powComplex, real part

Percentage Accurate: 41.5% → 80.8%

Time: 9.4s

Alternatives: 11

Speedup: 5.7×

• 35.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 80.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := e^{\mathsf{fma}\left(t\_0, y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}\\ \mathbf{if}\;y.im \leq -28.5:\\ \;\;\;\;t\_1 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+22}:\\ \;\;\;\;1 \cdot e^{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \cos \left(\mathsf{fma}\left(t\_0, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \end{array} \end{array} \]

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 (<= y.im -28.5)
     (* t_1 (cos (* y.re (atan2 x.im x.re))))
     (if (<= y.im 2.7e+22)
       (* 1.0 (exp (* (log (hypot x.im x.re)) y.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 (y_46_im <= -28.5) {
		tmp = t_1 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= 2.7e+22) {
		tmp = 1.0 * exp((log(hypot(x_46_im, x_46_re)) * y_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 (y_46_im <= -28.5)
		tmp = Float64(t_1 * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_im <= 2.7e+22)
		tmp = Float64(1.0 * exp(Float64(log(hypot(x_46_im, x_46_re)) * y_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[y$46$im, -28.5], N[(t$95$1 * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.7e+22], N[(1.0 * N[Exp[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $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 3 regimes

2. if y.im < -28.5

   1. Initial program 35.9%

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

   4. Applied rewrites 70.8%

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

   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. lift-atan2.f64 N/A

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

      2. lift-*.f64 69.9

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

   7. Applied rewrites 69.9%

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

   if -28.5 < y.im < 2.7000000000000002e22

   1. Initial program 46.3%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 91.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. sqrt-pow2 N/A

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

      7. pow-to-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-hypot.f64 91.3

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

   10. Applied rewrites 91.3%

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

   if 2.7000000000000002e22 < y.im

   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. Step-by-step derivation

   4. Applied rewrites 67.9%

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

Alternative 2: 81.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -28.5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+22}:\\ \;\;\;\;1 \cdot e^{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (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))))))
   (if (<= y.im -28.5)
     t_0
     (if (<= y.im 2.7e+22)
       (* 1.0 (exp (* (log (hypot x.im x.re)) y.re)))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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)));
	double tmp;
	if (y_46_im <= -28.5) {
		tmp = t_0;
	} else if (y_46_im <= 2.7e+22) {
		tmp = 1.0 * exp((log(hypot(x_46_im, x_46_re)) * y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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))))
	tmp = 0.0
	if (y_46_im <= -28.5)
		tmp = t_0;
	elseif (y_46_im <= 2.7e+22)
		tmp = Float64(1.0 * exp(Float64(log(hypot(x_46_im, x_46_re)) * y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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]}, If[LessEqual[y$46$im, -28.5], t$95$0, If[LessEqual[y$46$im, 2.7e+22], N[(1.0 * N[Exp[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -28.5 or 2.7000000000000002e22 < y.im

   1. Initial program 36.0%

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

   4. Applied rewrites 69.4%

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

   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. lift-atan2.f64 N/A

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

      2. lift-*.f64 69.4

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

   7. Applied rewrites 69.4%

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

   if -28.5 < y.im < 2.7000000000000002e22

   1. Initial program 46.3%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 91.3%

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

      1. lift-pow.f64 N/A

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

      2. lift-hypot.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. pow2 N/A

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

      5. pow2 N/A

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

      6. sqrt-pow2 N/A

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

      7. pow-to-exp N/A

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

      8. lower-exp.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. pow2 N/A

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

      11. pow2 N/A

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

      12. lower-log.f64 N/A

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

      13. lift-hypot.f64 91.3

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

   10. Applied rewrites 91.3%

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

Alternative 3: 76.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+29}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 0.32:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.5e+29)
   (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
   (if (<= y.re 0.32)
     (* (exp (- (* y.im (atan2 x.im x.re)))) (cos (* y.re (atan2 x.im x.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_re <= -8.5e+29) {
		tmp = 1.0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	} else if (y_46_re <= 0.32) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8.5e+29)
		tmp = Float64(1.0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	elseif (y_46_re <= 0.32)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8.5e+29], N[(1.0 * 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], If[LessEqual[y$46$re, 0.32], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 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 3 regimes

