powComplex, imaginary part

Percentage Accurate: 40.1% → 76.9%

Time: 5.4s

Alternatives: 11

Speedup: 2.1×

• 64.4% 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: 40.1% 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: 76.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := -1 \cdot \log \left(\frac{1}{x.im}\right)\\ t_3 := \log \left(-1 \cdot x.im\right)\\ t_4 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;x.im \leq \frac{-5}{4}:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq 330000000:\\ \;\;\;\;e^{t\_4 \cdot y.re - t\_0} \cdot \sin \left(t\_4 \cdot y.im + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \end{array} \]

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 -5/4)
    (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))
    (if (<= x.im 330000000)
      (* (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 <= -1.25) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 330000000.0) {
		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 <= (-1.25d0)) then
        tmp = exp(((t_3 * y_46re) - t_0)) * sin(((t_3 * y_46im) + t_1))
    else if (x_46im <= 330000000.0d0) 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 <= -1.25) {
		tmp = Math.exp(((t_3 * y_46_re) - t_0)) * Math.sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 330000000.0) {
		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 <= -1.25:
		tmp = math.exp(((t_3 * y_46_re) - t_0)) * math.sin(((t_3 * y_46_im) + t_1))
	elif x_46_im <= 330000000.0:
		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 <= -1.25)
		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 <= 330000000.0)
		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 <= -1.25)
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	elseif (x_46_im <= 330000000.0)
		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, -5/4], 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, 330000000], 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 < -1.25

   1. Initial program 40.1%

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

   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 18.4%

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

   4. Applied rewrites 18.4%

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

   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 31.8%

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

   7. Applied rewrites 31.8%

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

   if -1.25 < x.im < 3.3e8

   1. Initial program 40.1%

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

   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 18.4%

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

   4. Applied rewrites 18.4%

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

   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 33.3%

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

   7. Applied rewrites 33.3%

      \[\leadsto e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 33.3%

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

      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 33.3%

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

   9. Applied rewrites 33.3%

      \[\leadsto e^{\log \color{blue}{\left(\left|-x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 66.6%

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

       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 66.6%

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

   11. Applied rewrites 66.6%

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

   if 3.3e8 < x.im

   1. Initial program 40.1%

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

   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 17.4%

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

   4. Applied rewrites 17.4%

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

   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 31.5%

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

   7. Applied rewrites 31.5%

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

Alternative 2: 71.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \log \left(-1 \cdot x.im\right)\\ t_3 := \log \left(\left|-x.re\right|\right)\\ t_4 := e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{if}\;x.im \leq \frac{-5}{4}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq \frac{8014330305721943}{348449143727040986586495598010130648530944}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;x.im \leq 10399999999999999592378776594568490767468128112049153555772773438556226962999761077289336660074706146229471457949777021702398639486777005109428469786184006107887482060758036619779859367869057292057906809113935872:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \]

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))))
       (t_4
        (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))))
  (if (<= x.im -5/4)
    (* (exp (- (* t_2 y.re) t_0)) (sin (+ (* t_2 y.im) t_1)))
    (if (<=
         x.im
         8014330305721943/348449143727040986586495598010130648530944)
      t_4
      (if (<=
           x.im
           10399999999999999592378776594568490767468128112049153555772773438556226962999761077289336660074706146229471457949777021702398639486777005109428469786184006107887482060758036619779859367869057292057906809113935872)
        (*
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           t_0))
         (sin (* y.re (atan2 x.im x.re))))
        t_4)))))

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 t_4 = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	double tmp;
	if (x_46_im <= -1.25) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_im <= 2.3e-26) {
		tmp = t_4;
	} else if (x_46_im <= 1.04e+211) {
		tmp = 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)));
	} else {
		tmp = t_4;
	}
	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 = log(((-1.0d0) * x_46im))
    t_3 = log(abs(-x_46re))
    t_4 = exp(((t_3 * y_46re) - t_0)) * sin(((t_3 * y_46im) + t_1))
    if (x_46im <= (-1.25d0)) then
        tmp = exp(((t_2 * y_46re) - t_0)) * sin(((t_2 * y_46im) + t_1))
    else if (x_46im <= 2.3d-26) then
        tmp = t_4
    else if (x_46im <= 1.04d+211) then
        tmp = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - t_0)) * sin((y_46re * atan2(x_46im, x_46re)))
    else
        tmp = t_4
    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 t_4 = Math.exp(((t_3 * y_46_re) - t_0)) * Math.sin(((t_3 * y_46_im) + t_1));
	double tmp;
	if (x_46_im <= -1.25) {
		tmp = Math.exp(((t_2 * y_46_re) - t_0)) * Math.sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_im <= 2.3e-26) {
		tmp = t_4;
	} else if (x_46_im <= 1.04e+211) {
		tmp = 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)));
	} else {
		tmp = t_4;
	}
	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))
	t_4 = math.exp(((t_3 * y_46_re) - t_0)) * math.sin(((t_3 * y_46_im) + t_1))
	tmp = 0
	if x_46_im <= -1.25:
		tmp = math.exp(((t_2 * y_46_re) - t_0)) * math.sin(((t_2 * y_46_im) + t_1))
	elif x_46_im <= 2.3e-26:
		tmp = t_4
	elif x_46_im <= 1.04e+211:
		tmp = 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)))
	else:
		tmp = t_4
	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)))
	t_4 = Float64(exp(Float64(Float64(t_3 * y_46_re) - t_0)) * sin(Float64(Float64(t_3 * y_46_im) + t_1)))
	tmp = 0.0
	if (x_46_im <= -1.25)
		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 <= 2.3e-26)
		tmp = t_4;
	elseif (x_46_im <= 1.04e+211)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = t_4;
	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));
	t_4 = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	tmp = 0.0;
	if (x_46_im <= -1.25)
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	elseif (x_46_im <= 2.3e-26)
		tmp = t_4;
	elseif (x_46_im <= 1.04e+211)
		tmp = 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)));
	else
		tmp = t_4;
	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]}, Block[{t$95$4 = 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, -5/4], 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, 8014330305721943/348449143727040986586495598010130648530944], t$95$4, If[LessEqual[x$46$im, 10399999999999999592378776594568490767468128112049153555772773438556226962999761077289336660074706146229471457949777021702398639486777005109428469786184006107887482060758036619779859367869057292057906809113935872], 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], t$95$4]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.25

