powComplex, imaginary part

Percentage Accurate: 40.2% → 73.5%

Time: 8.9s

Alternatives: 14

Speedup: 2.0×

• 64.2% 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: 40.2% 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: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \log \left(\left|-1 \cdot x.im\right|\right)\\ t_3 := \log \left(-1 \cdot x.im\right)\\ t_4 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;x.im \leq -6.2 \cdot 10^{-137}:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq 3500000000:\\ \;\;\;\;e^{t\_4 \cdot y.re - t\_0} \cdot \sin \left(t\_4 \cdot y.im + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \end{array} \]

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 (log (fabs (* -1.0 x.im))))
       (t_3 (log (* -1.0 x.im)))
       (t_4 (log (fabs (- x.re)))))
  (if (<= x.im -6.2e-137)
    (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))
    (if (<= x.im 3500000000.0)
      (* (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 = log(fabs((-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 <= -6.2e-137) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 3500000000.0) {
		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 = log(abs(((-1.0d0) * x_46im)))
    t_3 = log(((-1.0d0) * x_46im))
    t_4 = log(abs(-x_46re))
    if (x_46im <= (-6.2d-137)) then
        tmp = exp(((t_3 * y_46re) - t_0)) * sin(((t_3 * y_46im) + t_1))
    else if (x_46im <= 3500000000.0d0) 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 = Math.log(Math.abs((-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 <= -6.2e-137) {
		tmp = Math.exp(((t_3 * y_46_re) - t_0)) * Math.sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 3500000000.0) {
		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 = math.log(math.fabs((-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 <= -6.2e-137:
		tmp = math.exp(((t_3 * y_46_re) - t_0)) * math.sin(((t_3 * y_46_im) + t_1))
	elif x_46_im <= 3500000000.0:
		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 = log(abs(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 <= -6.2e-137)
		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 <= 3500000000.0)
		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 = log(abs((-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 <= -6.2e-137)
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	elseif (x_46_im <= 3500000000.0)
		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[Log[N[Abs[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -6.2e-137], 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, 3500000000.0], 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 < -6.1999999999999996e-137

   1. Initial program 40.2%

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

   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 17.7%

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

   4. Applied rewrites 17.7%

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

   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 31.3%

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

   7. Applied rewrites 31.3%

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

   if -6.1999999999999996e-137 < x.im < 3.5e9

   1. Initial program 40.2%

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

   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 18.8%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.2%

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

   7. Applied rewrites 34.2%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.2%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.2%

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

   9. Applied rewrites 34.2%

      \[\leadsto e^{\log \color{blue}{\left(\left|-x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 3.5e9 < x.im

   1. Initial program 40.2%

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

   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 18.8%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.2%

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

   7. Applied rewrites 34.2%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.2%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.2%

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

   9. Applied rewrites 34.2%

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

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

       1. lower-*.f64 65.4%

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

   14. Applied rewrites 65.4%

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

   15. Taylor expanded in x.im around -inf

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

       1. lower-*.f64 63.1%

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

   17. Applied rewrites 63.1%

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

Alternative 2: 71.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \log \left(-1 \cdot x.im\right)\\ t_3 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;x.im \leq -6.2 \cdot 10^{-137}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq 1.8 \cdot 10^{-54}:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{+74}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\left(-1 \cdot \log \left(\frac{1}{x.im}\right)\right) \cdot y.im + t\_1\right)\\ \end{array} \]