2. if y.re < -8.5000000000000006e29

   1. Initial program 44.3%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 80.4

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

   5. Applied rewrites 80.4%

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

   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 82.9%

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

      1. lift-hypot.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 82.9

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

   10. Applied rewrites 82.9%

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

   if -8.5000000000000006e29 < y.re < 0.320000000000000007

   1. Initial program 42.4%

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

   3. Taylor expanded in y.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. lift-atan2.f64 50.7

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 78.0

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

   8. Applied rewrites 78.0%

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

   if 0.320000000000000007 < y.re

   1. Initial program 37.1%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 59.4

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

   5. Applied rewrites 59.4%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 68.2%

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

Alternative 4: 64.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -4100000:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.re \cdot x.re}\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+87}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4100000.0)
   (* (cos (* y.re (atan2 x.im x.re))) (pow (sqrt (* x.re x.re)) y.re))
   (if (<= y.im 3.4e+87)
     (* 1.0 (pow (hypot x.im x.re) y.re))
     (* 1.0 (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 tmp;
	if (y_46_im <= -4100000.0) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(sqrt((x_46_re * x_46_re)), y_46_re);
	} else if (y_46_im <= 3.4e+87) {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = 1.0 * 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)
	tmp = 0.0
	if (y_46_im <= -4100000.0)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (sqrt(Float64(x_46_re * x_46_re)) ^ y_46_re));
	elseif (y_46_im <= 3.4e+87)
		tmp = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(1.0 * (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_] := If[LessEqual[y$46$im, -4100000.0], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(x$46$re * x$46$re), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.4e+87], N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(1.0 * 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.im < -4.1e6

   1. Initial program 35.9%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 31.7

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

   5. Applied rewrites 31.7%

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

      1. lift-hypot.f64 N/A

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

      2. pow2 N/A

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

      3. pow2 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 38.5

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in x.re around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 34.1

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

   10. Applied rewrites 34.1%

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

   if -4.1e6 < y.im < 3.4000000000000002e87

   1. Initial program 45.2%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 81.9

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

   5. Applied rewrites 81.9%

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

   6. Taylor expanded in 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 86.2%

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

   if 3.4000000000000002e87 < y.im

   1. Initial program 36.6%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 29.2

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

   5. Applied rewrites 29.2%

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

   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 29.9%

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

      1. lift-hypot.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 35.4

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

   10. Applied rewrites 35.4%

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

Alternative 5: 65.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+87}:\\ \;\;\;\;1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.im -1.85e+25)
     t_0
     (if (<= y.im 3.4e+87) (* 1.0 (pow (hypot x.im x.re) y.re)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_im <= -1.85e+25) {
		tmp = t_0;
	} else if (y_46_im <= 3.4e+87) {
		tmp = 1.0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_im <= -1.85e+25)
		tmp = t_0;
	elseif (y_46_im <= 3.4e+87)
		tmp = Float64(1.0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.85e+25], t$95$0, If[LessEqual[y$46$im, 3.4e+87], N[(1.0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.8499999999999999e25 or 3.4000000000000002e87 < y.im

   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. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 30.7

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

   5. Applied rewrites 30.7%

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

   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 30.9%

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

      1. lift-hypot.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 37.1

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

   10. Applied rewrites 37.1%

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

   if -1.8499999999999999e25 < y.im < 3.4000000000000002e87

   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. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 80.7

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

   5. Applied rewrites 80.7%

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

   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 84.9%

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

Alternative 6: 62.5% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-32}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -2.8e-36) t_0 (if (<= y.re 7e-32) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -2.8e-36) {
		tmp = t_0;
	} else if (y_46_re <= 7e-32) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.8e-36)
		tmp = t_0;
	elseif (y_46_re <= 7e-32)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.8e-36], t$95$0, If[LessEqual[y$46$re, 7e-32], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.8000000000000001e-36 or 6.9999999999999997e-32 < y.re

   1. Initial program 40.8%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in 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 72.7%

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

      1. lift-hypot.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 71.0

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

   10. Applied rewrites 71.0%

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

   if -2.8000000000000001e-36 < y.re < 6.9999999999999997e-32

   1. Initial program 42.3%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 51.8

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

   5. Applied rewrites 51.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 51.8%

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

Alternative 7: 53.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {x.re}^{y.re}\\ \mathbf{if}\;y.re \leq -1.75 \cdot 10^{+109}:\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq -5.9 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1250:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow x.re y.re))))
   (if (<= y.re -1.75e+109)
     (* 1.0 (pow x.im y.re))
     (if (<= y.re -5.9e-5) t_0 (if (<= y.re 1250.0) 1.0 t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -1.75e+109) {
		tmp = 1.0 * pow(x_46_im, y_46_re);
	} else if (y_46_re <= -5.9e-5) {
		tmp = t_0;
	} else if (y_46_re <= 1250.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 * (x_46re ** y_46re)
    if (y_46re <= (-1.75d+109)) then
        tmp = 1.0d0 * (x_46im ** y_46re)
    else if (y_46re <= (-5.9d-5)) then
        tmp = t_0
    else if (y_46re <= 1250.0d0) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -1.75e+109) {
		tmp = 1.0 * Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= -5.9e-5) {
		tmp = t_0;
	} else if (y_46_re <= 1250.0) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(x_46_re, y_46_re)
	tmp = 0
	if y_46_re <= -1.75e+109:
		tmp = 1.0 * math.pow(x_46_im, y_46_re)
	elif y_46_re <= -5.9e-5:
		tmp = t_0
	elif y_46_re <= 1250.0:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (x_46_re ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.75e+109)
		tmp = Float64(1.0 * (x_46_im ^ y_46_re));
	elseif (y_46_re <= -5.9e-5)
		tmp = t_0;
	elseif (y_46_re <= 1250.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (x_46_re ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1.75e+109)
		tmp = 1.0 * (x_46_im ^ y_46_re);
	elseif (y_46_re <= -5.9e-5)
		tmp = t_0;
	elseif (y_46_re <= 1250.0)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.75e+109], N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.9e-5], t$95$0, If[LessEqual[y$46$re, 1250.0], 1.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.74999999999999992e109

