powComplex, imaginary part

Percentage Accurate: 40.7% → 75.7%

Time: 5.3s

Alternatives: 9

Speedup: 2.1×

• 64.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))
   (sin (+ (* 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))) * sin(((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))) * sin(((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.sin(((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.sin(((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))) * sin(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))) * sin(((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[Sin[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: 40.7% accurate, 1.0× speedup

Math

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

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)))
   (sin (+ (* 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))) * sin(((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))) * sin(((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.sin(((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.sin(((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))) * sin(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))) * sin(((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[Sin[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: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := -1 \cdot \log \left(\frac{1}{x.im}\right)\\ t_3 := \log \left(-1 \cdot x.im\right)\\ t_4 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;x.im \leq -33000000000000:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq 16200000000000000000000:\\ \;\;\;\;e^{t\_4 \cdot y.re - t\_0} \cdot \sin \left(t\_4 \cdot y.im + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.im))
       (t_1 (* (atan2 x.im x.re) y.re))
       (t_2 (* -1 (log (/ 1 x.im))))
       (t_3 (log (* -1 x.im)))
       (t_4 (log (fabs (- x.re)))))
  (if (<= x.im -33000000000000)
    (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))
    (if (<= x.im 16200000000000000000000)
      (* (exp (- (* t_4 y.re) t_0)) (sin (+ (* t_4 y.im) t_1)))
      (* (exp (- (* t_2 y.re) t_0)) (sin (+ (* t_2 y.im) 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 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = -1.0 * log((1.0 / x_46_im));
	double t_3 = log((-1.0 * x_46_im));
	double t_4 = log(fabs(-x_46_re));
	double tmp;
	if (x_46_im <= -33000000000000.0) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 1.62e+22) {
		tmp = exp(((t_4 * y_46_re) - t_0)) * sin(((t_4 * y_46_im) + t_1));
	} else {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + 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) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = atan2(x_46im, x_46re) * y_46re
    t_2 = (-1.0d0) * log((1.0d0 / x_46im))
    t_3 = log(((-1.0d0) * x_46im))
    t_4 = log(abs(-x_46re))
    if (x_46im <= (-33000000000000.0d0)) then
        tmp = exp(((t_3 * y_46re) - t_0)) * sin(((t_3 * y_46im) + t_1))
    else if (x_46im <= 1.62d+22) then
        tmp = exp(((t_4 * y_46re) - t_0)) * sin(((t_4 * y_46im) + t_1))
    else
        tmp = exp(((t_2 * y_46re) - t_0)) * sin(((t_2 * y_46im) + 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.atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = -1.0 * Math.log((1.0 / x_46_im));
	double t_3 = Math.log((-1.0 * x_46_im));
	double t_4 = Math.log(Math.abs(-x_46_re));
	double tmp;
	if (x_46_im <= -33000000000000.0) {
		tmp = Math.exp(((t_3 * y_46_re) - t_0)) * Math.sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 1.62e+22) {
		tmp = Math.exp(((t_4 * y_46_re) - t_0)) * Math.sin(((t_4 * y_46_im) + t_1));
	} else {
		tmp = Math.exp(((t_2 * y_46_re) - t_0)) * Math.sin(((t_2 * y_46_im) + 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.atan2(x_46_im, x_46_re) * y_46_re
	t_2 = -1.0 * math.log((1.0 / x_46_im))
	t_3 = math.log((-1.0 * x_46_im))
	t_4 = math.log(math.fabs(-x_46_re))
	tmp = 0
	if x_46_im <= -33000000000000.0:
		tmp = math.exp(((t_3 * y_46_re) - t_0)) * math.sin(((t_3 * y_46_im) + t_1))
	elif x_46_im <= 1.62e+22:
		tmp = math.exp(((t_4 * y_46_re) - t_0)) * math.sin(((t_4 * y_46_im) + t_1))
	else:
		tmp = math.exp(((t_2 * y_46_re) - t_0)) * math.sin(((t_2 * y_46_im) + 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 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = Float64(-1.0 * log(Float64(1.0 / x_46_im)))
	t_3 = log(Float64(-1.0 * x_46_im))
	t_4 = log(abs(Float64(-x_46_re)))
	tmp = 0.0
	if (x_46_im <= -33000000000000.0)
		tmp = Float64(exp(Float64(Float64(t_3 * y_46_re) - t_0)) * sin(Float64(Float64(t_3 * y_46_im) + t_1)));
	elseif (x_46_im <= 1.62e+22)
		tmp = Float64(exp(Float64(Float64(t_4 * y_46_re) - t_0)) * sin(Float64(Float64(t_4 * y_46_im) + t_1)));
	else
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + 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 = atan2(x_46_im, x_46_re) * y_46_re;
	t_2 = -1.0 * log((1.0 / x_46_im));
	t_3 = log((-1.0 * x_46_im));
	t_4 = log(abs(-x_46_re));
	tmp = 0.0;
	if (x_46_im <= -33000000000000.0)
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	elseif (x_46_im <= 1.62e+22)
		tmp = exp(((t_4 * y_46_re) - t_0)) * sin(((t_4 * y_46_im) + t_1));
	else
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[(-1 * N[Log[N[(1 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[(-1 * x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -33000000000000], N[(N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$3 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 16200000000000000000000], N[(N[Exp[N[(N[(t$95$4 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$4 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.3e13