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 (log (* -1.0 x.im)))
       (t_3 (log (fabs (- x.re)))))
  (if (<= x.im -6.2e-137)
    (* (exp (- (* t_2 y.re) t_0)) (sin (+ (* t_2 y.im) t_1)))
    (if (<= x.im 1.8e-54)
      (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))
      (if (<= x.im 9e+74)
        (*
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           t_0))
         (sin (* y.re (atan2 x.im x.re))))
        (*
         (exp (- (* y.im (atan2 x.im x.re))))
         (sin (+ (* (* -1.0 (log (/ 1.0 x.im))) 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 = log((-1.0 * x_46_im));
	double t_3 = log(fabs(-x_46_re));
	double tmp;
	if (x_46_im <= -6.2e-137) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_im <= 1.8e-54) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 9e+74) {
		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 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((((-1.0 * log((1.0 / x_46_im))) * 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) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = atan2(x_46im, x_46re) * y_46re
    t_2 = log(((-1.0d0) * x_46im))
    t_3 = log(abs(-x_46re))
    if (x_46im <= (-6.2d-137)) then
        tmp = exp(((t_2 * y_46re) - t_0)) * sin(((t_2 * y_46im) + t_1))
    else if (x_46im <= 1.8d-54) then
        tmp = exp(((t_3 * y_46re) - t_0)) * sin(((t_3 * y_46im) + t_1))
    else if (x_46im <= 9d+74) 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 = exp(-(y_46im * atan2(x_46im, x_46re))) * sin(((((-1.0d0) * log((1.0d0 / x_46im))) * 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 = Math.log((-1.0 * x_46_im));
	double t_3 = Math.log(Math.abs(-x_46_re));
	double tmp;
	if (x_46_im <= -6.2e-137) {
		tmp = Math.exp(((t_2 * y_46_re) - t_0)) * Math.sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_im <= 1.8e-54) {
		tmp = Math.exp(((t_3 * y_46_re) - t_0)) * Math.sin(((t_3 * y_46_im) + t_1));
	} else if (x_46_im <= 9e+74) {
		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 = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((((-1.0 * Math.log((1.0 / x_46_im))) * 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 = math.log((-1.0 * x_46_im))
	t_3 = math.log(math.fabs(-x_46_re))
	tmp = 0
	if x_46_im <= -6.2e-137:
		tmp = math.exp(((t_2 * y_46_re) - t_0)) * math.sin(((t_2 * y_46_im) + t_1))
	elif x_46_im <= 1.8e-54:
		tmp = math.exp(((t_3 * y_46_re) - t_0)) * math.sin(((t_3 * y_46_im) + t_1))
	elif x_46_im <= 9e+74:
		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 = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((((-1.0 * math.log((1.0 / x_46_im))) * 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 = log(Float64(-1.0 * x_46_im))
	t_3 = log(abs(Float64(-x_46_re)))
	tmp = 0.0
	if (x_46_im <= -6.2e-137)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + t_1)));
	elseif (x_46_im <= 1.8e-54)
		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 <= 9e+74)
		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 = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(Float64(Float64(-1.0 * log(Float64(1.0 / x_46_im))) * 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 = log((-1.0 * x_46_im));
	t_3 = log(abs(-x_46_re));
	tmp = 0.0;
	if (x_46_im <= -6.2e-137)
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	elseif (x_46_im <= 1.8e-54)
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	elseif (x_46_im <= 9e+74)
		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 = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((((-1.0 * log((1.0 / x_46_im))) * 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[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -6.2e-137], 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, 1.8e-54], 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, 9e+74], 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], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(N[(N[(-1.0 * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -6.1999999999999996e-137

   1. Initial program 40.2%

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

   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 17.7%

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

   4. Applied rewrites 17.7%

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

   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 31.3%

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

   7. Applied rewrites 31.3%

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

   if -6.1999999999999996e-137 < x.im < 1.7999999999999999e-54

   1. Initial program 40.2%

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

   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 18.8%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.2%

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

   7. Applied rewrites 34.2%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.2%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.2%

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

   9. Applied rewrites 34.2%

      \[\leadsto e^{\log \color{blue}{\left(\left|-x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 1.7999999999999999e-54 < x.im < 8.9999999999999999e74

   1. Initial program 40.2%

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

   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 53.5%

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

   4. Applied rewrites 53.5%

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

   if 8.9999999999999999e74 < x.im

   1. Initial program 40.2%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 26.9%

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

   4. Applied rewrites 26.9%

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

   5. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-log.f64 N/A

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

      3. lower-/.f64 22.9%

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

   7. Applied rewrites 22.9%

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

Alternative 3: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\left|-x.re\right|\right)\\ \mathbf{if}\;y.re \leq -980000000000:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin t\_1\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+149}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - t\_1\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.im))
       (t_1 (* y.re (atan2 x.im x.re)))
       (t_2 (log (fabs (- x.re)))))
  (if (<= y.re -980000000000.0)
    (*
     (exp
      (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
     (sin t_1))
    (if (<= y.re 7.8e+149)
      (*
       (exp (- (* t_2 y.re) t_0))
       (sin (+ (* t_2 y.im) (* (atan2 x.im x.re) y.re))))
      (*
       (sin (- PI t_1))
       (pow (sqrt (+ (pow x.im 2.0) (pow x.re 2.0))) 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(fabs(-x_46_re));
	double tmp;
	if (y_46_re <= -980000000000.0) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(t_1);
	} else if (y_46_re <= 7.8e+149) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = 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_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.log(Math.abs(-x_46_re));
	double tmp;
	if (y_46_re <= -980000000000.0) {
		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_1);
	} else if (y_46_re <= 7.8e+149) {
		tmp = Math.exp(((t_2 * y_46_re) - t_0)) * Math.sin(((t_2 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = 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_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.log(math.fabs(-x_46_re))
	tmp = 0
	if y_46_re <= -980000000000.0:
		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_1)
	elif y_46_re <= 7.8e+149:
		tmp = math.exp(((t_2 * y_46_re) - t_0)) * math.sin(((t_2 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))
	else:
		tmp = 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_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(abs(Float64(-x_46_re)))
	tmp = 0.0
	if (y_46_re <= -980000000000.0)
		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_1));
	elseif (y_46_re <= 7.8e+149)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	else
		tmp = 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_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = log(abs(-x_46_re));
	tmp = 0.0;
	if (y_46_re <= -980000000000.0)
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(t_1);
	elseif (y_46_re <= 7.8e+149)
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	else
		tmp = 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$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -980000000000.0], 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$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+149], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2.0], $MachinePrecision] + N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.8e11

