powComplex, imaginary part

Percentage Accurate: 40.8% → 80.3%

Time: 2.2min

Alternatives: 12

Speedup: N/A×

• 34.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (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: 40.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (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: 80.3% accurate, N/A× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (*
    (exp (fma t_0 y.re (* (- (atan2 x.im x.re)) y.im)))
    (sin (fma 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(hypot(x_46_re, x_46_im));
	return exp(fma(t_0, y_46_re, (-atan2(x_46_im, x_46_re) * y_46_im))) * sin(fma(t_0, y_46_im, (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(hypot(x_46_re, x_46_im))
	return Float64(exp(fma(t_0, y_46_re, Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_im))) * sin(fma(t_0, y_46_im, Float64(atan(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[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(t$95$0 * y$46$re + N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 38.0%

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

4. Applied rewrites 83.9%

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

Alternative 2: 78.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}\\ \mathbf{if}\;x.im \leq -5.6 \cdot 10^{+37}:\\ \;\;\;\;t\_0 \cdot \sin \left(\mathsf{fma}\left(\log \left(-x.im\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (fma (log (hypot x.re x.im)) y.re (* (- (atan2 x.im x.re)) y.im)))))
   (if (<= x.im -5.6e+37)
     (* t_0 (sin (fma (log (- x.im)) y.im (* (atan2 x.im x.re) y.re))))
     (* t_0 (fma y.im (log (hypot x.im x.re)) (* y.re (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, (-atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (x_46_im <= -5.6e+37) {
		tmp = t_0 * sin(fma(log(-x_46_im), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = t_0 * fma(y_46_im, log(hypot(x_46_im, x_46_re)), (y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_im)))
	tmp = 0.0
	if (x_46_im <= -5.6e+37)
		tmp = Float64(t_0 * sin(fma(log(Float64(-x_46_im)), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))));
	else
		tmp = Float64(t_0 * fma(y_46_im, log(hypot(x_46_im, x_46_re)), Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re + N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -5.6e+37], N[(t$95$0 * N[Sin[N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -5.5999999999999996e37

   1. Initial program 26.3%

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

   4. Applied rewrites 84.8%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if -5.5999999999999996e37 < x.im

   1. Initial program 41.1%

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

   4. Applied rewrites 83.7%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 39.8%

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

   8. Taylor expanded in y.im around 0

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

      1. lift-atan2.f64 N/A

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

      2. lift-*.f64 62.8

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

   10. Applied rewrites 62.8%

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

   11. Taylor expanded in y.im around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-log.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. lower-hypot.f64 N/A

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

       6. lift-atan2.f64 N/A

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

       7. lift-*.f64 81.2

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

   13. Applied rewrites 81.2%

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

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5.6 \cdot 10^{+37}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(-x.im\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)}\\ \mathbf{if}\;y.im \leq 3.2 \cdot 10^{+116}:\\ \;\;\;\;t\_1 \cdot \mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1
         (exp
          (fma (log (hypot x.re x.im)) y.re (* (- (atan2 x.im x.re)) y.im)))))
   (if (<= y.im 3.2e+116)
     (* t_1 (fma y.im (log (hypot x.im x.re)) t_0))
     (* t_1 t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, (-atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_im <= 3.2e+116) {
		tmp = t_1 * fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0);
	} else {
		tmp = t_1 * t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_im)))
	tmp = 0.0
	if (y_46_im <= 3.2e+116)
		tmp = Float64(t_1 * fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0));
	else
		tmp = Float64(t_1 * t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re + N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, 3.2e+116], N[(t$95$1 * N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 3.2e116