   1. Initial program 40.7%

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

   2. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 18.0%

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 30.8%

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

   7. Applied rewrites 30.8%

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

   if -3.3e13 < x.im < 1.62e22

   1. Initial program 40.7%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 19.1%

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.1%

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

   7. Applied rewrites 34.1%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.1%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.1%

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

   9. Applied rewrites 34.1%

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

       1. rem-exp-log N/A

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

       2. lift-log.f64 N/A

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

       3. exp-fabs N/A

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

       4. lift-log.f64 N/A

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

       5. rem-exp-log N/A

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

       6. lower-fabs.f64 66.3%

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

       7. lift-*.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 66.3%

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

   11. Applied rewrites 66.3%

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

   if 1.62e22 < x.im

   1. Initial program 40.7%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 18.5%

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 31.9%

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

   7. Applied rewrites 31.9%

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

Alternative 2: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \log \left(-1 \cdot x.im\right)\\ t_3 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;x.im \leq -33000000000000:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq 1349999999999999866755350528:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq 780000000000000039431392094629131259537187615337958198473589443865282578122104684994895452796553450117041846373011394003208751512059247217710127266905324318097408:\\ \;\;\;\;e^{\left(\frac{1}{2} \cdot \log \left(0 \cdot 0 + x.im \cdot x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(t\_1 + -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.im))
       (t_1 (* (atan2 x.im x.re) y.re))
       (t_2 (log (* -1 x.im)))
       (t_3 (log (fabs (- x.re)))))
  (if (<= x.im -33000000000000)
    (* (exp (- (* t_2 y.re) t_0)) (sin (+ (* t_2 y.im) t_1)))
    (if (<= x.im 1349999999999999866755350528)
      (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))
      (if (<=
           x.im
           780000000000000039431392094629131259537187615337958198473589443865282578122104684994895452796553450117041846373011394003208751512059247217710127266905324318097408)
        (*
         (exp
          (-
           (* (* 1/2 (log (+ (* 0 0) (* x.im x.im)))) y.re)
           (* (atan2 x.im 0) y.im)))
         (sin (* y.re (atan2 x.im 0))))
        (/
         (sin (+ t_1 (* -1 (* y.im (log (/ 1 x.im))))))
         (exp (* y.im (atan2 x.im x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = log((-1.0 * x_46_im));
	double t_3 = log(fabs(-x_46_re));
	double tmp;
	if (x_46_im <= -33000000000000.0) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_im <= 1.35e+27) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 7.8e+161) {
		tmp = exp((((0.5 * log(((0.0 * 0.0) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, 0.0) * y_46_im))) * sin((y_46_re * atan2(x_46_im, 0.0)));
	} else {
		tmp = sin((t_1 + (-1.0 * (y_46_im * log((1.0 / x_46_im)))))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = atan2(x_46im, x_46re) * y_46re
    t_2 = log(((-1.0d0) * x_46im))
    t_3 = log(abs(-x_46re))
    if (x_46im <= (-33000000000000.0d0)) then
        tmp = exp(((t_2 * y_46re) - t_0)) * sin(((t_2 * y_46im) + t_1))
    else if (x_46im <= 1.35d+27) then
        tmp = exp(((t_3 * y_46re) - t_0)) * sin(((t_3 * y_46im) + t_1))
    else if (x_46im <= 7.8d+161) then
        tmp = exp((((0.5d0 * log(((0.0d0 * 0.0d0) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, 0.0d0) * y_46im))) * sin((y_46re * atan2(x_46im, 0.0d0)))
