powComplex, real part

Percentage Accurate: 39.9% → 79.8%

Time: 7.4s

Alternatives: 9

Speedup: 2.7×

• 64.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 39.9% accurate, 1.0× speedup

Math

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

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: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{1}{x.im}\right)\\ t_3 := \log \left(\frac{-1}{x.im}\right)\\ \mathbf{if}\;x.im \leq -2.3 \cdot 10^{+20}:\\ \;\;\;\;\frac{\cos \left(\mathsf{fma}\left(-1, y.im \cdot t\_3, t\_0\right)\right)}{e^{t\_1 - -1 \cdot \left(y.re \cdot t\_3\right)}}\\ \mathbf{elif}\;x.im \leq 2.2 \cdot 10^{+74}:\\ \;\;\;\;e^{\log \left(\left|-x.re\right|\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-1, y.im \cdot t\_2, t\_0\right)\right) \cdot e^{-1 \cdot \left(y.re \cdot t\_2\right) - t\_1}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (* y.im (atan2 x.im x.re)))
       (t_2 (log (/ 1.0 x.im)))
       (t_3 (log (/ -1.0 x.im))))
  (if (<= x.im -2.3e+20)
    (/
     (cos (fma -1.0 (* y.im t_3) t_0))
     (exp (- t_1 (* -1.0 (* y.re t_3)))))
    (if (<= x.im 2.2e+74)
      (*
       (exp
        (- (* (log (fabs (- x.re))) y.re) (* (atan2 x.im x.re) y.im)))
       1.0)
      (*
       (cos (fma -1.0 (* y.im t_2) t_0))
       (exp (- (* -1.0 (* y.re t_2)) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = log((1.0 / x_46_im));
	double t_3 = log((-1.0 / x_46_im));
	double tmp;
	if (x_46_im <= -2.3e+20) {
		tmp = cos(fma(-1.0, (y_46_im * t_3), t_0)) / exp((t_1 - (-1.0 * (y_46_re * t_3))));
	} else if (x_46_im <= 2.2e+74) {
		tmp = exp(((log(fabs(-x_46_re)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
	} else {
		tmp = cos(fma(-1.0, (y_46_im * t_2), t_0)) * exp(((-1.0 * (y_46_re * t_2)) - t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = log(Float64(1.0 / x_46_im))
	t_3 = log(Float64(-1.0 / x_46_im))
	tmp = 0.0
	if (x_46_im <= -2.3e+20)
		tmp = Float64(cos(fma(-1.0, Float64(y_46_im * t_3), t_0)) / exp(Float64(t_1 - Float64(-1.0 * Float64(y_46_re * t_3)))));
	elseif (x_46_im <= 2.2e+74)
		tmp = Float64(exp(Float64(Float64(log(abs(Float64(-x_46_re))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * 1.0);
	else
		tmp = Float64(cos(fma(-1.0, Float64(y_46_im * t_2), t_0)) * exp(Float64(Float64(-1.0 * Float64(y_46_re * t_2)) - t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -2.3e+20], N[(N[Cos[N[(-1.0 * N[(y$46$im * t$95$3), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$1 - N[(-1.0 * N[(y$46$re * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.2e+74], N[(N[Exp[N[(N[(N[Log[N[Abs[(-x$46$re)], $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[(N[Cos[N[(-1.0 * N[(y$46$im * t$95$2), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(-1.0 * N[(y$46$re * t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.3e20