   1. Initial program 39.3%

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 38.5%

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

   8. Taylor expanded in y.im around 0

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

      1. lift-atan2.f64 N/A

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

      2. lift-*.f64 61.7

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

   10. Applied rewrites 61.7%

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

   11. Taylor expanded in y.im around 0

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

       1. lower-fma.f64 N/A

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

       2. lower-log.f64 N/A

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

       3. pow2 N/A

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

       4. pow2 N/A

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

       5. lower-hypot.f64 N/A

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

       6. lift-atan2.f64 N/A

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

       7. lift-*.f64 83.9

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

   13. Applied rewrites 83.9%

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

   if 3.2e116 < y.im

   1. Initial program 32.7%

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 30.6%

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

   8. Taylor expanded in y.im around 0

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

      1. lift-atan2.f64 N/A

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

      2. lift-*.f64 69.6

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

   10. Applied rewrites 69.6%

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

Alternative 4: 71.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \cos t\_1\\ t_3 := {t\_0}^{2}\\ t_4 := \sin t\_1\\ t_5 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_6 := \tan^{-1}_* \frac{x.im}{x.re} \cdot t\_5\\ t_7 := {\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot t\_5\\ t_8 := t\_4 \cdot t\_6\\ t_9 := t\_4 \cdot t\_5\\ \mathbf{if}\;y.im \leq -0.0035 \lor \neg \left(y.im \leq 9 \cdot 10^{-24}\right):\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, t\_8, \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, t\_2 \cdot \left(t\_0 \cdot t\_6\right), \mathsf{fma}\left(-0.5, t\_3 \cdot t\_9, \mathsf{fma}\left(0.5, t\_4 \cdot t\_7, y.im \cdot \mathsf{fma}\left(-0.16666666666666666, t\_2 \cdot \left({t\_0}^{3} \cdot t\_5\right), \mathsf{fma}\left(-0.16666666666666666, t\_4 \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot t\_5\right), \mathsf{fma}\left(0.5, t\_2 \cdot \left(t\_0 \cdot t\_7\right), 0.5 \cdot \left(t\_3 \cdot t\_8\right)\right)\right)\right)\right)\right)\right), t\_2 \cdot \left(t\_0 \cdot t\_5\right)\right)\right), t\_9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (cos t_1))
        (t_3 (pow t_0 2.0))
        (t_4 (sin t_1))
        (t_5 (pow (hypot x.im x.re) y.re))
        (t_6 (* (atan2 x.im x.re) t_5))
        (t_7 (* (pow (atan2 x.im x.re) 2.0) t_5))
        (t_8 (* t_4 t_6))
        (t_9 (* t_4 t_5)))
   (if (or (<= y.im -0.0035) (not (<= y.im 9e-24)))
     (*
      (exp (fma (log (hypot x.re x.im)) y.re (* (- (atan2 x.im x.re)) y.im)))
      t_1)
     (fma
      y.im
      (fma
       -1.0
       t_8
       (fma
        y.im
        (fma
         -1.0
         (* t_2 (* t_0 t_6))
         (fma
          -0.5
