14. Applied rewrites 39.9%

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

   if -1.8999999999999999e194 < y.im < 2.55e216

   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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 73.0

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

   5. Applied rewrites 73.0%

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

   if 2.55e216 < y.im

   1. Initial program 55.6%

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

   3. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-atan2.f64 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 66.7

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

   8. Applied rewrites 66.7%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+194}:\\ \;\;\;\;1 \cdot {\left(\frac{\mathsf{fma}\left(0.5, x.re \cdot x.re, x.im \cdot x.im\right)}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 2.55 \cdot 10^{+216}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.im \cdot \log x.re\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+194}:\\ \;\;\;\;1 \cdot {\left(\frac{\mathsf{fma}\left(0.5, x.re \cdot x.re, x.im \cdot x.im\right)}{x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.9e+194)
   (* 1.0 (pow (/ (fma 0.5 (* x.re x.re) (* x.im x.im)) x.im) y.re))
   (* (cos (* y.re (atan2 x.im x.re))) (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 tmp;
	if (y_46_im <= -1.9e+194) {
		tmp = 1.0 * pow((fma(0.5, (x_46_re * x_46_re), (x_46_im * x_46_im)) / x_46_im), y_46_re);
	} else {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * 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)
	tmp = 0.0
	if (y_46_im <= -1.9e+194)
		tmp = Float64(1.0 * (Float64(fma(0.5, Float64(x_46_re * x_46_re), Float64(x_46_im * x_46_im)) / x_46_im) ^ y_46_re));
	else
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (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_] := If[LessEqual[y$46$im, -1.9e+194], N[(1.0 * N[Power[N[(N[(0.5 * N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.8999999999999999e194

   1. Initial program 38.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 24.1

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

   5. Applied rewrites 24.1%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 17.3

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

   8. Applied rewrites 17.3%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 25.7%

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

   12. Taylor expanded in x.im around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lower-*.f64 39.9

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

   14. Applied rewrites 39.9%

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

   if -1.8999999999999999e194 < y.im

   1. Initial program 42.5%

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

   3. Taylor expanded in y.im around 0

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

      1. 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. lower-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 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 8: 62.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+124}:\\ \;\;\;\;1 \cdot {\left(0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.9e+124)
   (* 1.0 (pow (* 0.5 (/ (* x.re x.re) x.im)) 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 tmp;
	if (y_46_im <= -1.9e+124) {
		tmp = 1.0 * pow((0.5 * ((x_46_re * x_46_re) / x_46_im)), y_46_re);
	} else {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.9e+124) {
		tmp = 1.0 * Math.pow((0.5 * ((x_46_re * x_46_re) / x_46_im)), y_46_re);
	} else {
		tmp = 1.0 * Math.pow(Math.hypot(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_im <= -1.9e+124:
		tmp = 1.0 * math.pow((0.5 * ((x_46_re * x_46_re) / x_46_im)), y_46_re)
	else:
		tmp = 1.0 * math.pow(math.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)
	tmp = 0.0
	if (y_46_im <= -1.9e+124)
		tmp = Float64(1.0 * (Float64(0.5 * Float64(Float64(x_46_re * x_46_re) / x_46_im)) ^ y_46_re));
	else
		tmp = Float64(1.0 * (hypot(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_im <= -1.9e+124)
		tmp = 1.0 * ((0.5 * ((x_46_re * x_46_re) / x_46_im)) ^ y_46_re);
	else
		tmp = 1.0 * (hypot(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$46$im, -1.9e+124], N[(1.0 * N[Power[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision], y$46$re], $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.im < -1.8999999999999999e124