   1. Initial program 39.9%

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

      1. lift-exp.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-negate-rev N/A

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

      4. exp-neg N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

   3. Applied rewrites 39.9%

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

   4. Taylor expanded in x.im around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

   6. Applied rewrites 35.2%

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

   if -2.3e20 < x.im < 2.2000000000000001e74

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5%

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 37.5%

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

       4. rem-exp-log N/A

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

       5. lift-log.f64 N/A

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

       6. exp-fabs N/A

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

       7. lift-log.f64 N/A

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

       8. rem-exp-log N/A

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

       9. lower-fabs.f64 72.7%

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

   12. Applied rewrites 72.7%

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

   if 2.2000000000000001e74 < x.im

   1. Initial program 39.9%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

   4. Applied rewrites 35.7%

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

Alternative 2: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\frac{1}{x.im}\right)\\ t_2 := \log \left(-1 \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -2.3 \cdot 10^{+20}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \cos \left(t\_2 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 2.2 \cdot 10^{+74}:\\ \;\;\;\;e^{\log \left(\left|-x.re\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;\cos \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{-1 \cdot \left(y.re \cdot t\_1\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.im))
       (t_1 (log (/ 1.0 x.im)))
       (t_2 (log (* -1.0 x.im))))
  (if (<= x.im -2.3e+20)
    (*
     (exp (- (* t_2 y.re) t_0))
     (cos (+ (* t_2 y.im) (* (atan2 x.im x.re) y.re))))
    (if (<= x.im 2.2e+74)
      (* (exp (- (* (log (fabs (- x.re))) y.re) t_0)) 1.0)
      (*
       (cos (fma -1.0 (* y.im t_1) (* y.re (atan2 x.im x.re))))
       (exp (- (* -1.0 (* y.re t_1)) (* 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 = log((1.0 / x_46_im));
	double t_2 = log((-1.0 * x_46_im));
	double tmp;
	if (x_46_im <= -2.3e+20) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * cos(((t_2 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_im <= 2.2e+74) {
		tmp = exp(((log(fabs(-x_46_re)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = cos(fma(-1.0, (y_46_im * t_1), (y_46_re * atan2(x_46_im, x_46_re)))) * exp(((-1.0 * (y_46_re * t_1)) - (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 = log(Float64(1.0 / x_46_im))
	t_2 = log(Float64(-1.0 * x_46_im))
	tmp = 0.0
	if (x_46_im <= -2.3e+20)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * cos(Float64(Float64(t_2 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_im <= 2.2e+74)
		tmp = Float64(exp(Float64(Float64(log(abs(Float64(-x_46_re))) * y_46_re) - t_0)) * 1.0);
	else
		tmp = Float64(cos(fma(-1.0, Float64(y_46_im * t_1), Float64(y_46_re * atan(x_46_im, x_46_re)))) * exp(Float64(Float64(-1.0 * Float64(y_46_re * t_1)) - 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[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -2.3e+20], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.2e+74], N[(N[Exp[N[(N[(N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * 1.0), $MachinePrecision], N[(N[Cos[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(-1.0 * N[(y$46$re * t$95$1), $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.im < -2.3e20