          (* t_3 t_9)
          (fma
           0.5
           (* t_4 t_7)
           (*
            y.im
            (fma
             -0.16666666666666666
             (* t_2 (* (pow t_0 3.0) t_5))
             (fma
              -0.16666666666666666
              (* t_4 (* (pow (atan2 x.im x.re) 3.0) t_5))
              (fma 0.5 (* t_2 (* t_0 t_7)) (* 0.5 (* t_3 t_8)))))))))
        (* t_2 (* t_0 t_5))))
      t_9))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_im, x_46_re));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = cos(t_1);
	double t_3 = pow(t_0, 2.0);
	double t_4 = sin(t_1);
	double t_5 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_6 = atan2(x_46_im, x_46_re) * t_5;
	double t_7 = pow(atan2(x_46_im, x_46_re), 2.0) * t_5;
	double t_8 = t_4 * t_6;
	double t_9 = t_4 * t_5;
	double tmp;
	if ((y_46_im <= -0.0035) || !(y_46_im <= 9e-24)) {
		tmp = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, (-atan2(x_46_im, x_46_re) * y_46_im))) * t_1;
	} else {
		tmp = fma(y_46_im, fma(-1.0, t_8, fma(y_46_im, fma(-1.0, (t_2 * (t_0 * t_6)), fma(-0.5, (t_3 * t_9), fma(0.5, (t_4 * t_7), (y_46_im * fma(-0.16666666666666666, (t_2 * (pow(t_0, 3.0) * t_5)), fma(-0.16666666666666666, (t_4 * (pow(atan2(x_46_im, x_46_re), 3.0) * t_5)), fma(0.5, (t_2 * (t_0 * t_7)), (0.5 * (t_3 * t_8))))))))), (t_2 * (t_0 * t_5)))), t_9);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = cos(t_1)
	t_3 = t_0 ^ 2.0
	t_4 = sin(t_1)
	t_5 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_6 = Float64(atan(x_46_im, x_46_re) * t_5)
	t_7 = Float64((atan(x_46_im, x_46_re) ^ 2.0) * t_5)
	t_8 = Float64(t_4 * t_6)
	t_9 = Float64(t_4 * t_5)
	tmp = 0.0
	if ((y_46_im <= -0.0035) || !(y_46_im <= 9e-24))
		tmp = Float64(exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_im))) * t_1);
	else
		tmp = fma(y_46_im, fma(-1.0, t_8, fma(y_46_im, fma(-1.0, Float64(t_2 * Float64(t_0 * t_6)), fma(-0.5, Float64(t_3 * t_9), fma(0.5, Float64(t_4 * t_7), Float64(y_46_im * fma(-0.16666666666666666, Float64(t_2 * Float64((t_0 ^ 3.0) * t_5)), fma(-0.16666666666666666, Float64(t_4 * Float64((atan(x_46_im, x_46_re) ^ 3.0) * t_5)), fma(0.5, Float64(t_2 * Float64(t_0 * t_7)), Float64(0.5 * Float64(t_3 * t_8))))))))), Float64(t_2 * Float64(t_0 * t_5)))), t_9);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$6 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision] * t$95$5), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$4 * t$95$6), $MachinePrecision]}, Block[{t$95$9 = N[(t$95$4 * t$95$5), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -0.0035], N[Not[LessEqual[y$46$im, 9e-24]], $MachinePrecision]], N[(N[Exp[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re + N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], N[(y$46$im * N[(-1.0 * t$95$8 + N[(y$46$im * N[(-1.0 * N[(t$95$2 * N[(t$95$0 * t$95$6), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(t$95$3 * t$95$9), $MachinePrecision] + N[(0.5 * N[(t$95$4 * t$95$7), $MachinePrecision] + N[(y$46$im * N[(-0.16666666666666666 * N[(t$95$2 * N[(N[Power[t$95$0, 3.0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$4 * N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$2 * N[(t$95$0 * t$95$7), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$3 * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(t$95$0 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$9), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -0.00350000000000000007 or 8.9999999999999995e-24 < y.im