    else
        tmp = sin((t_1 + ((-1.0d0) * (y_46im * log((1.0d0 / x_46im)))))) / exp((y_46im * atan2(x_46im, x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = Math.log((-1.0 * x_46_im));
	double t_3 = Math.log(Math.abs(-x_46_re));
	double tmp;
	if (x_46_im <= -33000000000000.0) {
		tmp = Math.exp(((t_2 * y_46_re) - t_0)) * Math.sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_im <= 1.35e+27) {
		tmp = Math.exp(((t_3 * y_46_re) - t_0)) * Math.sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 7.8e+161) {
		tmp = Math.exp((((0.5 * Math.log(((0.0 * 0.0) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, 0.0) * y_46_im))) * Math.sin((y_46_re * Math.atan2(x_46_im, 0.0)));
	} else {
		tmp = Math.sin((t_1 + (-1.0 * (y_46_im * Math.log((1.0 / x_46_im)))))) / Math.exp((y_46_im * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_re
	t_2 = math.log((-1.0 * x_46_im))
	t_3 = math.log(math.fabs(-x_46_re))
	tmp = 0
	if x_46_im <= -33000000000000.0:
		tmp = math.exp(((t_2 * y_46_re) - t_0)) * math.sin(((t_2 * y_46_im) + t_1))
	elif x_46_im <= 1.35e+27:
		tmp = math.exp(((t_3 * y_46_re) - t_0)) * math.sin(((t_3 * y_46_im) + t_1))
	elif x_46_im <= 7.8e+161:
		tmp = math.exp((((0.5 * math.log(((0.0 * 0.0) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, 0.0) * y_46_im))) * math.sin((y_46_re * math.atan2(x_46_im, 0.0)))
	else:
		tmp = math.sin((t_1 + (-1.0 * (y_46_im * math.log((1.0 / x_46_im)))))) / math.exp((y_46_im * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = log(Float64(-1.0 * x_46_im))
	t_3 = log(abs(Float64(-x_46_re)))
	tmp = 0.0
	if (x_46_im <= -33000000000000.0)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + t_1)));
	elseif (x_46_im <= 1.35e+27)
		tmp = Float64(exp(Float64(Float64(t_3 * y_46_re) - t_0)) * sin(Float64(Float64(t_3 * y_46_im) + t_1)));
	elseif (x_46_im <= 7.8e+161)
		tmp = Float64(exp(Float64(Float64(Float64(0.5 * log(Float64(Float64(0.0 * 0.0) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, 0.0) * y_46_im))) * sin(Float64(y_46_re * atan(x_46_im, 0.0))));
	else
		tmp = Float64(sin(Float64(t_1 + Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_im)))))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	t_2 = log((-1.0 * x_46_im));
	t_3 = log(abs(-x_46_re));
	tmp = 0.0;
	if (x_46_im <= -33000000000000.0)
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	elseif (x_46_im <= 1.35e+27)
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	elseif (x_46_im <= 7.8e+161)
		tmp = exp((((0.5 * log(((0.0 * 0.0) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, 0.0) * y_46_im))) * sin((y_46_re * atan2(x_46_im, 0.0)));
	else
		tmp = sin((t_1 + (-1.0 * (y_46_im * log((1.0 / x_46_im)))))) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1 * x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -33000000000000], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1349999999999999866755350528], N[(N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$3 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 780000000000000039431392094629131259537187615337958198473589443865282578122104684994895452796553450117041846373011394003208751512059247217710127266905324318097408], N[(N[Exp[N[(N[(N[(1/2 * N[Log[N[(N[(0 * 0), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / 0], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / 0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$1 + N[(-1 * N[(y$46$im * N[Log[N[(1 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -3.3e13

   1. Initial program 40.7%

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

   2. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 18.0%

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

   4. Applied rewrites 18.0%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 30.8%

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

   7. Applied rewrites 30.8%

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

   if -3.3e13 < x.im < 1.3499999999999999e27

   1. Initial program 40.7%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 19.1%

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.1%

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

   7. Applied rewrites 34.1%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.1%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.1%