   1. Initial program 39.9%

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

   2. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 17.7%

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

   4. Applied rewrites 17.7%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 35.2%

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

   7. Applied rewrites 35.2%

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

   if -2.3e20 < x.im < 2.2000000000000001e74

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5%

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 37.5%

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

       4. rem-exp-log N/A

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

       5. lift-log.f64 N/A

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

       6. exp-fabs N/A

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

       7. lift-log.f64 N/A

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

       8. rem-exp-log N/A

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

       9. lower-fabs.f64 72.7%

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

   12. Applied rewrites 72.7%

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

   if 2.2000000000000001e74 < x.im

   1. Initial program 39.9%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-atan2.f64 N/A

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

   4. Applied rewrites 35.7%

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

Alternative 3: 79.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(-1 \cdot x.im\right)\\ \mathbf{if}\;x.im \leq -2.3 \cdot 10^{+20}:\\ \;\;\;\;e^{t\_1 \cdot y.re - t\_0} \cdot \cos \left(t\_1 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;e^{\log \left(\left|-x.re\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.im}\right)\right) \cdot y.re - t\_0} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.im)) (t_1 (log (* -1.0 x.im))))
  (if (<= x.im -2.3e+20)
    (*
     (exp (- (* t_1 y.re) t_0))
     (cos (+ (* t_1 y.im) (* (atan2 x.im x.re) y.re))))
    (if (<= x.im 4.5e-78)
      (* (exp (- (* (log (fabs (- x.re))) y.re) t_0)) 1.0)
      (*
       (exp (- (* (* -1.0 (log (/ 1.0 x.im))) y.re) t_0))
       (cos (* y.re (atan2 x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log((-1.0 * x_46_im));
	double tmp;
	if (x_46_im <= -2.3e+20) {
		tmp = exp(((t_1 * y_46_re) - t_0)) * cos(((t_1 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_im <= 4.5e-78) {
		tmp = exp(((log(fabs(-x_46_re)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp((((-1.0 * log((1.0 / x_46_im))) * y_46_re) - t_0)) * cos((y_46_re * 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = log(((-1.0d0) * x_46im))
    if (x_46im <= (-2.3d+20)) then
        tmp = exp(((t_1 * y_46re) - t_0)) * cos(((t_1 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
    else if (x_46im <= 4.5d-78) then
        tmp = exp(((log(abs(-x_46re)) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((((-1.0d0) * log((1.0d0 / x_46im))) * y_46re) - t_0)) * cos((y_46re * 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 t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.log((-1.0 * x_46_im));
	double tmp;
	if (x_46_im <= -2.3e+20) {
		tmp = Math.exp(((t_1 * y_46_re) - t_0)) * Math.cos(((t_1 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_im <= 4.5e-78) {
		tmp = Math.exp(((Math.log(Math.abs(-x_46_re)) * 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)) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.log((-1.0 * x_46_im))
	tmp = 0
	if x_46_im <= -2.3e+20:
		tmp = math.exp(((t_1 * y_46_re) - t_0)) * math.cos(((t_1 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))
	elif x_46_im <= 4.5e-78:
		tmp = math.exp(((math.log(math.fabs(-x_46_re)) * 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)) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(Float64(-1.0 * x_46_im))
	tmp = 0.0
	if (x_46_im <= -2.3e+20)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_0)) * cos(Float64(Float64(t_1 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_im <= 4.5e-78)
		tmp = Float64(exp(Float64(Float64(log(abs(Float64(-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_im))) * y_46_re) - t_0)) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = log((-1.0 * x_46_im));
	tmp = 0.0;
	if (x_46_im <= -2.3e+20)
		tmp = exp(((t_1 * y_46_re) - t_0)) * cos(((t_1 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	elseif (x_46_im <= 4.5e-78)
		tmp = exp(((log(abs(-x_46_re)) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp((((-1.0 * log((1.0 / x_46_im))) * y_46_re) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -2.3e+20], N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4.5e-78], N[(N[Exp[N[(N[(N[Log[N[Abs[(-x$46$re)], $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] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.3e20

   1. Initial program 39.9%

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

   2. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 17.7%

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

   4. Applied rewrites 17.7%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 35.2%

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

   7. Applied rewrites 35.2%

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

   if -2.3e20 < x.im < 4.5000000000000001e-78

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5%

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 37.5%

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

       4. rem-exp-log N/A

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

       5. lift-log.f64 N/A

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

       6. exp-fabs N/A

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

       7. lift-log.f64 N/A

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

       8. rem-exp-log N/A

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

       9. lower-fabs.f64 72.7%

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

   12. Applied rewrites 72.7%

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

   if 4.5000000000000001e-78 < x.im

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 35.4%

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

   7. Applied rewrites 35.4%

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

Alternative 4: 79.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -7.6 \cdot 10^{+42}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - t\_0} \cdot t\_1\\ \mathbf{elif}\;x.im \leq 4.5 \cdot 10^{-78}:\\ \;\;\;\;e^{\log \left(\left|-x.re\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.im}\right)\right) \cdot y.re - t\_0} \cdot t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.im))
       (t_1 (cos (* y.re (atan2 x.im x.re)))))
  (if (<= x.im -7.6e+42)
    (* (exp (- (* (log (* -1.0 x.im)) y.re) t_0)) t_1)
    (if (<= x.im 4.5e-78)
      (* (exp (- (* (log (fabs (- x.re))) y.re) t_0)) 1.0)
      (* (exp (- (* (* -1.0 (log (/ 1.0 x.im))) y.re) t_0)) t_1)))))