   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. 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. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 83.9%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 60.7%

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

   if -1.74999999999999992e109 < y.re < -5.8999999999999998e-5 or 1250 < 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. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in 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 70.4%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 53.8%

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

   if -5.8999999999999998e-5 < y.re < 1250

   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. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 52.1

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

   5. Applied rewrites 52.1%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 50.8%

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

Alternative 8: 56.0% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -1.35 \cdot 10^{-97}:\\ \;\;\;\;1 \cdot {\left(-1 \cdot x.re\right)}^{y.re}\\ \mathbf{elif}\;x.re \leq 1.1 \cdot 10^{-126}:\\ \;\;\;\;1 \cdot {\left(-1 \cdot x.im\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -1.35e-97)
   (* 1.0 (pow (* -1.0 x.re) y.re))
   (if (<= x.re 1.1e-126)
     (* 1.0 (pow (* -1.0 x.im) y.re))
     (* 1.0 (pow x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1.35e-97) {
		tmp = 1.0 * pow((-1.0 * x_46_re), y_46_re);
	} else if (x_46_re <= 1.1e-126) {
		tmp = 1.0 * pow((-1.0 * x_46_im), y_46_re);
	} else {
		tmp = 1.0 * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-1.35d-97)) then
        tmp = 1.0d0 * (((-1.0d0) * x_46re) ** y_46re)
    else if (x_46re <= 1.1d-126) then
        tmp = 1.0d0 * (((-1.0d0) * x_46im) ** y_46re)
    else
        tmp = 1.0d0 * (x_46re ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -1.35e-97) {
		tmp = 1.0 * Math.pow((-1.0 * x_46_re), y_46_re);
	} else if (x_46_re <= 1.1e-126) {
		tmp = 1.0 * Math.pow((-1.0 * x_46_im), y_46_re);
	} else {
		tmp = 1.0 * Math.pow(x_46_re, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -1.35e-97:
		tmp = 1.0 * math.pow((-1.0 * x_46_re), y_46_re)
	elif x_46_re <= 1.1e-126:
		tmp = 1.0 * math.pow((-1.0 * x_46_im), y_46_re)
	else:
		tmp = 1.0 * math.pow(x_46_re, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -1.35e-97)
		tmp = Float64(1.0 * (Float64(-1.0 * x_46_re) ^ y_46_re));
	elseif (x_46_re <= 1.1e-126)
		tmp = Float64(1.0 * (Float64(-1.0 * x_46_im) ^ y_46_re));
	else
		tmp = Float64(1.0 * (x_46_re ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -1.35e-97)
		tmp = 1.0 * ((-1.0 * x_46_re) ^ y_46_re);
	elseif (x_46_re <= 1.1e-126)
		tmp = 1.0 * ((-1.0 * x_46_im) ^ y_46_re);
	else
		tmp = 1.0 * (x_46_re ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -1.35e-97], N[(1.0 * N[Power[N[(-1.0 * x$46$re), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.1e-126], N[(1.0 * N[Power[N[(-1.0 * x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.34999999999999993e-97

   1. Initial program 38.2%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 58.0

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

   5. Applied rewrites 58.0%

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

   6. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 55.1

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

   8. Applied rewrites 55.1%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 57.4%

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

   if -1.34999999999999993e-97 < x.re < 1.10000000000000007e-126

   1. Initial program 48.0%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 60.7

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

   5. Applied rewrites 60.7%

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

   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 64.8%

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

   9. Taylor expanded in x.im around -inf

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

       1. lower-*.f64 50.9

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

   11. Applied rewrites 50.9%

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

   if 1.10000000000000007e-126 < x.re

   1. Initial program 38.7%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 64.5%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 59.1%