   1. Initial program 40.1%

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

   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 18.4%

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

   4. Applied rewrites 18.4%

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

   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 31.8%

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

   7. Applied rewrites 31.8%

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

   if -1.25 < x.im < 2.3000000000000001e-26 or 1.04e211 < x.im

   1. Initial program 40.1%

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

   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 18.4%

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

   4. Applied rewrites 18.4%

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

   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 33.3%

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

   7. Applied rewrites 33.3%

      \[\leadsto e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 33.3%

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

      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 33.3%

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

   9. Applied rewrites 33.3%

      \[\leadsto e^{\log \color{blue}{\left(\left|-x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 66.6%

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

       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 66.6%

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

   11. Applied rewrites 66.6%

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

   if 2.3000000000000001e-26 < x.im < 1.04e211

   1. Initial program 40.1%

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

   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%

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

Alternative 3: 69.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;y.re \leq \frac{-7269215601948759}{10384593717069655257060992658440192}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin t\_2\\ \mathbf{elif}\;y.re \leq \frac{7656353255721115}{21267647932558653966460912964485513216}:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;y.re \leq 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816:\\ \;\;\;\;{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot \sin t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - t\_2\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

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 (* y.re (atan2 x.im x.re)))
       (t_3 (log (fabs (- x.re)))))
  (if (<= y.re -7269215601948759/10384593717069655257060992658440192)
    (*
     (exp
      (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
     (sin t_2))
    (if (<=
         y.re
         7656353255721115/21267647932558653966460912964485513216)
      (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))
      (if (<=
           y.re
           2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816)
        (*
         (pow (+ (* x.im x.im) (* x.re x.re)) (* 1/2 y.re))
         (sin t_1))
        (*
         (sin (- PI t_2))
         (pow (sqrt (+ (pow x.im 2) (pow x.re 2))) y.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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = log(fabs(-x_46_re));
	double tmp;
	if (y_46_re <= -7e-19) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(t_2);
	} else if (y_46_re <= 3.6e-22) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	} else if (y_46_re <= 2.15e+207) {
		tmp = pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * sin(t_1);
	} else {
		tmp = sin((((double) M_PI) - t_2)) * pow(sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0))), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.log(Math.abs(-x_46_re));
	double tmp;
	if (y_46_re <= -7e-19) {
		tmp = 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(t_2);
	} else if (y_46_re <= 3.6e-22) {
		tmp = Math.exp(((t_3 * y_46_re) - t_0)) * Math.sin(((t_3 * y_46_im) + t_1));
	} else if (y_46_re <= 2.15e+207) {
		tmp = Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * Math.sin(t_1);
	} else {
		tmp = Math.sin((Math.PI - t_2)) * Math.pow(Math.sqrt((Math.pow(x_46_im, 2.0) + Math.pow(x_46_re, 2.0))), y_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 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = math.log(math.fabs(-x_46_re))
	tmp = 0
	if y_46_re <= -7e-19:
		tmp = 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(t_2)
	elif y_46_re <= 3.6e-22:
		tmp = math.exp(((t_3 * y_46_re) - t_0)) * math.sin(((t_3 * y_46_im) + t_1))
	elif y_46_re <= 2.15e+207:
		tmp = math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * math.sin(t_1)
	else:
		tmp = math.sin((math.pi - t_2)) * math.pow(math.sqrt((math.pow(x_46_im, 2.0) + math.pow(x_46_re, 2.0))), y_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 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = log(abs(Float64(-x_46_re)))
	tmp = 0.0
	if (y_46_re <= -7e-19)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(t_2));
	elseif (y_46_re <= 3.6e-22)
		tmp = Float64(exp(Float64(Float64(t_3 * y_46_re) - t_0)) * sin(Float64(Float64(t_3 * y_46_im) + t_1)));
	elseif (y_46_re <= 2.15e+207)
		tmp = Float64((Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(0.5 * y_46_re)) * sin(t_1));
	else
		tmp = Float64(sin(Float64(pi - t_2)) * (sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = log(abs(-x_46_re));
	tmp = 0.0;
	if (y_46_re <= -7e-19)
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(t_2);
	elseif (y_46_re <= 3.6e-22)
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	elseif (y_46_re <= 2.15e+207)
		tmp = (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (0.5 * y_46_re)) * sin(t_1);
	else
		tmp = sin((pi - t_2)) * (sqrt(((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -7269215601948759/10384593717069655257060992658440192], 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[t$95$2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7656353255721115/21267647932558653966460912964485513216], 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[y$46$re, 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816], 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[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2], $MachinePrecision] + N[Power[x$46$re, 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.0000000000000003e-19

   1. Initial program 40.1%

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

   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 -7.0000000000000003e-19 < y.re < 3.5999999999999998e-22