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

   9. Applied rewrites 34.1%

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

       1. rem-exp-log N/A

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

       2. lift-log.f64 N/A

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

       3. exp-fabs N/A

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

       4. lift-log.f64 N/A

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

       5. rem-exp-log N/A

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

       6. lower-fabs.f64 66.3%

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

       7. lift-*.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 66.3%

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

   11. Applied rewrites 66.3%

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

   if 1.3499999999999999e27 < x.im < 7.8000000000000004e161

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7%

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in undef-var around zero

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{0} \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 45.8%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{0} \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. Taylor expanded in undef-var around zero

      \[\leadsto e^{\log \left(\sqrt{0 \cdot \color{blue}{0} + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 45.8%

       \[\leadsto e^{\log \left(\sqrt{0 \cdot \color{blue}{0} + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Taylor expanded in undef-var around zero

       \[\leadsto e^{\log \left(\sqrt{0 \cdot 0 + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{\color{blue}{0}} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 45.8%

       \[\leadsto e^{\log \left(\sqrt{0 \cdot 0 + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{\color{blue}{0}} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   14. Taylor expanded in undef-var around zero

       \[\leadsto e^{\log \left(\sqrt{0 \cdot 0 + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]
   15. Step-by-step derivation

   16. Applied rewrites 43.7%

       \[\leadsto e^{\log \left(\sqrt{0 \cdot 0 + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]
   17. Step-by-step derivation

       1. lift-log.f64 N/A

          \[\leadsto e^{\color{blue}{\log \left(\sqrt{0 \cdot 0 + x.im \cdot x.im}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]

       2. lift-sqrt.f64 N/A

          \[\leadsto e^{\log \color{blue}{\left(\sqrt{0 \cdot 0 + x.im \cdot x.im}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]

       3. pow1/2 N/A

          \[\leadsto e^{\log \color{blue}{\left({\left(0 \cdot 0 + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]

       4. log-pow N/A

          \[\leadsto e^{\color{blue}{\left(\frac{1}{2} \cdot \log \left(0 \cdot 0 + x.im \cdot x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]

       5. lower-unsound-*.f64 N/A

          \[\leadsto e^{\color{blue}{\left(\frac{1}{2} \cdot \log \left(0 \cdot 0 + x.im \cdot x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]

       6. lower-unsound-log.f64 43.7%

          \[\leadsto e^{\left(\frac{1}{2} \cdot \color{blue}{\log \left(0 \cdot 0 + x.im \cdot x.im\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]

   18. Applied rewrites 43.7%

       \[\leadsto e^{\color{blue}{\left(\frac{1}{2} \cdot \log \left(0 \cdot 0 + x.im \cdot x.im\right)\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{0} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{0}\right) \]

   if 7.8000000000000004e161 < x.im

   1. Initial program 40.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. exp-neg N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