   1. Initial program 40.2%

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

   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 53.5%

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

   4. Applied rewrites 53.5%

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

   if -9.8e11 < y.re < 7.7999999999999998e149

   1. Initial program 40.2%

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

   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 18.8%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.2%

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

   7. Applied rewrites 34.2%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.2%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.2%

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

   9. Applied rewrites 34.2%

      \[\leadsto e^{\log \color{blue}{\left(\left|-x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-1 \cdot x.re\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 7.7999999999999998e149 < y.re

   1. Initial program 40.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 28.8%

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

   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 38.6%

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

   6. Applied rewrites 38.6%

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

Alternative 4: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -330000000000:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin t\_0\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-11}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\left|-x.re\right|\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - t\_0\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re))))
  (if (<= y.re -330000000000.0)
    (*
     (exp
      (-
       (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
       (* (atan2 x.im x.re) y.im)))
     (sin t_0))
    (if (<= y.re 2.2e-11)
      (*
       (exp (- (* y.im (atan2 x.im x.re))))
       (sin
        (+
         (* (log (fabs (- x.re))) y.im)
         (* (atan2 x.im x.re) y.re))))
      (*
       (sin (- PI t_0))
       (pow (sqrt (+ (pow x.im 2.0) (pow x.re 2.0))) 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 tmp;
	if (y_46_re <= -330000000000.0) {
		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))) * sin(t_0);
	} else if (y_46_re <= 2.2e-11) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(((log(fabs(-x_46_re)) * y_46_im) + (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 tmp;
	if (y_46_re <= -330000000000.0) {
		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))) * Math.sin(t_0);
	} else if (y_46_re <= 2.2e-11) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_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.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)
	tmp = 0
	if y_46_re <= -330000000000.0:
		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))) * math.sin(t_0)
	elif y_46_re <= 2.2e-11:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_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.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))
	tmp = 0.0
	if (y_46_re <= -330000000000.0)
		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))) * sin(t_0));
	elseif (y_46_re <= 2.2e-11)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_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(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);
	tmp = 0.0;
	if (y_46_re <= -330000000000.0)
		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))) * sin(t_0);
	elseif (y_46_re <= 2.2e-11)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(((log(abs(-x_46_re)) * y_46_im) + (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]}, If[LessEqual[y$46$re, -330000000000.0], 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] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.2e-11], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $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[Sin[N[(Pi - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2.0], $MachinePrecision] + N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.3e11

   1. Initial program 40.2%

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

   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 53.5%

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

   4. Applied rewrites 53.5%

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

   if -3.3e11 < y.re < 2.2000000000000002e-11

   1. Initial program 40.2%

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

   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 18.8%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.2%

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

   7. Applied rewrites 34.2%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.2%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.2%