   1. Initial program 40.1%

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

   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 18.4%

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

   4. Applied rewrites 18.4%

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

   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 33.3%

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

   7. Applied rewrites 33.3%

      \[\leadsto e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 33.3%

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

      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 33.3%

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

   9. Applied rewrites 33.3%

      \[\leadsto e^{\log \color{blue}{\left(\left|-x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 66.6%

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

       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 66.6%

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

   11. Applied rewrites 66.6%

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

   if 3.5999999999999998e-22 < y.re < 2.1499999999999999e207

   1. Initial program 40.1%

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

   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 44.3%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 44.3%

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

      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 44.3%

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

      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 44.3%

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

   if 2.1499999999999999e207 < y.re

   1. Initial program 40.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.6%

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

   4. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 39.8%

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

   6. Applied rewrites 39.8%

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

Alternative 4: 60.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t\_1\\ t_3 := x.im \cdot x.im + x.re \cdot x.re\\ t_4 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_5 := e^{-t\_4} \cdot t\_2\\ \mathbf{if}\;y.re \leq \frac{-3961408125713217}{1237940039285380274899124224}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_2\\ \mathbf{elif}\;y.re \leq \frac{-6205025112157247}{63316582777114760719488645381029680648993625369910231018000142359781689627272157995600998671678219517337003885060131670873949448782528309751691815706084650986651333670066978816}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;y.re \leq \frac{2337007168393775}{2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512}:\\ \;\;\;\;\frac{\sin \left(t\_0 + y.im \cdot \left(\frac{1}{2} \cdot \log t\_3\right)\right)}{e^{t\_4}}\\ \mathbf{elif}\;y.re \leq \frac{7656353255721115}{21267647932558653966460912964485513216}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;y.re \leq 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816:\\ \;\;\;\;{t\_3}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot \sin t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - t\_1\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.re))
       (t_1 (* y.re (atan2 x.im x.re)))
       (t_2 (sin t_1))
       (t_3 (+ (* x.im x.im) (* x.re x.re)))
       (t_4 (* y.im (atan2 x.im x.re)))
       (t_5 (* (exp (- t_4)) t_2)))
  (if (<= y.re -3961408125713217/1237940039285380274899124224)
    (*
     (exp
      (-
       (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
       (* (atan2 x.im x.re) y.im)))
     t_2)
    (if (<=
         y.re
         -6205025112157247/63316582777114760719488645381029680648993625369910231018000142359781689627272157995600998671678219517337003885060131670873949448782528309751691815706084650986651333670066978816)
      t_5
      (if (<=
           y.re
           2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512)
        (/ (sin (+ t_0 (* y.im (* 1/2 (log t_3))))) (exp t_4))
        (if (<=
             y.re
             7656353255721115/21267647932558653966460912964485513216)
          t_5
          (if (<=
               y.re
               2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816)
            (* (pow t_3 (* 1/2 y.re)) (sin t_0))
            (*
             (sin (- PI t_1))
             (pow (sqrt (+ (pow x.im 2) (pow x.re 2))) y.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_re;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(t_1);