   3. Applied rewrites 40.7%

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

   4. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 29.4%

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

   6. Applied rewrites 29.4%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 22.5%

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

   9. Applied rewrites 22.5%

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

Alternative 3: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;y.re \leq -580000000000000035318263647924383797362032640:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 225000000000000010377491622486033308756234866571085004518503806731814696755177007954468072900008083456:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.im))
       (t_1
        (*
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           t_0))
         (sin (* y.re (atan2 x.im x.re)))))
       (t_2 (log (fabs (- x.re)))))
  (if (<= y.re -580000000000000035318263647924383797362032640)
    t_1
    (if (<=
         y.re
         225000000000000010377491622486033308756234866571085004518503806731814696755177007954468072900008083456)
      (*
       (exp (- (* t_2 y.re) t_0))
       (sin (+ (* t_2 y.im) (* (atan2 x.im x.re) y.re))))
      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 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = log(fabs(-x_46_re));
	double tmp;
	if (y_46_re <= -5.8e+44) {
		tmp = t_1;
	} else if (y_46_re <= 2.25e+101) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = 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) :: t_2
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - t_0)) * sin((y_46re * atan2(x_46im, x_46re)))
    t_2 = log(abs(-x_46re))
    if (y_46re <= (-5.8d+44)) then
        tmp = t_1
    else if (y_46re <= 2.25d+101) then
        tmp = exp(((t_2 * y_46re) - t_0)) * sin(((t_2 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
    else
        tmp = 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.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_2 = Math.log(Math.abs(-x_46_re));
	double tmp;
	if (y_46_re <= -5.8e+44) {
		tmp = t_1;
	} else if (y_46_re <= 2.25e+101) {
		tmp = Math.exp(((t_2 * y_46_re) - t_0)) * Math.sin(((t_2 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = 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.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_2 = math.log(math.fabs(-x_46_re))
	tmp = 0
	if y_46_re <= -5.8e+44:
		tmp = t_1
	elif y_46_re <= 2.25e+101:
		tmp = math.exp(((t_2 * y_46_re) - t_0)) * math.sin(((t_2 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))
	else:
		tmp = 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 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))))
	t_2 = log(abs(Float64(-x_46_re)))
	tmp = 0.0
	if (y_46_re <= -5.8e+44)
		tmp = t_1;
	elseif (y_46_re <= 2.25e+101)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	else
		tmp = 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 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_2 = log(abs(-x_46_re));
	tmp = 0.0;
	if (y_46_re <= -5.8e+44)
		tmp = t_1;
	elseif (y_46_re <= 2.25e+101)
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = 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[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -580000000000000035318263647924383797362032640], t$95$1, If[LessEqual[y$46$re, 225000000000000010377491622486033308756234866571085004518503806731814696755177007954468072900008083456], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[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], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.8000000000000004e44 or 2.2500000000000001e101 < y.re

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7%

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

   4. Applied rewrites 53.7%

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

   if -5.8000000000000004e44 < y.re < 2.2500000000000001e101

   1. Initial program 40.7%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 19.1%

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

   4. Applied rewrites 19.1%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.1%

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

   7. Applied rewrites 34.1%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.1%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.1%