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

   9. Applied rewrites 34.2%

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

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

       1. lower-exp.f64 N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 48.0%

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

   14. Applied rewrites 48.0%

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

   if 2.2000000000000002e-11 < y.re

   1. Initial program 40.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 28.8%

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

   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 38.6%

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

   6. Applied rewrites 38.6%

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

Alternative 5: 68.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{if}\;y.re \leq -360000000000:\\ \;\;\;\;{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot \sin t\_0\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-11}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\left|-x.re\right|\right) \cdot y.im + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* (atan2 x.im x.re) y.re)))
  (if (<= y.re -360000000000.0)
    (* (pow (+ (* x.im x.im) (* x.re x.re)) (* 0.5 y.re)) (sin t_0))
    (if (<= y.re 2.2e-11)
      (*
       (exp (- (* y.im (atan2 x.im x.re))))
       (sin (+ (* (log (fabs (- x.re))) y.im) t_0)))
      (*
       (sin (- PI (* y.re (atan2 x.im x.re))))
       (pow (sqrt (+ (pow x.im 2.0) (pow x.re 2.0))) 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 tmp;
	if (y_46_re <= -360000000000.0) {
		tmp = pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * sin(t_0);
	} else if (y_46_re <= 2.2e-11) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(((log(fabs(-x_46_re)) * y_46_im) + t_0));
	} else {
		tmp = sin((((double) M_PI) - (y_46_re * atan2(x_46_im, x_46_re)))) * 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 tmp;
	if (y_46_re <= -360000000000.0) {
		tmp = Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * Math.sin(t_0);
	} else if (y_46_re <= 2.2e-11) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin(((Math.log(Math.abs(-x_46_re)) * y_46_im) + t_0));
	} else {
		tmp = Math.sin((Math.PI - (y_46_re * Math.atan2(x_46_im, x_46_re)))) * 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
	tmp = 0
	if y_46_re <= -360000000000.0:
		tmp = math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * math.sin(t_0)
	elif y_46_re <= 2.2e-11:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin(((math.log(math.fabs(-x_46_re)) * y_46_im) + t_0))
	else:
		tmp = math.sin((math.pi - (y_46_re * math.atan2(x_46_im, x_46_re)))) * 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)
	tmp = 0.0
	if (y_46_re <= -360000000000.0)
		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_0));
	elseif (y_46_re <= 2.2e-11)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(Float64(log(abs(Float64(-x_46_re))) * y_46_im) + t_0)));
	else
		tmp = Float64(sin(Float64(pi - Float64(y_46_re * atan(x_46_im, x_46_re)))) * (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;
	tmp = 0.0;
	if (y_46_re <= -360000000000.0)
		tmp = (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (0.5 * y_46_re)) * sin(t_0);
	elseif (y_46_re <= 2.2e-11)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(((log(abs(-x_46_re)) * y_46_im) + t_0));
	else
		tmp = sin((pi - (y_46_re * atan2(x_46_im, x_46_re)))) * (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]}, If[LessEqual[y$46$re, -360000000000.0], N[(N[Power[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.2e-11], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(N[(N[Log[N[Abs[(-x$46$re)], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi - N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2.0], $MachinePrecision] + N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.6e11

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 43.8%

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

   6. Applied rewrites 43.8%

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

   if -3.6e11 < y.re < 2.2000000000000002e-11

   1. Initial program 40.2%

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

   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 18.8%

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

   4. Applied rewrites 18.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.2%

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

   7. Applied rewrites 34.2%

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

      1. rem-exp-log N/A

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

      2. lift-log.f64 N/A

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

      3. exp-fabs N/A

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

      4. lift-log.f64 N/A

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

      5. rem-exp-log N/A

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

      6. lower-fabs.f64 34.2%

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

      7. lift-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 34.2%

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

   9. Applied rewrites 34.2%

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

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

       1. lower-exp.f64 N/A

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

       2. lower-neg.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 48.0%

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

   14. Applied rewrites 48.0%

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

   if 2.2000000000000002e-11 < y.re

   1. Initial program 40.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 28.8%

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

   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 38.6%

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

   6. Applied rewrites 38.6%

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

Alternative 6: 59.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -4.9 \cdot 10^{+19}:\\ \;\;\;\;{\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-11}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin t\_0\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\pi - t\_0\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re))))
  (if (<= y.re -4.9e+19)
    (*
     (pow (+ (* x.im x.im) (* x.re x.re)) (* 0.5 y.re))
     (sin (* (atan2 x.im x.re) y.re)))
    (if (<= y.re 2.2e-11)
      (* (exp (- (* y.im (atan2 x.im x.re)))) (sin t_0))
      (*
       (sin (- PI t_0))
       (pow (sqrt (+ (pow x.im 2.0) (pow x.re 2.0))) 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 tmp;
	if (y_46_re <= -4.9e+19) {
		tmp = 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));
	} else if (y_46_re <= 2.2e-11) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(t_0);
	} 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 tmp;
	if (y_46_re <= -4.9e+19) {
		tmp = 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));
	} else if (y_46_re <= 2.2e-11) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin(t_0);
	} 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)
	tmp = 0
	if y_46_re <= -4.9e+19:
		tmp = 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))
	elif y_46_re <= 2.2e-11:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin(t_0)
	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))
	tmp = 0.0
	if (y_46_re <= -4.9e+19)
		tmp = 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)));
	elseif (y_46_re <= 2.2e-11)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(t_0));
	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);
	tmp = 0.0;
	if (y_46_re <= -4.9e+19)
		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));
	elseif (y_46_re <= 2.2e-11)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin(t_0);
	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]}, If[LessEqual[y$46$re, -4.9e+19], N[(N[Power[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(0.5 * 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, 2.2e-11], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(Pi - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[Power[x$46$im, 2.0], $MachinePrecision] + N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.9e19

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 43.8%

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

   6. Applied rewrites 43.8%

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

   if -4.9e19 < y.re < 2.2000000000000002e-11

   1. Initial program 40.2%

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

   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 53.5%

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

   4. Applied rewrites 53.5%

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

   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 2.2000000000000002e-11 < y.re