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_4 = y_46_im * atan2(x_46_im, x_46_re);
	double t_5 = exp(-t_4) * t_2;
	double tmp;
	if (y_46_re <= -3.2e-12) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_2;
	} else if (y_46_re <= -9.8e-161) {
		tmp = t_5;
	} else if (y_46_re <= 1.1e-168) {
		tmp = sin((t_0 + (y_46_im * (0.5 * log(t_3))))) / exp(t_4);
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_5;
	} else if (y_46_re <= 2.15e+207) {
		tmp = pow(t_3, (0.5 * y_46_re)) * sin(t_0);
	} else {
		tmp = sin((((double) M_PI) - t_1)) * pow(sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0))), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.sin(t_1);
	double t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_4 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_5 = Math.exp(-t_4) * t_2;
	double tmp;
	if (y_46_re <= -3.2e-12) {
		tmp = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * t_2;
	} else if (y_46_re <= -9.8e-161) {
		tmp = t_5;
	} else if (y_46_re <= 1.1e-168) {
		tmp = Math.sin((t_0 + (y_46_im * (0.5 * Math.log(t_3))))) / Math.exp(t_4);
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_5;
	} else if (y_46_re <= 2.15e+207) {
		tmp = Math.pow(t_3, (0.5 * y_46_re)) * Math.sin(t_0);
	} else {
		tmp = Math.sin((Math.PI - t_1)) * Math.pow(Math.sqrt((Math.pow(x_46_im, 2.0) + Math.pow(x_46_re, 2.0))), y_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_re
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.sin(t_1)
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	t_4 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_5 = math.exp(-t_4) * t_2
	tmp = 0
	if y_46_re <= -3.2e-12:
		tmp = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * t_2
	elif y_46_re <= -9.8e-161:
		tmp = t_5
	elif y_46_re <= 1.1e-168:
		tmp = math.sin((t_0 + (y_46_im * (0.5 * math.log(t_3))))) / math.exp(t_4)
	elif y_46_re <= 3.6e-22:
		tmp = t_5
	elif y_46_re <= 2.15e+207:
		tmp = math.pow(t_3, (0.5 * y_46_re)) * math.sin(t_0)
	else:
		tmp = math.sin((math.pi - t_1)) * math.pow(math.sqrt((math.pow(x_46_im, 2.0) + math.pow(x_46_re, 2.0))), y_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_re)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = sin(t_1)
	t_3 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	t_4 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_5 = Float64(exp(Float64(-t_4)) * t_2)
	tmp = 0.0
	if (y_46_re <= -3.2e-12)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * t_2);
	elseif (y_46_re <= -9.8e-161)
		tmp = t_5;
	elseif (y_46_re <= 1.1e-168)
		tmp = Float64(sin(Float64(t_0 + Float64(y_46_im * Float64(0.5 * log(t_3))))) / exp(t_4));
	elseif (y_46_re <= 3.6e-22)
		tmp = t_5;
	elseif (y_46_re <= 2.15e+207)
		tmp = Float64((t_3 ^ Float64(0.5 * y_46_re)) * sin(t_0));
	else
		tmp = Float64(sin(Float64(pi - t_1)) * (sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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_re;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = sin(t_1);
	t_3 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	t_4 = y_46_im * atan2(x_46_im, x_46_re);
	t_5 = exp(-t_4) * t_2;
	tmp = 0.0;
	if (y_46_re <= -3.2e-12)
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_2;
	elseif (y_46_re <= -9.8e-161)
		tmp = t_5;
	elseif (y_46_re <= 1.1e-168)
		tmp = sin((t_0 + (y_46_im * (0.5 * log(t_3))))) / exp(t_4);
	elseif (y_46_re <= 3.6e-22)
		tmp = t_5;
	elseif (y_46_re <= 2.15e+207)
		tmp = (t_3 ^ (0.5 * y_46_re)) * sin(t_0);
	else
		tmp = sin((pi - t_1)) * (sqrt(((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Exp[(-t$95$4)], $MachinePrecision] * t$95$2), $MachinePrecision]}, If[LessEqual[y$46$re, -3961408125713217/1237940039285380274899124224], 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$2), $MachinePrecision], If[LessEqual[y$46$re, -6205025112157247/63316582777114760719488645381029680648993625369910231018000142359781689627272157995600998671678219517337003885060131670873949448782528309751691815706084650986651333670066978816], t$95$5, If[LessEqual[y$46$re, 2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512], N[(N[Sin[N[(t$95$0 + N[(y$46$im * N[(1/2 * N[Log[t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$4], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7656353255721115/21267647932558653966460912964485513216], t$95$5, If[LessEqual[y$46$re, 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816], N[(N[Power[t$95$3, N[(1/2 * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2], $MachinePrecision] + N[Power[x$46$re, 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.2000000000000001e-12