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

   9. Applied rewrites 34.1%

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

       1. rem-exp-log N/A

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

       2. lift-log.f64 N/A

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

       3. exp-fabs N/A

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

       4. lift-log.f64 N/A

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

       5. rem-exp-log N/A

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

       6. lower-fabs.f64 66.3%

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

       7. lift-*.f64 N/A

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

       8. mul-1-neg N/A

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

       9. lower-neg.f64 66.3%

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

   11. Applied rewrites 66.3%

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

Alternative 4: 61.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_0\\ t_2 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := e^{-t\_2} \cdot t\_0\\ \mathbf{if}\;y.re \leq -880000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq \frac{-5996419475438757}{363419362147803445274661903944002267176820680343659030140745099590319644056698961663095525356881782780381260803133088966767300814307328}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.re \leq \frac{6000448200243097}{342882754299605542703496015699200579379649539745770754382000124278512336359979559197823481221022674600830295333617006984059886491421540493951506482390354393725906168794375391533474387361995876540094533828897487199474622120556760561893297406274466013266278287285969349365133754612883980378790581378220032}:\\ \;\;\;\;\frac{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + y.im \cdot \left(\frac{1}{2} \cdot \log \left(x.im \cdot x.im + x.re \cdot x.re\right)\right)\right)}{e^{t\_2}}\\ \mathbf{elif}\;y.re \leq 13000:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
       (t_1
        (*
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im)))
         t_0))
       (t_2 (* y.im (atan2 x.im x.re)))
       (t_3 (* (exp (- t_2)) t_0)))
  (if (<= y.re -880000000)
    t_1
    (if (<=
         y.re
         -5996419475438757/363419362147803445274661903944002267176820680343659030140745099590319644056698961663095525356881782780381260803133088966767300814307328)
      t_3
      (if (<=
           y.re
           6000448200243097/342882754299605542703496015699200579379649539745770754382000124278512336359979559197823481221022674600830295333617006984059886491421540493951506482390354393725906168794375391533474387361995876540094533828897487199474622120556760561893297406274466013266278287285969349365133754612883980378790581378220032)
        (/
         (sin
          (+
           (* (atan2 x.im x.re) y.re)
           (* y.im (* 1/2 (log (+ (* x.im x.im) (* x.re x.re)))))))
         (exp t_2))
        (if (<= y.re 13000) t_3 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	double t_2 = y_46_im * atan2(x_46_im, x_46_re);
	double t_3 = exp(-t_2) * t_0;
	double tmp;
	if (y_46_re <= -880000000.0) {
		tmp = t_1;
	} else if (y_46_re <= -1.65e-119) {
		tmp = t_3;
	} else if (y_46_re <= 1.75e-287) {
		tmp = sin(((atan2(x_46_im, x_46_re) * y_46_re) + (y_46_im * (0.5 * log(((x_46_im * x_46_im) + (x_46_re * x_46_re))))))) / exp(t_2);
	} else if (y_46_re <= 13000.0) {
		tmp = t_3;
	} else {
		tmp = 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) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * t_0
    t_2 = y_46im * atan2(x_46im, x_46re)
    t_3 = exp(-t_2) * t_0
    if (y_46re <= (-880000000.0d0)) then
        tmp = t_1
    else if (y_46re <= (-1.65d-119)) then
        tmp = t_3
    else if (y_46re <= 1.75d-287) then
        tmp = sin(((atan2(x_46im, x_46re) * y_46re) + (y_46im * (0.5d0 * log(((x_46im * x_46im) + (x_46re * x_46re))))))) / exp(t_2)
    else if (y_46re <= 13000.0d0) then
        tmp = t_3
    else
        tmp = 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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	double t_2 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.exp(-t_2) * t_0;
	double tmp;
	if (y_46_re <= -880000000.0) {
		tmp = t_1;
	} else if (y_46_re <= -1.65e-119) {
		tmp = t_3;
	} else if (y_46_re <= 1.75e-287) {
		tmp = Math.sin(((Math.atan2(x_46_im, x_46_re) * y_46_re) + (y_46_im * (0.5 * Math.log(((x_46_im * x_46_im) + (x_46_re * x_46_re))))))) / Math.exp(t_2);
	} else if (y_46_re <= 13000.0) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * t_0
	t_2 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_3 = math.exp(-t_2) * t_0
	tmp = 0
	if y_46_re <= -880000000.0:
		tmp = t_1
	elif y_46_re <= -1.65e-119:
		tmp = t_3
	elif y_46_re <= 1.75e-287:
		tmp = math.sin(((math.atan2(x_46_im, x_46_re) * y_46_re) + (y_46_im * (0.5 * math.log(((x_46_im * x_46_im) + (x_46_re * x_46_re))))))) / math.exp(t_2)
	elif y_46_re <= 13000.0:
		tmp = t_3
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * t_0)
	t_2 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_3 = Float64(exp(Float64(-t_2)) * t_0)
	tmp = 0.0
	if (y_46_re <= -880000000.0)
		tmp = t_1;
	elseif (y_46_re <= -1.65e-119)
		tmp = t_3;
	elseif (y_46_re <= 1.75e-287)
		tmp = Float64(sin(Float64(Float64(atan(x_46_im, x_46_re) * y_46_re) + Float64(y_46_im * Float64(0.5 * log(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))))) / exp(t_2));
	elseif (y_46_re <= 13000.0)
		tmp = t_3;
	else
		tmp = 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	t_2 = y_46_im * atan2(x_46_im, x_46_re);
	t_3 = exp(-t_2) * t_0;
	tmp = 0.0;
	if (y_46_re <= -880000000.0)
		tmp = t_1;
	elseif (y_46_re <= -1.65e-119)
		tmp = t_3;
	elseif (y_46_re <= 1.75e-287)
		tmp = sin(((atan2(x_46_im, x_46_re) * y_46_re) + (y_46_im * (0.5 * log(((x_46_im * x_46_im) + (x_46_re * x_46_re))))))) / exp(t_2);
	elseif (y_46_re <= 13000.0)
		tmp = t_3;
	else
		tmp = 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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[(-t$95$2)], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -880000000], t$95$1, If[LessEqual[y$46$re, -5996419475438757/363419362147803445274661903944002267176820680343659030140745099590319644056698961663095525356881782780381260803133088966767300814307328], t$95$3, If[LessEqual[y$46$re, 6000448200243097/342882754299605542703496015699200579379649539745770754382000124278512336359979559197823481221022674600830295333617006984059886491421540493951506482390354393725906168794375391533474387361995876540094533828897487199474622120556760561893297406274466013266278287285969349365133754612883980378790581378220032], N[(N[Sin[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] + N[(y$46$im * N[(1/2 * N[Log[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 13000], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.8e8 or 13000 < y.re

