powComplex, real part

Percentage Accurate: 41.0% → 79.0%

Time: 14.3s

Alternatives: 11

Speedup: 2.5×

• 35.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 41.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -1 \cdot 10^{-21}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{-1}{x.re}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{elif}\;x.re \leq 4.8 \cdot 10^{-53}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \cos t\_1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-1, y.im \cdot t\_2, t\_1\right)\right) \cdot e^{-1 \cdot \left(y.re \cdot t\_2\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ 1.0 x.re))))
   (if (<= x.re -1e-21)
     (* (exp (- (* (* -1.0 (log (/ -1.0 x.re))) y.re) t_0)) 1.0)
     (if (<= x.re 4.8e-53)
       (*
        (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
        (cos t_1))
       (*
        (cos (fma -1.0 (* y.im t_2) t_1))
        (exp (- (* -1.0 (* y.re t_2)) (* y.im (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log((1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -1e-21) {
		tmp = exp((((-1.0 * log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_re <= 4.8e-53) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * cos(t_1);
	} else {
		tmp = cos(fma(-1.0, (y_46_im * t_2), t_1)) * exp(((-1.0 * (y_46_re * t_2)) - (y_46_im * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(Float64(1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -1e-21)
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0);
	elseif (x_46_re <= 4.8e-53)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * cos(t_1));
	else
		tmp = Float64(cos(fma(-1.0, Float64(y_46_im * t_2), t_1)) * exp(Float64(Float64(-1.0 * Float64(y_46_re * t_2)) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1e-21], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 4.8e-53], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(-1.0 * N[(y$46$im * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(-1.0 * N[(y$46$re * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -9.99999999999999908e-22

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.re around -inf

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

   10. Applied rewrites 38.2%

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

   if -9.99999999999999908e-22 < x.re < 4.80000000000000015e-53

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   if 4.80000000000000015e-53 < x.re

   1. Initial program 41.0%

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

   2. Taylor expanded in x.re around inf

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

   4. Applied rewrites 34.8%

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

Alternative 2: 77.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \cos t\_1\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-78}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\sqrt{0.5}\right), y.im, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)}\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)) (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -7.6e-17)
     (*
      (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
      (cos t_1))
     (if (<= y.re 1.25e-78)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (cos (fma (log (sqrt 0.5)) y.im t_1)))
       (*
        (exp (- (* (log (sqrt (fma x.re x.re (* x.im x.im)))) y.re) t_0))
        1.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -7.6e-17) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * cos(t_1);
	} else if (y_46_re <= 1.25e-78) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos(fma(log(sqrt(0.5)), y_46_im, t_1));
	} else {
		tmp = exp(((log(sqrt(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -7.6e-17)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * cos(t_1));
	elseif (y_46_re <= 1.25e-78)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(fma(log(sqrt(0.5)), y_46_im, t_1)));
	else
		tmp = Float64(exp(Float64(Float64(log(sqrt(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7.6e-17], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.25e-78], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.6000000000000002e-17

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   if -7.6000000000000002e-17 < y.re < 1.2499999999999999e-78

   1. Initial program 41.0%

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

   2. Taylor expanded in y.re around 0

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

   4. Applied rewrites 27.5%

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

   6. Applied rewrites 10.6%

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 53.2%

       \[\leadsto e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\sqrt{\color{blue}{0.5}}\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   if 1.2499999999999999e-78 < y.re

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 64.5%

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

Alternative 3: 77.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{if}\;y.re \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-78}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\sqrt{0.5}\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (-
            (* (log (sqrt (fma x.re x.re (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          1.0)))
   (if (<= y.re -7.6e-17)
     t_0
     (if (<= y.re 1.25e-78)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (cos (fma (log (sqrt 0.5)) y.im (* y.re (atan2 x.im x.re)))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	double tmp;
	if (y_46_re <= -7.6e-17) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-78) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * cos(fma(log(sqrt(0.5)), y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(exp(Float64(Float64(log(sqrt(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0)
	tmp = 0.0
	if (y_46_re <= -7.6e-17)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-78)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * cos(fma(log(sqrt(0.5)), y_46_im, Float64(y_46_re * atan(x_46_im, x_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[(N[Log[N[Sqrt[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -7.6e-17], t$95$0, If[LessEqual[y$46$re, 1.25e-78], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Log[N[Sqrt[0.5], $MachinePrecision]], $MachinePrecision] * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.6000000000000002e-17 or 1.2499999999999999e-78 < y.re