   1. Initial program 40.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 28.8%

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

   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 38.6%

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

   6. Applied rewrites 38.6%

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

Alternative 7: 58.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -4.9 \cdot 10^{+19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{-17}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0
        (*
         (pow (+ (* x.im x.im) (* x.re x.re)) (* 0.5 y.re))
         (sin (* (atan2 x.im x.re) y.re)))))
  (if (<= y.re -4.9e+19)
    t_0
    (if (<= y.re 8.2e-17)
      (*
       (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 <= -4.9e+19) {
		tmp = t_0;
	} else if (y_46_re <= 8.2e-17) {
		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 <= (-4.9d+19)) then
        tmp = t_0
    else if (y_46re <= 8.2d-17) 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 <= -4.9e+19) {
		tmp = t_0;
	} else if (y_46_re <= 8.2e-17) {
		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 <= -4.9e+19:
		tmp = t_0
	elif y_46_re <= 8.2e-17:
		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 <= -4.9e+19)
		tmp = t_0;
	elseif (y_46_re <= 8.2e-17)
		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 <= -4.9e+19)
		tmp = t_0;
	elseif (y_46_re <= 8.2e-17)
		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[(0.5 * 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, -4.9e+19], t$95$0, If[LessEqual[y$46$re, 8.2e-17], 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 < -4.9e19 or 8.2000000000000001e-17 < y.re

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 43.8%

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

   6. Applied rewrites 43.8%

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

   if -4.9e19 < y.re < 8.2000000000000001e-17

   1. Initial program 40.2%

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

   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 53.5%

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

   4. Applied rewrites 53.5%

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

   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 8: 46.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_1 := x.re \cdot x.re + x.im \cdot x.im\\ t_2 := {\left(\sqrt{t\_1 \cdot t\_1}\right)}^{\left(0.5 \cdot y.re\right)} \cdot t\_0\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{-103}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-61}:\\ \;\;\;\;t\_0 \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* (atan2 x.im x.re) y.re)))
       (t_1 (+ (* x.re x.re) (* x.im x.im)))
       (t_2 (* (pow (sqrt (* t_1 t_1)) (* 0.5 y.re)) t_0)))
  (if (<= y.im -2e-103)
    t_2
    (if (<= y.im 2.6e-61) (* t_0 (pow (fabs x.im) y.re)) t_2))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_2 = pow(sqrt((t_1 * t_1)), (0.5 * y_46_re)) * t_0;
	double tmp;
	if (y_46_im <= -2e-103) {
		tmp = t_2;
	} else if (y_46_im <= 2.6e-61) {
		tmp = t_0 * pow(fabs(x_46_im), y_46_re);
	} 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 = sin((atan2(x_46im, x_46re) * y_46re))
    t_1 = (x_46re * x_46re) + (x_46im * x_46im)
    t_2 = (sqrt((t_1 * t_1)) ** (0.5d0 * y_46re)) * t_0
    if (y_46im <= (-2d-103)) then
        tmp = t_2
    else if (y_46im <= 2.6d-61) then
        tmp = t_0 * (abs(x_46im) ** y_46re)
    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.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	double t_2 = Math.pow(Math.sqrt((t_1 * t_1)), (0.5 * y_46_re)) * t_0;
	double tmp;
	if (y_46_im <= -2e-103) {
		tmp = t_2;
	} else if (y_46_im <= 2.6e-61) {
		tmp = t_0 * Math.pow(Math.abs(x_46_im), y_46_re);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im)
	t_2 = math.pow(math.sqrt((t_1 * t_1)), (0.5 * y_46_re)) * t_0
	tmp = 0
	if y_46_im <= -2e-103:
		tmp = t_2
	elif y_46_im <= 2.6e-61:
		tmp = t_0 * math.pow(math.fabs(x_46_im), y_46_re)
	else:
		tmp = t_2
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	t_1 = Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))
	t_2 = Float64((sqrt(Float64(t_1 * t_1)) ^ Float64(0.5 * y_46_re)) * t_0)
	tmp = 0.0
	if (y_46_im <= -2e-103)
		tmp = t_2;
	elseif (y_46_im <= 2.6e-61)
		tmp = Float64(t_0 * (abs(x_46_im) ^ y_46_re));
	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 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	t_1 = (x_46_re * x_46_re) + (x_46_im * x_46_im);
	t_2 = (sqrt((t_1 * t_1)) ^ (0.5 * y_46_re)) * t_0;
	tmp = 0.0;
	if (y_46_im <= -2e-103)
		tmp = t_2;
	elseif (y_46_im <= 2.6e-61)
		tmp = t_0 * (abs(x_46_im) ^ y_46_re);
	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[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sqrt[N[(t$95$1 * t$95$1), $MachinePrecision]], $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$im, -2e-103], t$95$2, If[LessEqual[y$46$im, 2.6e-61], N[(t$95$0 * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.9999999999999999e-103 or 2.6000000000000001e-61 < y.im