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7%

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

   4. Applied rewrites 53.7%

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

   if -8.8e8 < y.re < -1.65e-119 or 1.75e-287 < y.re < 13000

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7%

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 39.5%

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

   7. Applied rewrites 39.5%

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

   if -1.65e-119 < y.re < 1.75e-287

   1. Initial program 40.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. exp-neg N/A

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

      7. sub-negate-rev N/A

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

      8. lift--.f64 N/A

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

   3. Applied rewrites 40.7%

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

   4. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 27.6%

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

   6. Applied rewrites 27.6%

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

Alternative 5: 60.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_0\\ \mathbf{if}\;y.re \leq -880000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 13000:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
       (t_1
        (*
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im)))
         t_0)))
  (if (<= y.re -880000000)
    t_1
    (if (<= y.re 13000)
      (* (exp (- (* y.im (atan2 x.im x.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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	double tmp;
	if (y_46_re <= -880000000.0) {
		tmp = t_1;
	} else if (y_46_re <= 13000.0) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = 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 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * t_0
    if (y_46re <= (-880000000.0d0)) then
        tmp = t_1
    else if (y_46re <= 13000.0d0) then
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * t_0
    else
        tmp = 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.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	double tmp;
	if (y_46_re <= -880000000.0) {
		tmp = t_1;
	} else if (y_46_re <= 13000.0) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * t_0
	tmp = 0
	if y_46_re <= -880000000.0:
		tmp = t_1
	elif y_46_re <= 13000.0:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * t_0)
	tmp = 0.0
	if (y_46_re <= -880000000.0)
		tmp = t_1;
	elseif (y_46_re <= 13000.0)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_0);
	else
		tmp = 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	tmp = 0.0;
	if (y_46_re <= -880000000.0)
		tmp = t_1;
	elseif (y_46_re <= 13000.0)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	else
		tmp = 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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -880000000], t$95$1, If[LessEqual[y$46$re, 13000], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.8e8 or 13000 < y.re

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7%

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

   4. Applied rewrites 53.7%

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

   if -8.8e8 < y.re < 13000

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7%

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 39.5%

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

   7. Applied rewrites 39.5%

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

Alternative 6: 59.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -880000000:\\ \;\;\;\;{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 13500:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)} + x.re \cdot x.re\right)}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
  (if (<= y.re -880000000)
    (* (pow (+ (* x.im x.im) (* x.re x.re)) (* 1/2 y.re)) t_0)
    (if (<= y.re 13500)
      (*
       (exp (- (* y.im (atan2 x.im x.re))))
       (sin (* y.re (atan2 x.im x.re))))
      (*
       (pow
        (+ (sqrt (* (* x.im x.im) (* x.im x.im))) (* x.re x.re))
        (* 1/2 y.re))
       t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -880000000.0) {
		tmp = pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	} else if (y_46_re <= 13500.0) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = pow((sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    if (y_46re <= (-880000000.0d0)) then
        tmp = (((x_46im * x_46im) + (x_46re * x_46re)) ** (0.5d0 * y_46re)) * t_0
    else if (y_46re <= 13500.0d0) then
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * sin((y_46re * atan2(x_46im, x_46re)))
    else
        tmp = ((sqrt(((x_46im * x_46im) * (x_46im * x_46im))) + (x_46re * x_46re)) ** (0.5d0 * y_46re)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -880000000.0) {
		tmp = Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	} else if (y_46_re <= 13500.0) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = Math.pow((Math.sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -880000000.0:
		tmp = math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0
	elif y_46_re <= 13500.0:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = math.pow((math.sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -880000000.0)
		tmp = Float64((Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(0.5 * y_46_re)) * t_0);
	elseif (y_46_re <= 13500.0)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64((Float64(sqrt(Float64(Float64(x_46_im * x_46_im) * Float64(x_46_im * x_46_im))) + Float64(x_46_re * x_46_re)) ^ Float64(0.5 * y_46_re)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -880000000.0)
		tmp = (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (0.5 * y_46_re)) * t_0;
	elseif (y_46_re <= 13500.0)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	else
		tmp = ((sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)) ^ (0.5 * y_46_re)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -880000000], N[(N[Power[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(1/2 * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 13500], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(1/2 * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.8e8

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 44.1%

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

   4. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.1%

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

      4. lift-pow.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-pow2 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 44.1%