   1. Initial program 40.1%

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

   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 -3.2000000000000001e-12 < y.re < -9.8000000000000007e-161 or 1.0999999999999999e-168 < y.re < 3.5999999999999998e-22

   1. Initial program 40.1%

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

   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 40.3%

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

   7. Applied rewrites 40.3%

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

   if -9.8000000000000007e-161 < y.re < 1.0999999999999999e-168

   1. Initial program 40.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 40.0%

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

   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 26.7%

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

   6. Applied rewrites 26.7%

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

   if 3.5999999999999998e-22 < y.re < 2.1499999999999999e207

   1. Initial program 40.1%

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

   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 44.3%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 44.3%

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

      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 44.3%

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

      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 44.3%

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

   if 2.1499999999999999e207 < y.re

   1. Initial program 40.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.6%

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

   4. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 39.8%

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

   6. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\sin \left(\pi - y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 5: 60.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ t_2 := x.im \cdot x.im + x.re \cdot x.re\\ t_3 := e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_1\\ \mathbf{if}\;y.re \leq \frac{-3961408125713217}{1237940039285380274899124224}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_1\\ \mathbf{elif}\;y.re \leq \frac{-1558003967662813}{69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.re \leq \frac{2337007168393775}{2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512}:\\ \;\;\;\;\sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log t\_2\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq \frac{7656353255721115}{21267647932558653966460912964485513216}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.re \leq 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816:\\ \;\;\;\;{t\_2}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - t\_0\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (sin t_0))
       (t_2 (+ (* x.im x.im) (* x.re x.re)))
       (t_3 (* (exp (- (* y.im (atan2 x.im x.re)))) t_1)))
  (if (<= y.re -3961408125713217/1237940039285380274899124224)
    (*
     (exp
      (-
       (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
       (* (atan2 x.im x.re) y.im)))
     t_1)
    (if (<=
         y.re
         -1558003967662813/69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192)
      t_3
      (if (<=
           y.re
           2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512)
        (*
         (sin (* (* 1/2 y.im) (log t_2)))
         (exp (* (- y.im) (atan2 x.im x.re))))
        (if (<=
             y.re
             7656353255721115/21267647932558653966460912964485513216)
          t_3
          (if (<=
               y.re
               2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816)
            (*
             (pow t_2 (* 1/2 y.re))
             (sin (* (atan2 x.im x.re) y.re)))
            (*
             (sin (- PI t_0))
             (pow (sqrt (+ (pow x.im 2) (pow x.re 2))) y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_3 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	double tmp;
	if (y_46_re <= -3.2e-12) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_1;
	} else if (y_46_re <= -2.25e-260) {
		tmp = t_3;
	} else if (y_46_re <= 1.1e-168) {
		tmp = sin(((0.5 * y_46_im) * log(t_2))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_3;
	} else if (y_46_re <= 2.15e+207) {
		tmp = pow(t_2, (0.5 * y_46_re)) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	} else {
		tmp = sin((((double) M_PI) - t_0)) * pow(sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0))), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_3 = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * t_1;
	double tmp;
	if (y_46_re <= -3.2e-12) {
		tmp = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * t_1;
	} else if (y_46_re <= -2.25e-260) {
		tmp = t_3;
	} else if (y_46_re <= 1.1e-168) {
		tmp = Math.sin(((0.5 * y_46_im) * Math.log(t_2))) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_3;
	} else if (y_46_re <= 2.15e+207) {
		tmp = Math.pow(t_2, (0.5 * y_46_re)) * Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	} else {
		tmp = Math.sin((Math.PI - t_0)) * Math.pow(Math.sqrt((Math.pow(x_46_im, 2.0) + Math.pow(x_46_re, 2.0))), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin(t_0)
	t_2 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	t_3 = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * t_1
	tmp = 0
	if y_46_re <= -3.2e-12:
		tmp = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * t_1
	elif y_46_re <= -2.25e-260:
		tmp = t_3
	elif y_46_re <= 1.1e-168:
		tmp = math.sin(((0.5 * y_46_im) * math.log(t_2))) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 3.6e-22:
		tmp = t_3
	elif y_46_re <= 2.15e+207:
		tmp = math.pow(t_2, (0.5 * y_46_re)) * math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	else:
		tmp = math.sin((math.pi - t_0)) * math.pow(math.sqrt((math.pow(x_46_im, 2.0) + math.pow(x_46_re, 2.0))), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	t_3 = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_1)
	tmp = 0.0
	if (y_46_re <= -3.2e-12)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * t_1);
	elseif (y_46_re <= -2.25e-260)
		tmp = t_3;
	elseif (y_46_re <= 1.1e-168)
		tmp = Float64(sin(Float64(Float64(0.5 * y_46_im) * log(t_2))) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 3.6e-22)
		tmp = t_3;
	elseif (y_46_re <= 2.15e+207)
		tmp = Float64((t_2 ^ Float64(0.5 * y_46_re)) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = Float64(sin(Float64(pi - t_0)) * (sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	t_2 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	t_3 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	tmp = 0.0;
	if (y_46_re <= -3.2e-12)
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_1;
	elseif (y_46_re <= -2.25e-260)
		tmp = t_3;
	elseif (y_46_re <= 1.1e-168)
		tmp = sin(((0.5 * y_46_im) * log(t_2))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 3.6e-22)
		tmp = t_3;
	elseif (y_46_re <= 2.15e+207)
		tmp = (t_2 ^ (0.5 * y_46_re)) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	else
		tmp = sin((pi - t_0)) * (sqrt(((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -3961408125713217/1237940039285380274899124224], 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$1), $MachinePrecision], If[LessEqual[y$46$re, -1558003967662813/69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192], t$95$3, If[LessEqual[y$46$re, 2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512], N[(N[Sin[N[(N[(1/2 * y$46$im), $MachinePrecision] * N[Log[t$95$2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7656353255721115/21267647932558653966460912964485513216], t$95$3, If[LessEqual[y$46$re, 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816], N[(N[Power[t$95$2, 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], N[(N[Sin[N[(Pi - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2], $MachinePrecision] + N[Power[x$46$re, 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.2000000000000001e-12