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 43.8%

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

   6. Applied rewrites 43.8%

      \[\leadsto {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot \color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-*.f64 43.4%

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-+.f64 43.4%

         \[\leadsto {\left(\sqrt{\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im + x.re \cdot x.re\right)}\right)}^{\left(0.5 \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lift-+.f64 43.4%

          \[\leadsto {\left(\sqrt{\left(x.re \cdot x.re + x.im \cdot x.im\right) \cdot \left(x.re \cdot x.re + x.im \cdot x.im\right)}\right)}^{\left(0.5 \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Applied rewrites 43.4%

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

   if -1.9999999999999999e-103 < y.im < 2.6000000000000001e-61

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 36.9%

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

   10. Applied rewrites 36.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 36.9%

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

       4. lift-sqrt.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. pow2 N/A

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

       7. rem-sqrt-square-rev N/A

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

       8. lower-fabs.f64 37.5%

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

   12. Applied rewrites 37.5%

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

Alternative 9: 45.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_1 := {\left(\sqrt{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)} + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot t\_0\\ \mathbf{if}\;y.im \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-61}:\\ \;\;\;\;t\_0 \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* (atan2 x.im x.re) y.re)))
       (t_1
        (*
         (pow
          (+ (sqrt (* (* x.im x.im) (* x.im x.im))) (* x.re x.re))
          (* 0.5 y.re))
         t_0)))
  (if (<= y.im -1.02e-103)
    t_1
    (if (<= y.im 2.6e-61) (* t_0 (pow (fabs x.im) y.re)) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = pow((sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	double tmp;
	if (y_46_im <= -1.02e-103) {
		tmp = t_1;
	} else if (y_46_im <= 2.6e-61) {
		tmp = t_0 * pow(fabs(x_46_im), y_46_re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    t_1 = ((sqrt(((x_46im * x_46im) * (x_46im * x_46im))) + (x_46re * x_46re)) ** (0.5d0 * y_46re)) * t_0
    if (y_46im <= (-1.02d-103)) then
        tmp = t_1
    else if (y_46im <= 2.6d-61) then
        tmp = t_0 * (abs(x_46im) ** y_46re)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = Math.pow((Math.sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	double tmp;
	if (y_46_im <= -1.02e-103) {
		tmp = t_1;
	} else if (y_46_im <= 2.6e-61) {
		tmp = t_0 * Math.pow(Math.abs(x_46_im), y_46_re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	t_1 = math.pow((math.sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0
	tmp = 0
	if y_46_im <= -1.02e-103:
		tmp = t_1
	elif y_46_im <= 2.6e-61:
		tmp = t_0 * math.pow(math.fabs(x_46_im), y_46_re)
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	t_1 = Float64((Float64(sqrt(Float64(Float64(x_46_im * x_46_im) * Float64(x_46_im * x_46_im))) + Float64(x_46_re * x_46_re)) ^ Float64(0.5 * y_46_re)) * t_0)
	tmp = 0.0
	if (y_46_im <= -1.02e-103)
		tmp = t_1;
	elseif (y_46_im <= 2.6e-61)
		tmp = Float64(t_0 * (abs(x_46_im) ^ y_46_re));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	t_1 = ((sqrt(((x_46_im * x_46_im) * (x_46_im * x_46_im))) + (x_46_re * x_46_re)) ^ (0.5 * y_46_re)) * t_0;
	tmp = 0.0;
	if (y_46_im <= -1.02e-103)
		tmp = t_1;
	elseif (y_46_im <= 2.6e-61)
		tmp = t_0 * (abs(x_46_im) ^ y_46_re);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] * N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$im, -1.02e-103], t$95$1, If[LessEqual[y$46$im, 2.6e-61], N[(t$95$0 * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.02e-103 or 2.6000000000000001e-61 < y.im

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 43.8%

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

   6. Applied rewrites 43.8%

      \[\leadsto {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot \color{blue}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)} \]
   7. Step-by-step derivation

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lower-*.f32 N/A

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

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

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

      5. lower-sqrt.f64 N/A

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

      6. lower-unsound-*.f64 43.3%

         \[\leadsto {\left(\sqrt{\left(x.im \cdot x.im\right) \cdot \left(x.im \cdot x.im\right)} + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   8. Applied rewrites 43.3%