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 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -7.6e+42) {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * t_1;
	} else if (x_46_im <= 4.5e-78) {
		tmp = exp(((log(fabs(-x_46_re)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp((((-1.0 * log((1.0 / x_46_im))) * y_46_re) - t_0)) * t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = cos((y_46re * atan2(x_46im, x_46re)))
    if (x_46im <= (-7.6d+42)) then
        tmp = exp(((log(((-1.0d0) * x_46im)) * y_46re) - t_0)) * t_1
    else if (x_46im <= 4.5d-78) then
        tmp = exp(((log(abs(-x_46re)) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((((-1.0d0) * log((1.0d0 / x_46im))) * y_46re) - t_0)) * t_1
    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 t_1 = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -7.6e+42) {
		tmp = Math.exp(((Math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * t_1;
	} else if (x_46_im <= 4.5e-78) {
		tmp = Math.exp(((Math.log(Math.abs(-x_46_re)) * 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)) * t_1;
	}
	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
	t_1 = math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -7.6e+42:
		tmp = math.exp(((math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * t_1
	elif x_46_im <= 4.5e-78:
		tmp = math.exp(((math.log(math.fabs(-x_46_re)) * 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)) * t_1
	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 = cos(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -7.6e+42)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_re) - t_0)) * t_1);
	elseif (x_46_im <= 4.5e-78)
		tmp = Float64(exp(Float64(Float64(log(abs(Float64(-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_im))) * y_46_re) - t_0)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = cos((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -7.6e+42)
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * t_1;
	elseif (x_46_im <= 4.5e-78)
		tmp = exp(((log(abs(-x_46_re)) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp((((-1.0 * log((1.0 / x_46_im))) * y_46_re) - t_0)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -7.6e+42], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x$46$im, 4.5e-78], N[(N[Exp[N[(N[(N[Log[N[Abs[(-x$46$re)], $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] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -7.5999999999999997e42

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 35.4%

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

   7. Applied rewrites 35.4%

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

   if -7.5999999999999997e42 < x.im < 4.5000000000000001e-78

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5%

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 37.5%

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

       4. rem-exp-log N/A

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

       5. lift-log.f64 N/A

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

       6. exp-fabs N/A

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

       7. lift-log.f64 N/A

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

       8. rem-exp-log N/A

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

       9. lower-fabs.f64 72.7%

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

   12. Applied rewrites 72.7%

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

   if 4.5000000000000001e-78 < x.im

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 35.4%

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

   7. Applied rewrites 35.4%

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

Alternative 5: 78.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -7.6 \cdot 10^{+42}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - t\_0} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq 4.3 \cdot 10^{-78}:\\ \;\;\;\;e^{\log \left(\left|-x.re\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.im}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \]

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 -7.6e+42)
    (*
     (exp (- (* (log (* -1.0 x.im)) y.re) t_0))
     (cos (* y.re (atan2 x.im x.re))))
    (if (<= x.im 4.3e-78)
      (* (exp (- (* (log (fabs (- x.re))) 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 <= -7.6e+42) {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (x_46_im <= 4.3e-78) {
		tmp = exp(((log(fabs(-x_46_re)) * 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 <= (-7.6d+42)) then
        tmp = exp(((log(((-1.0d0) * x_46im)) * y_46re) - t_0)) * cos((y_46re * atan2(x_46im, x_46re)))
    else if (x_46im <= 4.3d-78) then
        tmp = exp(((log(abs(-x_46re)) * 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 <= -7.6e+42) {
		tmp = Math.exp(((Math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (x_46_im <= 4.3e-78) {
		tmp = Math.exp(((Math.log(Math.abs(-x_46_re)) * 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 <= -7.6e+42:
		tmp = math.exp(((math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif x_46_im <= 4.3e-78:
		tmp = math.exp(((math.log(math.fabs(-x_46_re)) * 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 <= -7.6e+42)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_re) - t_0)) * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (x_46_im <= 4.3e-78)
		tmp = Float64(exp(Float64(Float64(log(abs(Float64(-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_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 <= -7.6e+42)
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * cos((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (x_46_im <= 4.3e-78)
		tmp = exp(((log(abs(-x_46_re)) * 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, -7.6e+42], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4.3e-78], N[(N[Exp[N[(N[(N[Log[N[Abs[(-x$46$re)], $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 3 regimes