   1. Initial program 40.1%

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

   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 -3.2000000000000001e-12 < y.re < -2.2499999999999999e-260 or 1.0999999999999999e-168 < y.re < 3.5999999999999998e-22

   1. Initial program 40.1%

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

   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 40.3%

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

   7. Applied rewrites 40.3%

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

   if -2.2499999999999999e-260 < y.re < 1.0999999999999999e-168

   1. Initial program 40.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 40.0%

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

   4. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 22.0%

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

   6. Applied rewrites 22.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 22.0%

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

   if 3.5999999999999998e-22 < y.re < 2.1499999999999999e207

   1. Initial program 40.1%

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

   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 44.3%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 44.3%

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

      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 44.3%

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

      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 44.3%

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

   if 2.1499999999999999e207 < y.re

   1. Initial program 40.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.6%

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

   4. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 39.8%

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

   6. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\sin \left(\pi - y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 6: 59.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := x.im \cdot x.im + x.re \cdot x.re\\ t_1 := {t\_0}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin t\_2\\ \mathbf{if}\;y.re \leq \frac{-3771848557197643}{4835703278458516698824704}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq \frac{-1558003967662813}{69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.re \leq \frac{2337007168393775}{2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512}:\\ \;\;\;\;\sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log t\_0\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq \frac{7656353255721115}{21267647932558653966460912964485513216}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.re \leq 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - t\_2\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (+ (* x.im x.im) (* x.re x.re)))
       (t_1
        (* (pow t_0 (* 1/2 y.re)) (sin (* (atan2 x.im x.re) y.re))))
       (t_2 (* y.re (atan2 x.im x.re)))
       (t_3 (* (exp (- (* y.im (atan2 x.im x.re)))) (sin t_2))))
  (if (<= y.re -3771848557197643/4835703278458516698824704)
    t_1
    (if (<=
         y.re
         -1558003967662813/69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192)
      t_3
      (if (<=
           y.re
           2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512)
        (*
         (sin (* (* 1/2 y.im) (log t_0)))
         (exp (* (- y.im) (atan2 x.im x.re))))
        (if (<=
             y.re
             7656353255721115/21267647932558653966460912964485513216)
          t_3
          (if (<=
               y.re
               2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816)
            t_1
            (*
             (sin (- PI t_2))
             (pow (sqrt (+ (pow x.im 2) (pow x.re 2))) y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = pow(t_0, (0.5 * y_46_re)) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(t_2);
	double tmp;
	if (y_46_re <= -7.8e-10) {
		tmp = t_1;
	} else if (y_46_re <= -2.25e-260) {
		tmp = t_3;
	} else if (y_46_re <= 1.1e-168) {
		tmp = sin(((0.5 * y_46_im) * log(t_0))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_3;
	} else if (y_46_re <= 2.15e+207) {
		tmp = t_1;
	} else {
		tmp = sin((((double) M_PI) - t_2)) * pow(sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0))), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_1 = Math.pow(t_0, (0.5 * y_46_re)) * Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double t_2 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_3 = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin(t_2);
	double tmp;
	if (y_46_re <= -7.8e-10) {
		tmp = t_1;
	} else if (y_46_re <= -2.25e-260) {
		tmp = t_3;
	} else if (y_46_re <= 1.1e-168) {
		tmp = Math.sin(((0.5 * y_46_im) * Math.log(t_0))) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_3;
	} else if (y_46_re <= 2.15e+207) {
		tmp = t_1;
	} else {
		tmp = Math.sin((Math.PI - t_2)) * Math.pow(Math.sqrt((Math.pow(x_46_im, 2.0) + Math.pow(x_46_re, 2.0))), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	t_1 = math.pow(t_0, (0.5 * y_46_re)) * math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	t_2 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_3 = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin(t_2)
	tmp = 0
	if y_46_re <= -7.8e-10:
		tmp = t_1
	elif y_46_re <= -2.25e-260:
		tmp = t_3
	elif y_46_re <= 1.1e-168:
		tmp = math.sin(((0.5 * y_46_im) * math.log(t_0))) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 3.6e-22:
		tmp = t_3
	elif y_46_re <= 2.15e+207:
		tmp = t_1
	else:
		tmp = math.sin((math.pi - t_2)) * math.pow(math.sqrt((math.pow(x_46_im, 2.0) + math.pow(x_46_re, 2.0))), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	t_1 = Float64((t_0 ^ Float64(0.5 * y_46_re)) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(t_2))
	tmp = 0.0
	if (y_46_re <= -7.8e-10)
		tmp = t_1;
	elseif (y_46_re <= -2.25e-260)
		tmp = t_3;
	elseif (y_46_re <= 1.1e-168)
		tmp = Float64(sin(Float64(Float64(0.5 * y_46_im) * log(t_0))) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 3.6e-22)
		tmp = t_3;
	elseif (y_46_re <= 2.15e+207)
		tmp = t_1;
	else
		tmp = Float64(sin(Float64(pi - t_2)) * (sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	t_1 = (t_0 ^ (0.5 * y_46_re)) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	t_2 = y_46_re * atan2(x_46_im, x_46_re);
	t_3 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(t_2);
	tmp = 0.0;
	if (y_46_re <= -7.8e-10)
		tmp = t_1;
	elseif (y_46_re <= -2.25e-260)
		tmp = t_3;
	elseif (y_46_re <= 1.1e-168)
		tmp = sin(((0.5 * y_46_im) * log(t_0))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 3.6e-22)
		tmp = t_3;
	elseif (y_46_re <= 2.15e+207)
		tmp = t_1;
	else
		tmp = sin((pi - t_2)) * (sqrt(((x_46_im ^ 2.0) + (x_46_re ^ 2.0))) ^ y_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[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[t$95$0, 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]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3771848557197643/4835703278458516698824704], t$95$1, If[LessEqual[y$46$re, -1558003967662813/69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192], t$95$3, If[LessEqual[y$46$re, 2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512], N[(N[Sin[N[(N[(1/2 * y$46$im), $MachinePrecision] * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7656353255721115/21267647932558653966460912964485513216], t$95$3, If[LessEqual[y$46$re, 2149999999999999869756787783799634253505931842645687432317414935889116081322126003877695797662618087362855866204951721140234906883065064887028361080783626049526844242667474211895928439227183308580700412706816], t$95$1, N[(N[Sin[N[(Pi - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2], $MachinePrecision] + N[Power[x$46$re, 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.7999999999999999e-10 or 3.5999999999999998e-22 < y.re < 2.1499999999999999e207

   1. Initial program 40.1%

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

   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 44.3%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 44.3%

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

      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 44.3%

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

      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 44.3%

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

   if -7.7999999999999999e-10 < y.re < -2.2499999999999999e-260 or 1.0999999999999999e-168 < y.re < 3.5999999999999998e-22