   1. Initial program 37.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 22.0

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

   5. Applied rewrites 22.0%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 16.4

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

   8. Applied rewrites 16.4%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 20.4%

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

   12. Taylor expanded in x.re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 32.3

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

   14. Applied rewrites 32.3%

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

   if -1.8999999999999999e124 < y.im

   1. Initial program 43.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 74.0

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

   5. Applied rewrites 74.0%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 71.8%

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

Alternative 9: 50.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+103}:\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \lor \neg \left(y.re \leq 1.05\right):\\ \;\;\;\;1 \cdot {\left(0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.1e+103)
   (* 1.0 (pow x.im y.re))
   (if (or (<= y.re -1.6) (not (<= y.re 1.05)))
     (* 1.0 (pow (* 0.5 (/ (* x.re x.re) x.im)) y.re))
     1.0)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.1e+103) {
		tmp = 1.0 * pow(x_46_im, y_46_re);
	} else if ((y_46_re <= -1.6) || !(y_46_re <= 1.05)) {
		tmp = 1.0 * pow((0.5 * ((x_46_re * x_46_re) / x_46_im)), y_46_re);
	} else {
		tmp = 1.0;
	}
	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) :: tmp
    if (y_46re <= (-3.1d+103)) then
        tmp = 1.0d0 * (x_46im ** y_46re)
    else if ((y_46re <= (-1.6d0)) .or. (.not. (y_46re <= 1.05d0))) then
        tmp = 1.0d0 * ((0.5d0 * ((x_46re * x_46re) / x_46im)) ** y_46re)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.1e+103) {
		tmp = 1.0 * Math.pow(x_46_im, y_46_re);
	} else if ((y_46_re <= -1.6) || !(y_46_re <= 1.05)) {
		tmp = 1.0 * Math.pow((0.5 * ((x_46_re * x_46_re) / x_46_im)), y_46_re);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.1e+103:
		tmp = 1.0 * math.pow(x_46_im, y_46_re)
	elif (y_46_re <= -1.6) or not (y_46_re <= 1.05):
		tmp = 1.0 * math.pow((0.5 * ((x_46_re * x_46_re) / x_46_im)), y_46_re)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.1e+103)
		tmp = Float64(1.0 * (x_46_im ^ y_46_re));
	elseif ((y_46_re <= -1.6) || !(y_46_re <= 1.05))
		tmp = Float64(1.0 * (Float64(0.5 * Float64(Float64(x_46_re * x_46_re) / x_46_im)) ^ y_46_re));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.1e+103)
		tmp = 1.0 * (x_46_im ^ y_46_re);
	elseif ((y_46_re <= -1.6) || ~((y_46_re <= 1.05)))
		tmp = 1.0 * ((0.5 * ((x_46_re * x_46_re) / x_46_im)) ^ y_46_re);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.1e+103], N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$re, -1.6], N[Not[LessEqual[y$46$re, 1.05]], $MachinePrecision]], N[(1.0 * N[Power[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.1000000000000002e103

   1. Initial program 48.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 69.5

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

   8. Applied rewrites 69.5%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 66.9%

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

   12. Taylor expanded in x.re around 0

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

   14. Applied rewrites 72.1%

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

   if -3.1000000000000002e103 < y.re < -1.6000000000000001 or 1.05000000000000004 < 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. Add Preprocessing

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 64.1

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

   5. Applied rewrites 64.1%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 50.8

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

   8. Applied rewrites 50.8%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 50.8%

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

   12. Taylor expanded in x.re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 59.5

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

   14. Applied rewrites 59.5%

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

   if -1.6000000000000001 < y.re < 1.05000000000000004

   1. Initial program 43.6%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 56.4

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

   5. Applied rewrites 56.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 55.0%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+103}:\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \lor \neg \left(y.re \leq 1.05\right):\\ \;\;\;\;1 \cdot {\left(0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.2% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.4 \cdot 10^{-12}:\\ \;\;\;\;1 \cdot {\left(\frac{\mathsf{fma}\left(0.5, x.re \cdot x.re, x.im \cdot x.im\right)}{x.im}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.05:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(0.5 \cdot \frac{x.re \cdot x.re}{x.im}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5.4e-12)
   (* 1.0 (pow (/ (fma 0.5 (* x.re x.re) (* x.im x.im)) x.im) y.re))
   (if (<= y.re 1.05) 1.0 (* 1.0 (pow (* 0.5 (/ (* x.re x.re) x.im)) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -5.4e-12) {
		tmp = 1.0 * pow((fma(0.5, (x_46_re * x_46_re), (x_46_im * x_46_im)) / x_46_im), y_46_re);
	} else if (y_46_re <= 1.05) {
		tmp = 1.0;
	} else {
		tmp = 1.0 * pow((0.5 * ((x_46_re * x_46_re) / x_46_im)), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -5.4e-12)
		tmp = Float64(1.0 * (Float64(fma(0.5, Float64(x_46_re * x_46_re), Float64(x_46_im * x_46_im)) / x_46_im) ^ y_46_re));
	elseif (y_46_re <= 1.05)
		tmp = 1.0;
	else
		tmp = Float64(1.0 * (Float64(0.5 * Float64(Float64(x_46_re * x_46_re) / x_46_im)) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -5.4e-12], N[(1.0 * N[Power[N[(N[(0.5 * N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05], 1.0, N[(1.0 * N[Power[N[(0.5 * N[(N[(x$46$re * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.39999999999999961e-12