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

   if -1.02e-103 < y.im < 2.6000000000000001e-61

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 36.9%

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

   10. Applied rewrites 36.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 36.9%

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

       4. lift-sqrt.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. pow2 N/A

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

       7. rem-sqrt-square-rev N/A

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

       8. lower-fabs.f64 37.5%

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

   12. Applied rewrites 37.5%

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

Alternative 10: 43.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_1 := {\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\left(0.5 \cdot y.re\right)} \cdot t\_0\\ \mathbf{if}\;y.re \leq -6.6 \cdot 10^{+26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-191}:\\ \;\;\;\;t\_0 \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* (atan2 x.im x.re) y.re)))
       (t_1
        (* (pow (+ (* x.im x.im) (* x.re x.re)) (* 0.5 y.re)) t_0)))
  (if (<= y.re -6.6e+26)
    t_1
    (if (<= y.re 1.05e-191) (* t_0 (pow (fabs x.im) y.re)) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	double tmp;
	if (y_46_re <= -6.6e+26) {
		tmp = t_1;
	} else if (y_46_re <= 1.05e-191) {
		tmp = t_0 * pow(fabs(x_46_im), y_46_re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    t_1 = (((x_46im * x_46im) + (x_46re * x_46re)) ** (0.5d0 * y_46re)) * t_0
    if (y_46re <= (-6.6d+26)) then
        tmp = t_1
    else if (y_46re <= 1.05d-191) then
        tmp = t_0 * (abs(x_46im) ** y_46re)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = Math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0;
	double tmp;
	if (y_46_re <= -6.6e+26) {
		tmp = t_1;
	} else if (y_46_re <= 1.05e-191) {
		tmp = t_0 * Math.pow(Math.abs(x_46_im), y_46_re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	t_1 = math.pow(((x_46_im * x_46_im) + (x_46_re * x_46_re)), (0.5 * y_46_re)) * t_0
	tmp = 0
	if y_46_re <= -6.6e+26:
		tmp = t_1
	elif y_46_re <= 1.05e-191:
		tmp = t_0 * math.pow(math.fabs(x_46_im), y_46_re)
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	t_1 = Float64((Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)) ^ Float64(0.5 * y_46_re)) * t_0)
	tmp = 0.0
	if (y_46_re <= -6.6e+26)
		tmp = t_1;
	elseif (y_46_re <= 1.05e-191)
		tmp = Float64(t_0 * (abs(x_46_im) ^ y_46_re));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	t_1 = (((x_46_im * x_46_im) + (x_46_re * x_46_re)) ^ (0.5 * y_46_re)) * t_0;
	tmp = 0.0;
	if (y_46_re <= -6.6e+26)
		tmp = t_1;
	elseif (y_46_re <= 1.05e-191)
		tmp = t_0 * (abs(x_46_im) ^ y_46_re);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -6.6e+26], t$95$1, If[LessEqual[y$46$re, 1.05e-191], N[(t$95$0 * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.5999999999999999e26 or 1.0499999999999999e-191 < y.re

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \]
   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 43.8%

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

   6. Applied rewrites 43.8%

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

   if -6.5999999999999999e26 < y.re < 1.0499999999999999e-191

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 36.9%

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

   10. Applied rewrites 36.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 36.9%

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

       4. lift-sqrt.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. pow2 N/A

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

       7. rem-sqrt-square-rev N/A

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

       8. lower-fabs.f64 37.5%

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

   12. Applied rewrites 37.5%

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

Alternative 11: 39.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y.im \leq 2.2 \cdot 10^{+82}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (if (<= y.im 2.2e+82)
  (* (sin (* (atan2 x.im x.re) y.re)) (pow (fabs x.im) y.re))
  (*
   (sin (* y.re (atan2 x.im x.re)))
   (pow (* x.im x.im) (* 0.5 y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 2.2e+82) {
		tmp = sin((atan2(x_46_im, x_46_re) * y_46_re)) * pow(fabs(x_46_im), y_46_re);
	} else {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow((x_46_im * x_46_im), (0.5 * y_46_re));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= 2.2d+82) then
        tmp = sin((atan2(x_46im, x_46re) * y_46re)) * (abs(x_46im) ** y_46re)
    else
        tmp = sin((y_46re * atan2(x_46im, x_46re))) * ((x_46im * x_46im) ** (0.5d0 * y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 2.2e+82) {
		tmp = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.pow(Math.abs(x_46_im), y_46_re);
	} else {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow((x_46_im * x_46_im), (0.5 * y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= 2.2e+82:
		tmp = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.pow(math.fabs(x_46_im), y_46_re)
	else:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow((x_46_im * x_46_im), (0.5 * y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 2.2e+82)
		tmp = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * (abs(x_46_im) ^ y_46_re));
	else
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (Float64(x_46_im * x_46_im) ^ Float64(0.5 * y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= 2.2e+82)
		tmp = sin((atan2(x_46_im, x_46_re) * y_46_re)) * (abs(x_46_im) ^ y_46_re);
	else
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * ((x_46_im * x_46_im) ^ (0.5 * y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 2.2e+82], N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[(x$46$im * x$46$im), $MachinePrecision], N[(0.5 * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 2.2000000000000001e82