   1. Initial program 40.1%

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

   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 40.3%

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

   7. Applied rewrites 40.3%

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

   if -2.2499999999999999e-260 < y.re < 1.0999999999999999e-168

   1. Initial program 40.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 40.0%

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

   4. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 22.0%

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

   6. Applied rewrites 22.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 22.0%

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

   if 2.1499999999999999e207 < y.re

   1. Initial program 40.1%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 28.6%

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

   4. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lower-pow.f64 39.8%

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

   6. Applied rewrites 39.8%

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

Alternative 7: 59.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := x.im \cdot x.im + x.re \cdot x.re\\ t_2 := {t\_1}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq \frac{-3771848557197643}{4835703278458516698824704}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq \frac{-1558003967662813}{69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq \frac{2337007168393775}{2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512}:\\ \;\;\;\;\sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log t\_1\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq \frac{7656353255721115}{21267647932558653966460912964485513216}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0
        (*
         (exp (- (* y.im (atan2 x.im x.re))))
         (sin (* y.re (atan2 x.im x.re)))))
       (t_1 (+ (* x.im x.im) (* x.re x.re)))
       (t_2
        (* (pow t_1 (* 1/2 y.re)) (sin (* (atan2 x.im x.re) y.re)))))
  (if (<= y.re -3771848557197643/4835703278458516698824704)
    t_2
    (if (<=
         y.re
         -1558003967662813/69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192)
      t_0
      (if (<=
           y.re
           2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512)
        (*
         (sin (* (* 1/2 y.im) (log t_1)))
         (exp (* (- y.im) (atan2 x.im x.re))))
        (if (<=
             y.re
             7656353255721115/21267647932558653966460912964485513216)
          t_0
          t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_2 = pow(t_1, (0.5 * y_46_re)) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -7.8e-10) {
		tmp = t_2;
	} else if (y_46_re <= -2.25e-260) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-168) {
		tmp = sin(((0.5 * y_46_im) * log(t_1))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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 = exp(-(y_46im * atan2(x_46im, x_46re))) * sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = (x_46im * x_46im) + (x_46re * x_46re)
    t_2 = (t_1 ** (0.5d0 * y_46re)) * sin((atan2(x_46im, x_46re) * y_46re))
    if (y_46re <= (-7.8d-10)) then
        tmp = t_2
    else if (y_46re <= (-2.25d-260)) then
        tmp = t_0
    else if (y_46re <= 1.1d-168) then
        tmp = sin(((0.5d0 * y_46im) * log(t_1))) * exp((-y_46im * atan2(x_46im, x_46re)))
    else if (y_46re <= 3.6d-22) then
        tmp = t_0
    else
        tmp = t_2
    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.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)));
	double t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	double t_2 = Math.pow(t_1, (0.5 * y_46_re)) * Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -7.8e-10) {
		tmp = t_2;
	} else if (y_46_re <= -2.25e-260) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-168) {
		tmp = Math.sin(((0.5 * y_46_im) * Math.log(t_1))) * Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 3.6e-22) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 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)))
	t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re)
	t_2 = math.pow(t_1, (0.5 * y_46_re)) * math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -7.8e-10:
		tmp = t_2
	elif y_46_re <= -2.25e-260:
		tmp = t_0
	elif y_46_re <= 1.1e-168:
		tmp = math.sin(((0.5 * y_46_im) * math.log(t_1))) * math.exp((-y_46_im * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 3.6e-22:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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))))
	t_1 = Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))
	t_2 = Float64((t_1 ^ Float64(0.5 * y_46_re)) * sin(Float64(atan(x_46_im, x_46_re) * y_46_re)))
	tmp = 0.0
	if (y_46_re <= -7.8e-10)
		tmp = t_2;
	elseif (y_46_re <= -2.25e-260)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-168)
		tmp = Float64(sin(Float64(Float64(0.5 * y_46_im) * log(t_1))) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 3.6e-22)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = (x_46_im * x_46_im) + (x_46_re * x_46_re);
	t_2 = (t_1 ^ (0.5 * y_46_re)) * sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -7.8e-10)
		tmp = t_2;
	elseif (y_46_re <= -2.25e-260)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-168)
		tmp = sin(((0.5 * y_46_im) * log(t_1))) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 3.6e-22)
		tmp = t_0;
	else
		tmp = t_2;
	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[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]}, Block[{t$95$1 = N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[t$95$1, 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]}, If[LessEqual[y$46$re, -3771848557197643/4835703278458516698824704], t$95$2, If[LessEqual[y$46$re, -1558003967662813/69244620785013915169893880288577664649638356718206520148356066053544201755915040781030818563707002931730762300556198370198467059319135998738502644985094416224536059445117594392163977484884566481701992255745056886508490364387532700233026358315193005046436800875428314917896192], t$95$0, If[LessEqual[y$46$re, 2337007168393775/2124551971267068394758352826209874509318372470908127692797776552801614239443408970956650009060917142675557317944986004061386317350610828957638079915066349407775325083341572876126912512], N[(N[Sin[N[(N[(1/2 * y$46$im), $MachinePrecision] * N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7656353255721115/21267647932558653966460912964485513216], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.7999999999999999e-10 or 3.5999999999999998e-22 < y.re

   1. Initial program 40.1%

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

   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 44.3%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 44.3%

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

      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 44.3%

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

      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 44.3%

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

   if -7.7999999999999999e-10 < y.re < -2.2499999999999999e-260 or 1.0999999999999999e-168 < y.re < 3.5999999999999998e-22

   1. Initial program 40.1%

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

   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 40.3%

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

   7. Applied rewrites 40.3%

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

   if -2.2499999999999999e-260 < y.re < 1.0999999999999999e-168

   1. Initial program 40.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 40.0%

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

   4. Taylor expanded in y.re around 0

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-exp.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-atan2.f64 22.0%

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

   6. Applied rewrites 22.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip-rev N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. rec-exp N/A