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 64.5%

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

   if -7.6000000000000002e-17 < y.re < 1.2499999999999999e-78

   1. Initial program 41.0%

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

   2. Taylor expanded in y.re around 0

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

   4. Applied rewrites 27.5%

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

   6. Applied rewrites 10.6%

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in x.re around inf

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

   11. Applied rewrites 53.2%

       \[\leadsto e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \cos \left(\mathsf{fma}\left(\log \left(\sqrt{\color{blue}{0.5}}\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 74.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{if}\;y.re \leq -7.6 \cdot 10^{-17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{-34}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\sqrt{\frac{x.re - 1}{2 \cdot x.re - 1}}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (-
            (* (log (sqrt (fma x.re x.re (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          1.0)))
   (if (<= y.re -7.6e-17)
     t_0
     (if (<= y.re 9e-34)
       (/
        (cos (* y.im (log (sqrt (/ (- x.re 1.0) (- (* 2.0 x.re) 1.0))))))
        (exp (* y.im (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((log(sqrt(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	double tmp;
	if (y_46_re <= -7.6e-17) {
		tmp = t_0;
	} else if (y_46_re <= 9e-34) {
		tmp = cos((y_46_im * log(sqrt(((x_46_re - 1.0) / ((2.0 * x_46_re) - 1.0)))))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(exp(Float64(Float64(log(sqrt(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0)
	tmp = 0.0
	if (y_46_re <= -7.6e-17)
		tmp = t_0;
	elseif (y_46_re <= 9e-34)
		tmp = Float64(cos(Float64(y_46_im * log(sqrt(Float64(Float64(x_46_re - 1.0) / Float64(Float64(2.0 * x_46_re) - 1.0)))))) / exp(Float64(y_46_im * atan(x_46_im, x_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[(N[Log[N[Sqrt[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]}, If[LessEqual[y$46$re, -7.6e-17], t$95$0, If[LessEqual[y$46$re, 9e-34], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[N[(N[(x$46$re - 1.0), $MachinePrecision] / N[(N[(2.0 * x$46$re), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.6000000000000002e-17 or 9.00000000000000085e-34 < y.re

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 64.5%

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

   if -7.6000000000000002e-17 < y.re < 9.00000000000000085e-34

   1. Initial program 41.0%

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

   2. Taylor expanded in y.re around 0

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

   4. Applied rewrites 27.5%

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

   6. Applied rewrites 10.6%

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

   8. Applied rewrites 53.1%

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

   9. Taylor expanded in y.re around 0

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

   11. Applied rewrites 53.0%

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

Alternative 5: 73.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{+21}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{-1}{x.re}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{elif}\;x.re \leq 3.8 \cdot 10^{-102}:\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)}\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.re}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -2e+21)
     (* (exp (- (* (* -1.0 (log (/ -1.0 x.re))) y.re) t_0)) 1.0)
     (if (<= x.re 3.8e-102)
       (*
        (exp (- (* (log (sqrt (fma x.re x.re (* x.im x.im)))) y.re) t_0))
        1.0)
       (* (exp (- (* (* -1.0 (log (/ 1.0 x.re))) y.re) t_0)) 1.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -2e+21) {
		tmp = exp((((-1.0 * log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_re <= 3.8e-102) {
		tmp = exp(((log(sqrt(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp((((-1.0 * log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -2e+21)
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0);
	elseif (x_46_re <= 3.8e-102)
		tmp = Float64(exp(Float64(Float64(log(sqrt(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -2e+21], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], If[LessEqual[x$46$re, 3.8e-102], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2e21

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.re around -inf

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

   10. Applied rewrites 38.2%

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

   if -2e21 < x.re < 3.80000000000000026e-102

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   9. Applied rewrites 64.5%

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

   if 3.80000000000000026e-102 < x.re

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.re around inf

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

   10. Applied rewrites 34.9%

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

Alternative 6: 73.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -5 \cdot 10^{-310}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{-1}{x.re}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.re}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -5e-310)
     (* (exp (- (* (* -1.0 (log (/ -1.0 x.re))) y.re) t_0)) 1.0)
     (* (exp (- (* (* -1.0 (log (/ 1.0 x.re))) y.re) t_0)) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -5e-310) {
		tmp = exp((((-1.0 * log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp((((-1.0 * log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= (-5d-310)) then
        tmp = exp(((((-1.0d0) * log(((-1.0d0) / x_46re))) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((((-1.0d0) * log((1.0d0 / x_46re))) * y_46re) - t_0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -5e-310) {
		tmp = Math.exp((((-1.0 * Math.log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = Math.exp((((-1.0 * Math.log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= -5e-310:
		tmp = math.exp((((-1.0 * math.log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0
	else:
		tmp = math.exp((((-1.0 * math.log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -5e-310)
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= -5e-310)
		tmp = exp((((-1.0 * log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp((((-1.0 * log((1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -5e-310], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.999999999999985e-310

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.re around -inf

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

   10. Applied rewrites 38.2%

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

   if -4.999999999999985e-310 < x.re

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.re around inf

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

   10. Applied rewrites 34.9%

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

Alternative 7: 72.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -2 \cdot 10^{-310}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.im}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -2e-310)
     (* (exp (- (* (log (* -1.0 x.im)) y.re) t_0)) 1.0)
     (* (exp (- (* (* -1.0 (log (/ 1.0 x.im))) y.re) t_0)) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -2e-310) {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp((((-1.0 * log((1.0 / x_46_im))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46im <= (-2d-310)) then
        tmp = exp(((log(((-1.0d0) * x_46im)) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((((-1.0d0) * log((1.0d0 / x_46im))) * y_46re) - t_0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -2e-310) {
		tmp = Math.exp(((Math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = Math.exp((((-1.0 * Math.log((1.0 / x_46_im))) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -2e-310:
		tmp = math.exp(((math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0
	else:
		tmp = math.exp((((-1.0 * math.log((1.0 / x_46_im))) * y_46_re) - t_0)) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -2e-310)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(1.0 / x_46_im))) * y_46_re) - t_0)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -2e-310)
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp((((-1.0 * log((1.0 / x_46_im))) * y_46_re) - t_0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -2e-310], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.999999999999994e-310