   1. Initial program 41.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 81.1

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

   5. Applied rewrites 81.1%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 63.7

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

   8. Applied rewrites 63.7%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 60.6%

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

   12. Taylor expanded in x.im around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 N/A

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

       5. pow2 N/A

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

       6. lower-*.f64 73.5

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

   14. Applied rewrites 73.5%

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

   if -5.39999999999999961e-12 < y.re < 1.05000000000000004

   1. Initial program 45.0%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 56.5%

      \[\leadsto 1 \]

   if 1.05000000000000004 < y.re

   1. Initial program 36.5%

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

   3. Taylor expanded in y.im around 0

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

      1. 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. lower-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 60.6

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

   5. Applied rewrites 60.6%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 47.9

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

   8. Applied rewrites 47.9%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 47.9%

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

   12. Taylor expanded in x.re around inf

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 N/A

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

       3. pow2 N/A

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

       4. lower-*.f64 57.6

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

   14. Applied rewrites 57.6%

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

Alternative 11: 50.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.65 \cdot 10^{-9} \lor \neg \left(y.re \leq 1.7 \cdot 10^{+29}\right):\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.65e-9) (not (<= y.re 1.7e+29)))
   (* 1.0 (pow x.im y.re))
   1.0))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.65e-9) || !(y_46_re <= 1.7e+29)) {
		tmp = 1.0 * pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0;
	}
	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) :: tmp
    if ((y_46re <= (-3.65d-9)) .or. (.not. (y_46re <= 1.7d+29))) then
        tmp = 1.0d0 * (x_46im ** y_46re)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.65e-9) || !(y_46_re <= 1.7e+29)) {
		tmp = 1.0 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -3.65e-9) or not (y_46_re <= 1.7e+29):
		tmp = 1.0 * math.pow(x_46_im, y_46_re)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -3.65e-9) || !(y_46_re <= 1.7e+29))
		tmp = Float64(1.0 * (x_46_im ^ y_46_re));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.65e-9) || ~((y_46_re <= 1.7e+29)))
		tmp = 1.0 * (x_46_im ^ y_46_re);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.65e-9], N[Not[LessEqual[y$46$re, 1.7e+29]], $MachinePrecision]], N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.65000000000000001e-9 or 1.69999999999999991e29 < y.re

   1. Initial program 39.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      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. lower-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 72.9

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

   5. Applied rewrites 72.9%

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

   6. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. pow2 N/A

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

      6. lower-*.f64 N/A

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

      7. pow2 N/A

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

      8. lower-*.f64 57.2

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

   8. Applied rewrites 57.2%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 55.6%

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

   12. Taylor expanded in x.re around 0

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

   14. Applied rewrites 51.4%

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

   if -3.65000000000000001e-9 < y.re < 1.69999999999999991e29

   1. Initial program 44.0%

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

   3. Taylor expanded in y.im around 0

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

      1. 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. lower-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 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 54.5%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.65 \cdot 10^{-9} \lor \neg \left(y.re \leq 1.7 \cdot 10^{+29}\right):\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 25.5% accurate, 680.0× speedup

Math

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

FPCore

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

C

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

Fortran

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 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. Add Preprocessing

3. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   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. lower-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 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 30.0%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025044 
(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)))))