2. if x.im < -7.5999999999999997e42

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 35.4%

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

   7. Applied rewrites 35.4%

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

   if -7.5999999999999997e42 < x.im < 4.2999999999999999e-78

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5%

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 37.5%

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

       4. rem-exp-log N/A

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

       5. lift-log.f64 N/A

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

       6. exp-fabs N/A

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

       7. lift-log.f64 N/A

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

       8. rem-exp-log N/A

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

       9. lower-fabs.f64 72.7%

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

   12. Applied rewrites 72.7%

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

   if 4.2999999999999999e-78 < x.im

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 36.4%

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

   10. Applied rewrites 36.4%

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

Alternative 6: 78.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{+46}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{elif}\;x.im \leq 4.3 \cdot 10^{-78}:\\ \;\;\;\;e^{\log \left(\left|-x.re\right|\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\left(-1 \cdot \log \left(\frac{1}{x.im}\right)\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \]

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 -1e+46)
    (* (exp (- (* (log (* -1.0 x.im)) y.re) t_0)) 1.0)
    (if (<= x.im 4.3e-78)
      (* (exp (- (* (log (fabs (- x.re))) 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 <= -1e+46) {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_im <= 4.3e-78) {
		tmp = exp(((log(fabs(-x_46_re)) * 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 <= (-1d+46)) then
        tmp = exp(((log(((-1.0d0) * x_46im)) * y_46re) - t_0)) * 1.0d0
    else if (x_46im <= 4.3d-78) then
        tmp = exp(((log(abs(-x_46re)) * 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 <= -1e+46) {
		tmp = Math.exp(((Math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else if (x_46_im <= 4.3e-78) {
		tmp = Math.exp(((Math.log(Math.abs(-x_46_re)) * 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 <= -1e+46:
		tmp = math.exp(((math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0
	elif x_46_im <= 4.3e-78:
		tmp = math.exp(((math.log(math.fabs(-x_46_re)) * 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 <= -1e+46)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0);
	elseif (x_46_im <= 4.3e-78)
		tmp = Float64(exp(Float64(Float64(log(abs(Float64(-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_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 <= -1e+46)
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	elseif (x_46_im <= 4.3e-78)
		tmp = exp(((log(abs(-x_46_re)) * 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, -1e+46], 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], If[LessEqual[x$46$im, 4.3e-78], N[(N[Exp[N[(N[(N[Log[N[Abs[(-x$46$re)], $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 3 regimes

2. if x.im < -9.9999999999999999e45

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 28.4%

         \[\leadsto e^{\log \left(-1 \cdot \left(x.im \cdot \left(1 + 0.5 \cdot \frac{{x.re}^{2}}{{x.im}^{\color{blue}{2}}}\right)\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   10. Applied rewrites 28.4%

       \[\leadsto e^{\log \color{blue}{\left(-1 \cdot \left(x.im \cdot \left(1 + 0.5 \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 35.9%

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

   if -9.9999999999999999e45 < x.im < 4.2999999999999999e-78

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5%

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 37.5%

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

       4. rem-exp-log N/A

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

       5. lift-log.f64 N/A

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

       6. exp-fabs N/A

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

       7. lift-log.f64 N/A

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

       8. rem-exp-log N/A

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

       9. lower-fabs.f64 72.7%

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

   12. Applied rewrites 72.7%

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

   if 4.2999999999999999e-78 < x.im

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 36.4%

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

   10. Applied rewrites 36.4%

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

Alternative 7: 76.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{+46}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - t\_0} \cdot 1\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left|-x.re\right|\right) \cdot y.re - t\_0} \cdot 1\\ \end{array} \]