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

      8. lower-*.f64 N/A

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

   8. Applied rewrites 22.0%

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

Alternative 8: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -135000000000000008876890202931671646615711665638753885453721428438309064336429232947200:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot t\_0\\ \mathbf{elif}\;y.im \leq 18999999999999999427829825536:\\ \;\;\;\;{\left(\left|\sqrt{{x.im}^{2} + {x.re}^{2}}\right|\right)}^{y.re} \cdot \sin \left(\log \left(\left|-x.re\right|\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
  (if (<=
       y.im
       -135000000000000008876890202931671646615711665638753885453721428438309064336429232947200)
    (*
     (exp
      (-
       (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
       (* (atan2 x.im x.re) y.im)))
     t_0)
    (if (<= y.im 18999999999999999427829825536)
      (*
       (pow (fabs (sqrt (+ (pow x.im 2) (pow x.re 2)))) y.re)
       (sin
        (+
         (* (log (fabs (- x.re))) y.im)
         (* (atan2 x.im x.re) y.re))))
      (* (exp (- (* y.im (atan2 x.im x.re)))) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -1.35e+86) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	} else if (y_46_im <= 1.9e+28) {
		tmp = pow(fabs(sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0)))), y_46_re) * sin(((log(fabs(-x_46_re)) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    if (y_46im <= (-1.35d+86)) then
        tmp = exp(((log(sqrt(((x_46re * x_46re) + (x_46im * x_46im)))) * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * t_0
    else if (y_46im <= 1.9d+28) then
        tmp = (abs(sqrt(((x_46im ** 2.0d0) + (x_46re ** 2.0d0)))) ** y_46re) * sin(((log(abs(-x_46re)) * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
    else
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -1.35e+86) {
		tmp = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	} else if (y_46_im <= 1.9e+28) {
		tmp = Math.pow(Math.abs(Math.sqrt((Math.pow(x_46_im, 2.0) + Math.pow(x_46_re, 2.0)))), y_46_re) * Math.sin(((Math.log(Math.abs(-x_46_re)) * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -1.35e+86:
		tmp = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * t_0
	elif y_46_im <= 1.9e+28:
		tmp = math.pow(math.fabs(math.sqrt((math.pow(x_46_im, 2.0) + math.pow(x_46_re, 2.0)))), y_46_re) * math.sin(((math.log(math.fabs(-x_46_re)) * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))
	else:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -1.35e+86)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * t_0);
	elseif (y_46_im <= 1.9e+28)
		tmp = Float64((abs(sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0)))) ^ y_46_re) * sin(Float64(Float64(log(abs(Float64(-x_46_re))) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	else
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -1.35e+86)
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * t_0;
	elseif (y_46_im <= 1.9e+28)
		tmp = (abs(sqrt(((x_46_im ^ 2.0) + (x_46_re ^ 2.0)))) ^ y_46_re) * sin(((log(abs(-x_46_re)) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -135000000000000008876890202931671646615711665638753885453721428438309064336429232947200], 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$46$im, 18999999999999999427829825536], N[(N[Power[N[Abs[N[Sqrt[N[(N[Power[x$46$im, 2], $MachinePrecision] + N[Power[x$46$re, 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(N[(N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.3500000000000001e86

   1. Initial program 40.1%

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

   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 -1.3500000000000001e86 < y.im < 1.8999999999999999e28

   1. Initial program 40.1%

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

   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 18.4%

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

   4. Applied rewrites 18.4%

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

   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 33.3%

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

   7. Applied rewrites 33.3%

      \[\leadsto e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \color{blue}{\left(-1 \cdot x.re\right)} \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 33.3%

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

      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 33.3%

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

   9. Applied rewrites 33.3%

      \[\leadsto e^{\log \color{blue}{\left(\left|-x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 66.6%

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

       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 66.6%

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

   11. Applied rewrites 66.6%

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

   12. Taylor expanded in y.im around 0

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

       1. lower-pow.f64 N/A

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

       2. lower-fabs.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-+.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-pow.f64 49.5%

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

   14. Applied rewrites 49.5%

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

   if 1.8999999999999999e28 < y.im

   1. Initial program 40.1%

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

   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 40.3%

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

   7. Applied rewrites 40.3%

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

Alternative 9: 57.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(\frac{1}{2} \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq \frac{-3771848557197643}{4835703278458516698824704}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq \frac{7656353255721115}{21267647932558653966460912964485513216}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0
        (*
         (pow (+ (* x.im x.im) (* x.re x.re)) (* 1/2 y.re))
         (sin (* (atan2 x.im x.re) y.re)))))
  (if (<= y.re -3771848557197643/4835703278458516698824704)
    t_0
    (if (<=
         y.re
         7656353255721115/21267647932558653966460912964485513216)
      (*
       (exp (- (* y.im (atan2 x.im x.re))))
       (sin (* y.re (atan2 x.im x.re))))
      t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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));
	double tmp;
	if (y_46_re <= -7.8e-10) {
		tmp = t_0;
	} else if (y_46_re <= 3.6e-22) {
		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 = 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 = (((x_46im * x_46im) + (x_46re * x_46re)) ** (0.5d0 * y_46re)) * sin((atan2(x_46im, x_46re) * y_46re))
    if (y_46re <= (-7.8d-10)) then
        tmp = t_0
    else if (y_46re <= 3.6d-22) then
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * sin((y_46re * atan2(x_46im, x_46re)))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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));
	double tmp;
	if (y_46_re <= -7.8e-10) {
		tmp = t_0;
	} else if (y_46_re <= 3.6e-22) {
		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 = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 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))
	tmp = 0
	if y_46_re <= -7.8e-10:
		tmp = t_0
	elif y_46_re <= 3.6e-22:
		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 = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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)))
	tmp = 0.0
	if (y_46_re <= -7.8e-10)
		tmp = t_0;
	elseif (y_46_re <= 3.6e-22)
		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 = 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 = (((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));
	tmp = 0.0;
	if (y_46_re <= -7.8e-10)
		tmp = t_0;
	elseif (y_46_re <= 3.6e-22)
		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 = 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[(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]}, If[LessEqual[y$46$re, -3771848557197643/4835703278458516698824704], t$95$0, If[LessEqual[y$46$re, 7656353255721115/21267647932558653966460912964485513216], 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], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -7.7999999999999999e-10 or 3.5999999999999998e-22 < y.re

   1. Initial program 40.1%

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

   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 44.3%

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

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 44.3%

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

      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 44.3%

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

      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 44.3%

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

   if -7.7999999999999999e-10 < y.re < 3.5999999999999998e-22

   1. Initial program 40.1%

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

   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 40.3%

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

   7. Applied rewrites 40.3%

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

Alternative 10: 44.3% 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 40.1%

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

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 44.3%

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

4. Applied rewrites 44.3%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
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 44.3%

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

   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 44.3%

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

   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 44.3%

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

Alternative 11: 14.2% 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 40.1%

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

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 44.3%

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

4. Applied rewrites 44.3%

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

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 14.2%

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

Reproduce

herbie shell --seed 2025285 -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)))))