   1. Initial program 32.5%

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

   4. Applied rewrites 74.8%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-*.f64 N/A

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

   7. Applied rewrites 31.8%

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

   8. Taylor expanded in y.im around 0

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

      1. lift-atan2.f64 N/A

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

      2. lift-*.f64 68.2

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

   10. Applied rewrites 68.2%

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

   if -0.00350000000000000007 < y.im < 8.9999999999999995e-24

   1. Initial program 44.1%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(y.im \cdot \left(-1 \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)\right) + \left(\frac{1}{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + y.im \cdot \left(\frac{-1}{6} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{-1}{6} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \frac{1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \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. Applied rewrites 82.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), \mathsf{fma}\left(-0.5, {\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), y.im \cdot \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), 0.5 \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -0.0035 \lor \neg \left(y.im \leq 9 \cdot 10^{-24}\right):\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.im\right)} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), \mathsf{fma}\left(-0.5, {\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), y.im \cdot \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), 0.5 \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;x.im \leq -1 \cdot 10^{+38}:\\ \;\;\;\;t\_1 \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 8.8 \cdot 10^{-39}:\\ \;\;\;\;t\_1 \cdot \left(\sin t\_0 + y.im \cdot \left(\cos t\_0 \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1
         (exp
          (-
           (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
           (* (atan2 x.im x.re) y.im)))))
   (if (<= x.im -1e+38)
     (* t_1 (sin (+ (* (log (- x.im)) y.im) (* (atan2 x.im x.re) y.re))))
     (if (<= x.im 8.8e-39)
       (* t_1 (+ (sin t_0) (* y.im (* (cos t_0) (log (hypot x.im x.re))))))
       (*
        (exp (- (* y.re (log x.im)) (* y.im (atan2 x.im x.re))))
        (sin (fma y.im (log x.im) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = 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)));
	double tmp;
	if (x_46_im <= -1e+38) {
		tmp = t_1 * sin(((log(-x_46_im) * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_im <= 8.8e-39) {
		tmp = t_1 * (sin(t_0) + (y_46_im * (cos(t_0) * log(hypot(x_46_im, x_46_re)))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = 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)))
	tmp = 0.0
	if (x_46_im <= -1e+38)
		tmp = Float64(t_1 * sin(Float64(Float64(log(Float64(-x_46_im)) * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_im <= 8.8e-39)
		tmp = Float64(t_1 * Float64(sin(t_0) + Float64(y_46_im * Float64(cos(t_0) * log(hypot(x_46_im, x_46_re))))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[x$46$im, -1e+38], N[(t$95$1 * N[Sin[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 8.8e-39], N[(t$95$1 * N[(N[Sin[t$95$0], $MachinePrecision] + N[(y$46$im * N[(N[Cos[t$95$0], $MachinePrecision] * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -9.99999999999999977e37

   1. Initial program 26.3%

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

   3. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 72.1

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

   5. Applied rewrites 72.1%

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

   if -9.99999999999999977e37 < x.im < 8.80000000000000003e-39

   1. Initial program 48.7%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 66.9

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

   5. Applied rewrites 66.9%

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

   if 8.80000000000000003e-39 < x.im

   1. Initial program 30.5%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 76.4

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

   5. Applied rewrites 76.4%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1 \cdot 10^{+38}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(-x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.im \leq 8.8 \cdot 10^{-39}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + y.im \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq 8.8 \cdot 10^{-39}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin t\_0 + y.im \cdot \left(\cos t\_0 \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= x.im 8.8e-39)
     (*
      (exp
       (-
        (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
        (* (atan2 x.im x.re) y.im)))
      (+ (sin t_0) (* y.im (* (cos t_0) (log (hypot x.im x.re))))))
     (*
      (exp (- (* y.re (log x.im)) (* y.im (atan2 x.im x.re))))
      (sin (fma y.im (log x.im) t_0))))))

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 (x_46_im <= 8.8e-39) {
		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) + (y_46_im * (cos(t_0) * log(hypot(x_46_im, x_46_re)))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_0));
	}
	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 (x_46_im <= 8.8e-39)
		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))) * Float64(sin(t_0) + Float64(y_46_im * Float64(cos(t_0) * log(hypot(x_46_im, x_46_re))))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, 8.8e-39], 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[(N[Sin[t$95$0], $MachinePrecision] + N[(y$46$im * N[(N[Cos[t$95$0], $MachinePrecision] * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 8.80000000000000003e-39

   1. Initial program 41.8%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-atan2.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. pow2 N/A