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

Alternative 9: 56.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -1.15 \cdot 10^{-33}:\\ \;\;\;\;1 \cdot {\left(-1 \cdot x.im\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 0.036:\\ \;\;\;\;1 \cdot {x.re}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -1.15e-33)
   (* 1.0 (pow (* -1.0 x.im) y.re))
   (if (<= x.im 0.036) (* 1.0 (pow x.re y.re)) (* 1.0 (pow x.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1.15e-33) {
		tmp = 1.0 * pow((-1.0 * x_46_im), y_46_re);
	} else if (x_46_im <= 0.036) {
		tmp = 1.0 * pow(x_46_re, y_46_re);
	} else {
		tmp = 1.0 * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-1.15d-33)) then
        tmp = 1.0d0 * (((-1.0d0) * x_46im) ** y_46re)
    else if (x_46im <= 0.036d0) then
        tmp = 1.0d0 * (x_46re ** y_46re)
    else
        tmp = 1.0d0 * (x_46im ** y_46re)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -1.15e-33) {
		tmp = 1.0 * Math.pow((-1.0 * x_46_im), y_46_re);
	} else if (x_46_im <= 0.036) {
		tmp = 1.0 * Math.pow(x_46_re, y_46_re);
	} else {
		tmp = 1.0 * Math.pow(x_46_im, y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -1.15e-33:
		tmp = 1.0 * math.pow((-1.0 * x_46_im), y_46_re)
	elif x_46_im <= 0.036:
		tmp = 1.0 * math.pow(x_46_re, y_46_re)
	else:
		tmp = 1.0 * math.pow(x_46_im, y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.15e-33)
		tmp = Float64(1.0 * (Float64(-1.0 * x_46_im) ^ y_46_re));
	elseif (x_46_im <= 0.036)
		tmp = Float64(1.0 * (x_46_re ^ y_46_re));
	else
		tmp = Float64(1.0 * (x_46_im ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -1.15e-33)
		tmp = 1.0 * ((-1.0 * x_46_im) ^ y_46_re);
	elseif (x_46_im <= 0.036)
		tmp = 1.0 * (x_46_re ^ y_46_re);
	else
		tmp = 1.0 * (x_46_im ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -1.15e-33], N[(1.0 * N[Power[N[(-1.0 * x$46$im), $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 0.036], N[(1.0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision], N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.14999999999999993e-33

   1. Initial program 31.6%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 56.9

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

   5. Applied rewrites 56.9%

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

   6. Taylor expanded in 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 60.6%

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

   9. Taylor expanded in x.im around -inf

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

       1. lower-*.f64 58.8

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

   11. Applied rewrites 58.8%

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

   if -1.14999999999999993e-33 < x.im < 0.0359999999999999973

   1. Initial program 51.5%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 62.5

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

   5. Applied rewrites 62.5%

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

   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 64.4%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 50.6%

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

   if 0.0359999999999999973 < x.im

   1. Initial program 32.9%

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

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 61.8

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

   5. Applied rewrites 61.8%

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

   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 64.7%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 64.5%

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

Alternative 10: 52.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-13}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (pow x.im y.re))))
   (if (<= y.re -5.8e+19) t_0 (if (<= y.re 1.8e-13) 1.0 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * pow(x_46_im, y_46_re);
	double tmp;
	if (y_46_re <= -5.8e+19) {
		tmp = t_0;
	} else if (y_46_re <= 1.8e-13) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 * (x_46im ** y_46re)
    if (y_46re <= (-5.8d+19)) then
        tmp = t_0
    else if (y_46re <= 1.8d-13) then
        tmp = 1.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * Math.pow(x_46_im, y_46_re);
	double tmp;
	if (y_46_re <= -5.8e+19) {
		tmp = t_0;
	} else if (y_46_re <= 1.8e-13) {
		tmp = 1.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 * math.pow(x_46_im, y_46_re)
	tmp = 0
	if y_46_re <= -5.8e+19:
		tmp = t_0
	elif y_46_re <= 1.8e-13:
		tmp = 1.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * (x_46_im ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -5.8e+19)
		tmp = t_0;
	elseif (y_46_re <= 1.8e-13)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 * (x_46_im ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -5.8e+19)
		tmp = t_0;
	elseif (y_46_re <= 1.8e-13)
		tmp = 1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e+19], t$95$0, If[LessEqual[y$46$re, 1.8e-13], 1.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.8e19 or 1.7999999999999999e-13 < y.re

   1. Initial program 40.4%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 68.9

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

   5. Applied rewrites 68.9%

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

   6. Taylor expanded in 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 74.6%

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

   9. Taylor expanded in x.re around 0

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

   11. Applied rewrites 55.7%

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

   if -5.8e19 < y.re < 1.7999999999999999e-13

   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. lift-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. lower-hypot.f64 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 49.5%

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

Alternative 11: 26.6% accurate, 680.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.5%

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

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

   5. lower-pow.f64 N/A

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

   6. pow2 N/A

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

   7. pow2 N/A

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

   8. lower-hypot.f64 60.8

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

5. Applied rewrites 60.8%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 26.6%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025089 
(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)))))