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.im around -inf

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

   10. Applied rewrites 35.8%

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

   if -1.999999999999994e-310 < x.im

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.im around inf

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

   10. Applied rewrites 36.5%

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

Alternative 8: 55.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{-309}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{-1}{x.re}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -4e-309)
     (* (exp (- (* (* -1.0 (log (/ -1.0 x.re))) y.re) t_0)) 1.0)
     (* (exp (- (* (log (* -1.0 x.im)) y.re) t_0)) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -4e-309) {
		tmp = exp((((-1.0 * log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= (-4d-309)) then
        tmp = exp(((((-1.0d0) * log(((-1.0d0) / x_46re))) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((log(((-1.0d0) * x_46im)) * y_46re) - t_0)) * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -4e-309) {
		tmp = Math.exp((((-1.0 * Math.log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = Math.exp(((Math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= -4e-309:
		tmp = math.exp((((-1.0 * math.log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0
	else:
		tmp = math.exp(((math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -4e-309)
		tmp = Float64(exp(Float64(Float64(Float64(-1.0 * log(Float64(-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= -4e-309)
		tmp = exp((((-1.0 * log((-1.0 / x_46_re))) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -4e-309], N[(N[Exp[N[(N[(N[(-1.0 * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -3.9999999999999977e-309

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.re around -inf

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

   10. Applied rewrites 38.2%

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

   if -3.9999999999999977e-309 < x.re

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.im around -inf

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

   10. Applied rewrites 35.8%

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

Alternative 9: 49.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{-309}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;1 + -1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -8e-309)
   (* (exp (- (* (log (* -1.0 x.im)) y.re) (* (atan2 x.im x.re) y.im))) 1.0)
   (+ 1.0 (* -1.0 (* y.im (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -8e-309) {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	} else {
		tmp = 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_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 <= (-8d-309)) then
        tmp = exp(((log(((-1.0d0) * x_46im)) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 1.0d0
    else
        tmp = 1.0d0 + ((-1.0d0) * (y_46im * atan2(x_46im, x_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 <= -8e-309) {
		tmp = Math.exp(((Math.log((-1.0 * x_46_im)) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	} else {
		tmp = 1.0 + (-1.0 * (y_46_im * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -8e-309:
		tmp = math.exp(((math.log((-1.0 * x_46_im)) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0
	else:
		tmp = 1.0 + (-1.0 * (y_46_im * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -8e-309)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0);
	else
		tmp = Float64(1.0 + Float64(-1.0 * Float64(y_46_im * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -8e-309)
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	else
		tmp = 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -8e-309], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(1.0 + N[(-1.0 * N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -8.0000000000000003e-309

   1. Initial program 41.0%

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

   2. Taylor expanded in y.im around 0

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

   4. Applied rewrites 62.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.5%

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

   8. Taylor expanded in x.im around -inf

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

   10. Applied rewrites 35.8%

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

   if -8.0000000000000003e-309 < x.im

   1. Initial program 41.0%

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

   2. Taylor expanded in x.im around inf

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

   4. Applied rewrites 35.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 27.1%

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

   8. Taylor expanded in y.im around 0

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

   10. Applied rewrites 26.4%

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

Alternative 10: 26.4% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (+ 1.0 (* -1.0 (* y.im (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_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
    code = 1.0d0 + ((-1.0d0) * (y_46im * atan2(x_46im, x_46re)))
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 + (-1.0 * (y_46_im * Math.atan2(x_46_im, x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 + (-1.0 * (y_46_im * math.atan2(x_46_im, x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 + Float64(-1.0 * Float64(y_46_im * atan(x_46_im, x_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 + (-1.0 * (y_46_im * atan2(x_46_im, x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 + N[(-1.0 * N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 41.0%

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

2. Taylor expanded in x.im around inf

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

4. Applied rewrites 35.9%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 27.1%

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

8. Taylor expanded in y.im around 0

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

10. Applied rewrites 26.4%

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

Alternative 11: 13.6% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\log x.im, y.re, 1\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Log[x$46$im], $MachinePrecision] * y$46$re + 1.0), $MachinePrecision]

Derivation

1. Initial program 41.0%

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

2. Taylor expanded in x.im around inf

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

4. Applied rewrites 35.9%

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

5. Taylor expanded in y.im around 0

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

7. Applied rewrites 26.2%

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

8. Taylor expanded in y.re around 0

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

10. Applied rewrites 13.6%

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

12. Applied rewrites 13.6%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\log x.im, y.re, 1\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(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)))))