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

      12. pow2 N/A

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

      13. lower-hypot.f64 66.1

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

   5. Applied rewrites 66.1%

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

   if 8.80000000000000003e-39 < x.im

   1. Initial program 30.5%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 76.4

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

   5. Applied rewrites 76.4%

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

Alternative 7: 58.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot t\_0\\ t_2 := {\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot t\_0\\ t_3 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_4 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_5 := \cos t\_4\\ t_6 := \sin t\_4\\ t_7 := t\_6 \cdot t\_0\\ t_8 := t\_6 \cdot t\_1\\ t_9 := {t\_3}^{2}\\ \mathbf{if}\;x.im \leq 230000000:\\ \;\;\;\;\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, t\_8, \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, t\_5 \cdot \left(t\_3 \cdot t\_1\right), \mathsf{fma}\left(-0.5, t\_9 \cdot t\_7, \mathsf{fma}\left(0.5, t\_6 \cdot t\_2, y.im \cdot \mathsf{fma}\left(-0.16666666666666666, t\_5 \cdot \left({t\_3}^{3} \cdot t\_0\right), \mathsf{fma}\left(-0.16666666666666666, t\_6 \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot t\_0\right), \mathsf{fma}\left(0.5, t\_5 \cdot \left(t\_3 \cdot t\_2\right), 0.5 \cdot \left(t\_9 \cdot t\_8\right)\right)\right)\right)\right)\right)\right), t\_5 \cdot \left(t\_3 \cdot t\_0\right)\right)\right), t\_7\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_4\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (* (atan2 x.im x.re) t_0))
        (t_2 (* (pow (atan2 x.im x.re) 2.0) t_0))
        (t_3 (log (hypot x.im x.re)))
        (t_4 (* y.re (atan2 x.im x.re)))
        (t_5 (cos t_4))
        (t_6 (sin t_4))
        (t_7 (* t_6 t_0))
        (t_8 (* t_6 t_1))
        (t_9 (pow t_3 2.0)))
   (if (<= x.im 230000000.0)
     (fma
      y.im
      (fma
       -1.0
       t_8
       (fma
        y.im
        (fma
         -1.0
         (* t_5 (* t_3 t_1))
         (fma
          -0.5
          (* t_9 t_7)
          (fma
           0.5
           (* t_6 t_2)
           (*
            y.im
            (fma
             -0.16666666666666666
             (* t_5 (* (pow t_3 3.0) t_0))
             (fma
              -0.16666666666666666
              (* t_6 (* (pow (atan2 x.im x.re) 3.0) t_0))
              (fma 0.5 (* t_5 (* t_3 t_2)) (* 0.5 (* t_9 t_8)))))))))
        (* t_5 (* t_3 t_0))))
      t_7)
     (*
      (exp (- (* y.re (log x.im)) (* y.im (atan2 x.im x.re))))