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 36.9%

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

   10. Applied rewrites 36.9%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 36.9%

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

       4. lift-sqrt.f64 N/A

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

       5. lift-pow.f64 N/A

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

       6. pow2 N/A

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

       7. rem-sqrt-square-rev N/A

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

       8. lower-fabs.f64 37.5%

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

   12. Applied rewrites 37.5%

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

   if 2.2000000000000001e82 < y.im

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 36.9%

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

   10. Applied rewrites 36.9%

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

       1. lift-pow.f64 N/A

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

       2. lift-sqrt.f64 N/A

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

       3. lift-pow.f64 N/A

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

       4. pow2 N/A

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

       5. lift-*.f64 N/A

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

       6. sqrt-pow2 N/A

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

       7. lower-pow.f64 N/A

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

       8. mult-flip N/A

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

       9. metadata-eval N/A

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

       10. *-commutative N/A

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

       11. lift-*.f64 36.9%

           \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(x.im \cdot x.im\right)}^{\left(0.5 \cdot y.re\right)} \]

   12. Applied rewrites 36.9%

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

Alternative 12: 37.5% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 40.2%

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

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 43.8%

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

4. Applied rewrites 43.8%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 13.9%

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

8. Taylor expanded in x.re around 0

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

   1. lower-pow.f64 N/A

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

   2. lower-sqrt.f64 N/A

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

   3. lower-pow.f64 36.9%

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

10. Applied rewrites 36.9%

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

    1. lift-*.f64 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f64 36.9%

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

    4. lift-sqrt.f64 N/A

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

    5. lift-pow.f64 N/A

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

    6. pow2 N/A

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

    7. rem-sqrt-square-rev N/A

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

    8. lower-fabs.f64 37.5%

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

12. Applied rewrites 37.5%

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

Alternative 13: 36.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot {x.im}^{y.re}\\ \mathbf{if}\;y.re \leq -1150000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 60000000000000:\\ \;\;\;\;t\_0 \cdot 1\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
       (t_1 (* t_0 (pow x.im y.re))))
  (if (<= y.re -1150000000.0)
    t_1
    (if (<= y.re 60000000000000.0) (* t_0 1.0) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * pow(x_46_im, y_46_re);
	double tmp;
	if (y_46_re <= -1150000000.0) {
		tmp = t_1;
	} else if (y_46_re <= 60000000000000.0) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = t_0 * (x_46im ** y_46re)
    if (y_46re <= (-1150000000.0d0)) then
        tmp = t_1
    else if (y_46re <= 60000000000000.0d0) then
        tmp = t_0 * 1.0d0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * Math.pow(x_46_im, y_46_re);
	double tmp;
	if (y_46_re <= -1150000000.0) {
		tmp = t_1;
	} else if (y_46_re <= 60000000000000.0) {
		tmp = t_0 * 1.0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = t_0 * math.pow(x_46_im, y_46_re)
	tmp = 0
	if y_46_re <= -1150000000.0:
		tmp = t_1
	elif y_46_re <= 60000000000000.0:
		tmp = t_0 * 1.0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_0 * (x_46_im ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1150000000.0)
		tmp = t_1;
	elseif (y_46_re <= 60000000000000.0)
		tmp = Float64(t_0 * 1.0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = t_0 * (x_46_im ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -1150000000.0)
		tmp = t_1;
	elseif (y_46_re <= 60000000000000.0)
		tmp = t_0 * 1.0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1150000000.0], t$95$1, If[LessEqual[y$46$re, 60000000000000.0], N[(t$95$0 * 1.0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.15e9 or 6e13 < y.re

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

   8. Taylor expanded in x.re around 0

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

      1. lower-pow.f64 N/A

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

      2. lower-sqrt.f64 N/A

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

      3. lower-pow.f64 36.9%

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

   10. Applied rewrites 36.9%

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

   11. Taylor expanded in x.im around 0

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

       1. lower-pow.f64 30.8%

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

   13. Applied rewrites 30.8%

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

   if -1.15e9 < y.re < 6e13

   1. Initial program 40.2%

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

   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 43.8%

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

   4. Applied rewrites 43.8%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 13.9%

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

Alternative 14: 13.9% 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))) 1.0))

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] * 1.0), $MachinePrecision]

Derivation

1. Initial program 40.2%

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

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 43.8%

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

4. Applied rewrites 43.8%

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

5. Taylor expanded in y.re around 0

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

7. Applied rewrites 13.9%

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

Reproduce

herbie shell --seed 2025258 
(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)))))