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 44.1%

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

   if -8.8e8 < y.re < 13500

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.7%

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

   4. Applied rewrites 53.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 39.5%

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

   7. Applied rewrites 39.5%

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

   if 13500 < y.re

   1. Initial program 40.7%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-pow.f64 44.1%

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

   4. Applied rewrites 44.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.1%

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

      4. lift-pow.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-pow2 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-pow.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-+.f64 N/A

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

      16. lower-pow.f64 N/A

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

      17. lift-+.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-+.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 44.1%

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

      24. lift-*.f64 N/A

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

      25. *-commutative N/A

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

   6. Applied rewrites 44.1%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-*.f32 N/A

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

      4. lower-unsound-*.f32 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-unsound-*.f64 43.9%

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

   8. Applied rewrites 43.9%

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

Alternative 7: 44.1% accurate, 1.9× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (*
 (pow
  (+ (* x.im x.im) (sqrt (* (* x.re x.re) (* x.re x.re))))
  (* 1/2 y.re))
 (sin (* (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) {
	return pow(((x_46_im * x_46_im) + sqrt(((x_46_re * x_46_re) * (x_46_re * x_46_re)))), (0.5 * y_46_re)) * sin((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
    code = (((x_46im * x_46im) + sqrt(((x_46re * x_46re) * (x_46re * x_46re)))) ** (0.5d0 * y_46re)) * sin((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) {
	return Math.pow(((x_46_im * x_46_im) + Math.sqrt(((x_46_re * x_46_re) * (x_46_re * x_46_re)))), (0.5 * y_46_re)) * Math.sin((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):
	return math.pow(((x_46_im * x_46_im) + math.sqrt(((x_46_re * x_46_re) * (x_46_re * x_46_re)))), (0.5 * y_46_re)) * math.sin((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)
	return Float64((Float64(Float64(x_46_im * x_46_im) + sqrt(Float64(Float64(x_46_re * x_46_re) * Float64(x_46_re * x_46_re)))) ^ Float64(0.5 * y_46_re)) * sin(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)
	tmp = (((x_46_im * x_46_im) + sqrt(((x_46_re * x_46_re) * (x_46_re * x_46_re)))) ^ (0.5 * y_46_re)) * sin((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_] := N[(N[Power[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] * N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1/2 * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

2. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 44.1%

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

4. Applied rewrites 44.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 44.1%

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

   4. lift-pow.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-pow2 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 44.1%

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

6. Applied rewrites 44.1%

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

   1. rem-square-sqrt N/A

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

   2. sqrt-unprod N/A

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

   3. lower-*.f32 N/A

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

   4. lower-unsound-*.f32 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-unsound-*.f64 43.9%

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

8. Applied rewrites 43.9%

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

Alternative 8: 43.9% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (*
 (pow (+ (* x.im x.im) (* x.re x.re)) (* 1/2 y.re))
 (sin (* (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) {
	return pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * sin((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
    code = (((x_46im * x_46im) + (x_46re * x_46re)) ** (0.5d0 * y_46re)) * sin((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) {
	return Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * Math.sin((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):
	return math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * math.sin((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)
	return Float64((Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(0.5 * y_46_re)) * sin(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)
	tmp = (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (0.5 * y_46_re)) * sin((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_] := N[(N[Power[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(1/2 * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

2. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 44.1%

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

4. Applied rewrites 44.1%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 44.1%

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

   4. lift-pow.f64 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. sqrt-pow2 N/A

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

   7. lift-+.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-pow.f64 N/A

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

   12. pow2 N/A

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

   13. lift-*.f64 N/A

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

   14. +-commutative N/A

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

   15. lift-+.f64 N/A

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

   16. lower-pow.f64 N/A

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

   17. lift-+.f64 N/A

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

   18. +-commutative N/A

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

   19. lift-+.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 44.1%

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

   24. lift-*.f64 N/A

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

   25. *-commutative N/A

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

6. Applied rewrites 44.1%

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

Alternative 9: 13.8% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (* (sin (* y.re (atan2 x.im x.re))) 1))

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 1), $MachinePrecision]

Derivation

1. Initial program 40.7%

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

2. Taylor expanded in y.im around 0

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

   1. lower-*.f64 N/A

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

   2. lower-sin.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-atan2.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 44.1%

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

4. Applied rewrites 44.1%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 13.8%

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

Reproduce

herbie shell --seed 2025274 -o generate:evaluate
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, imaginary part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (sin (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))