      (sin (fma y.im (log x.im) t_4))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = atan2(x_46_im, x_46_re) * t_0;
	double t_2 = pow(atan2(x_46_im, x_46_re), 2.0) * t_0;
	double t_3 = log(hypot(x_46_im, x_46_re));
	double t_4 = y_46_re * atan2(x_46_im, x_46_re);
	double t_5 = cos(t_4);
	double t_6 = sin(t_4);
	double t_7 = t_6 * t_0;
	double t_8 = t_6 * t_1;
	double t_9 = pow(t_3, 2.0);
	double tmp;
	if (x_46_im <= 230000000.0) {
		tmp = fma(y_46_im, fma(-1.0, t_8, fma(y_46_im, fma(-1.0, (t_5 * (t_3 * t_1)), fma(-0.5, (t_9 * t_7), fma(0.5, (t_6 * t_2), (y_46_im * fma(-0.16666666666666666, (t_5 * (pow(t_3, 3.0) * t_0)), fma(-0.16666666666666666, (t_6 * (pow(atan2(x_46_im, x_46_re), 3.0) * t_0)), fma(0.5, (t_5 * (t_3 * t_2)), (0.5 * (t_9 * t_8))))))))), (t_5 * (t_3 * t_0)))), t_7);
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_4));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = Float64(atan(x_46_im, x_46_re) * t_0)
	t_2 = Float64((atan(x_46_im, x_46_re) ^ 2.0) * t_0)
	t_3 = log(hypot(x_46_im, x_46_re))
	t_4 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_5 = cos(t_4)
	t_6 = sin(t_4)
	t_7 = Float64(t_6 * t_0)
	t_8 = Float64(t_6 * t_1)
	t_9 = t_3 ^ 2.0
	tmp = 0.0
	if (x_46_im <= 230000000.0)
		tmp = fma(y_46_im, fma(-1.0, t_8, fma(y_46_im, fma(-1.0, Float64(t_5 * Float64(t_3 * t_1)), fma(-0.5, Float64(t_9 * t_7), fma(0.5, Float64(t_6 * t_2), Float64(y_46_im * fma(-0.16666666666666666, Float64(t_5 * Float64((t_3 ^ 3.0) * t_0)), fma(-0.16666666666666666, Float64(t_6 * Float64((atan(x_46_im, x_46_re) ^ 3.0) * t_0)), fma(0.5, Float64(t_5 * Float64(t_3 * t_2)), Float64(0.5 * Float64(t_9 * t_8))))))))), Float64(t_5 * Float64(t_3 * t_0)))), t_7);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_4)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$4], $MachinePrecision]}, Block[{t$95$6 = N[Sin[t$95$4], $MachinePrecision]}, Block[{t$95$7 = N[(t$95$6 * t$95$0), $MachinePrecision]}, Block[{t$95$8 = N[(t$95$6 * t$95$1), $MachinePrecision]}, Block[{t$95$9 = N[Power[t$95$3, 2.0], $MachinePrecision]}, If[LessEqual[x$46$im, 230000000.0], N[(y$46$im * N[(-1.0 * t$95$8 + N[(y$46$im * N[(-1.0 * N[(t$95$5 * N[(t$95$3 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(t$95$9 * t$95$7), $MachinePrecision] + N[(0.5 * N[(t$95$6 * t$95$2), $MachinePrecision] + N[(y$46$im * N[(-0.16666666666666666 * N[(t$95$5 * N[(N[Power[t$95$3, 3.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$6 * N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$5 * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$9 * t$95$8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$7), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.3e8