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 -1e+46)
    (* (exp (- (* (log (* -1.0 x.im)) y.re) t_0)) 1.0)
    (* (exp (- (* (log (fabs (- 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_im <= -1e+46) {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = exp(((log(fabs(-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_46im <= (-1d+46)) then
        tmp = exp(((log(((-1.0d0) * x_46im)) * y_46re) - t_0)) * 1.0d0
    else
        tmp = exp(((log(abs(-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_im <= -1e+46) {
		tmp = Math.exp(((Math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	} else {
		tmp = Math.exp(((Math.log(Math.abs(-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_im <= -1e+46:
		tmp = math.exp(((math.log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0
	else:
		tmp = math.exp(((math.log(math.fabs(-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_im <= -1e+46)
		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(log(abs(Float64(-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_im <= -1e+46)
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - t_0)) * 1.0;
	else
		tmp = exp(((log(abs(-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$46$im, -1e+46], 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[Log[N[Abs[(-x$46$re)], $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 < -9.9999999999999999e45

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. lower-pow.f64 28.4%

         \[\leadsto e^{\log \left(-1 \cdot \left(x.im \cdot \left(1 + 0.5 \cdot \frac{{x.re}^{2}}{{x.im}^{\color{blue}{2}}}\right)\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   10. Applied rewrites 28.4%

       \[\leadsto e^{\log \color{blue}{\left(-1 \cdot \left(x.im \cdot \left(1 + 0.5 \cdot \frac{{x.re}^{2}}{{x.im}^{2}}\right)\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot 1 \]

   11. Taylor expanded in x.re around 0

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

   13. Applied rewrites 35.9%

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

   if -9.9999999999999999e45 < x.im

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-cos.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.9%

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

   4. Applied rewrites 61.9%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 64.0%

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

   8. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 37.5%

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

   10. Applied rewrites 37.5%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lift-neg.f64 37.5%

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

       4. rem-exp-log N/A

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

       5. lift-log.f64 N/A

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

       6. exp-fabs N/A

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

       7. lift-log.f64 N/A

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

       8. rem-exp-log N/A

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

       9. lower-fabs.f64 72.7%

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

   12. Applied rewrites 72.7%

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

Alternative 8: 72.7% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (*
 (exp (- (* (log (fabs (- x.re))) y.re) (* (atan2 x.im x.re) y.im)))
 1.0))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp(((log(fabs(-x_46_re)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 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 = exp(((log(abs(-x_46re)) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 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 Math.exp(((Math.log(Math.abs(-x_46_re)) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Exp[N[(N[(N[Log[N[Abs[(-x$46$re)], $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]

Derivation

1. Initial program 39.9%

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

2. Taylor expanded in y.im around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-atan2.f64 61.9%

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

4. Applied rewrites 61.9%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 64.0%

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

8. Taylor expanded in x.re around -inf

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

   1. lower-*.f64 37.5%

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

10. Applied rewrites 37.5%

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

    1. lift-*.f64 N/A

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

    2. mul-1-neg N/A

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

    3. lift-neg.f64 37.5%

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

    4. rem-exp-log N/A

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

    5. lift-log.f64 N/A

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

    6. exp-fabs N/A

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

    7. lift-log.f64 N/A

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

    8. rem-exp-log N/A

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

    9. lower-fabs.f64 72.7%

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

12. Applied rewrites 72.7%

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

Alternative 9: 37.5% accurate, 2.8× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return exp(((log(-x_46_re) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * 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 = exp(((log(-x_46re) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * 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 Math.exp(((Math.log(-x_46_re) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * 1.0;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

2. Taylor expanded in y.im around 0

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

   1. lower-cos.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-atan2.f64 61.9%

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

4. Applied rewrites 61.9%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 64.0%

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

8. Taylor expanded in x.re around -inf

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

   1. lower-*.f64 37.5%

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

10. Applied rewrites 37.5%

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

    1. lift-*.f64 N/A

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

    2. mul-1-neg N/A

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

    3. lift-neg.f64 37.5%

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

12. Applied rewrites 37.5%

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

Reproduce

herbie shell --seed 2025259 
(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)))))