   1. Initial program 41.3%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(y.im \cdot \left(-1 \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)\right) + \left(\frac{1}{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + y.im \cdot \left(\frac{-1}{6} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{-1}{6} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \frac{1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \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. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), \mathsf{fma}\left(-0.5, {\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), y.im \cdot \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), 0.5 \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if 2.3e8 < x.im

   1. Initial program 30.2%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 81.1

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

   5. Applied rewrites 81.1%

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

Alternative 8: 57.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot t\_0\\ t_2 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := \cos t\_3\\ t_5 := \sin t\_3\\ t_6 := t\_5 \cdot t\_0\\ \mathbf{if}\;x.im \leq 230000000:\\ \;\;\;\;\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, t\_5 \cdot t\_1, \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, t\_4 \cdot \left(t\_2 \cdot t\_1\right), \mathsf{fma}\left(-0.5, {t\_2}^{2} \cdot t\_6, 0.5 \cdot \left(t\_5 \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot t\_0\right)\right)\right)\right), t\_4 \cdot \left(t\_2 \cdot t\_0\right)\right)\right), t\_6\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_3\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re))
        (t_1 (* (atan2 x.im x.re) t_0))
        (t_2 (log (hypot x.im x.re)))
        (t_3 (* y.re (atan2 x.im x.re)))
        (t_4 (cos t_3))
        (t_5 (sin t_3))
        (t_6 (* t_5 t_0)))
   (if (<= x.im 230000000.0)
     (fma
      y.im
      (fma
       -1.0
       (* t_5 t_1)
       (fma
        y.im
        (fma
         -1.0
         (* t_4 (* t_2 t_1))
         (fma
          -0.5
          (* (pow t_2 2.0) t_6)
          (* 0.5 (* t_5 (* (pow (atan2 x.im x.re) 2.0) t_0)))))
        (* t_4 (* t_2 t_0))))
      t_6)
     (*
      (exp (- (* y.re (log x.im)) (* y.im (atan2 x.im x.re))))
      (sin (fma y.im (log x.im) t_3))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double t_1 = atan2(x_46_im, x_46_re) * t_0;
	double t_2 = log(hypot(x_46_im, x_46_re));
	double t_3 = y_46_re * atan2(x_46_im, x_46_re);
	double t_4 = cos(t_3);
	double t_5 = sin(t_3);
	double t_6 = t_5 * t_0;
	double tmp;
	if (x_46_im <= 230000000.0) {
		tmp = fma(y_46_im, fma(-1.0, (t_5 * t_1), fma(y_46_im, fma(-1.0, (t_4 * (t_2 * t_1)), fma(-0.5, (pow(t_2, 2.0) * t_6), (0.5 * (t_5 * (pow(atan2(x_46_im, x_46_re), 2.0) * t_0))))), (t_4 * (t_2 * t_0)))), t_6);
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_3));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	t_1 = Float64(atan(x_46_im, x_46_re) * t_0)
	t_2 = log(hypot(x_46_im, x_46_re))
	t_3 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_4 = cos(t_3)
	t_5 = sin(t_3)
	t_6 = Float64(t_5 * t_0)
	tmp = 0.0
	if (x_46_im <= 230000000.0)
		tmp = fma(y_46_im, fma(-1.0, Float64(t_5 * t_1), fma(y_46_im, fma(-1.0, Float64(t_4 * Float64(t_2 * t_1)), fma(-0.5, Float64((t_2 ^ 2.0) * t_6), Float64(0.5 * Float64(t_5 * Float64((atan(x_46_im, x_46_re) ^ 2.0) * t_0))))), Float64(t_4 * Float64(t_2 * t_0)))), t_6);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_3)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * t$95$0), $MachinePrecision]}, If[LessEqual[x$46$im, 230000000.0], N[(y$46$im * N[(-1.0 * N[(t$95$5 * t$95$1), $MachinePrecision] + N[(y$46$im * N[(-1.0 * N[(t$95$4 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[Power[t$95$2, 2.0], $MachinePrecision] * t$95$6), $MachinePrecision] + N[(0.5 * N[(t$95$5 * N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[(t$95$2 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$6), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 2.3e8

   1. Initial program 41.3%

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

   3. Taylor expanded in y.im around 0

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

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), \mathsf{fma}\left(-0.5, {\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), 0.5 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   if 2.3e8 < x.im

   1. Initial program 30.2%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 81.1

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

   5. Applied rewrites 81.1%

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

Alternative 9: 38.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := \sqrt{{x.im}^{2} + {x.re}^{2}}\\ t_4 := {t\_3}^{y.re}\\ t_5 := t\_2 \cdot t\_4\\ t_6 := \tan^{-1}_* \frac{x.im}{x.re} \cdot t\_4\\ t_7 := t\_2 \cdot t\_6\\ t_8 := \log t\_3\\ t_9 := {\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot t\_4\\ t_10 := {t\_8}^{2}\\ \mathbf{if}\;x.im \leq -1.18 \cdot 10^{-306}:\\ \;\;\;\;{y.im}^{3} \cdot \mathsf{fma}\left(-1, \frac{t\_1 \cdot \left(t\_8 \cdot t\_6\right)}{y.im}, \mathsf{fma}\left(-1, \frac{t\_7}{{y.im}^{2}}, \mathsf{fma}\left(-0.5, \frac{t\_10 \cdot t\_5}{y.im}, \mathsf{fma}\left(-0.16666666666666666, t\_1 \cdot \left({t\_8}^{3} \cdot t\_4\right), \mathsf{fma}\left(-0.16666666666666666, t\_2 \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot t\_4\right), \mathsf{fma}\left(0.5, t\_1 \cdot \left(t\_8 \cdot t\_9\right), \mathsf{fma}\left(0.5, t\_10 \cdot t\_7, \mathsf{fma}\left(0.5, \frac{t\_2 \cdot t\_9}{y.im}, \frac{t\_1 \cdot \left(t\_8 \cdot t\_4\right)}{{y.im}^{2}} + \frac{t\_5}{{y.im}^{3}}\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3 (sqrt (+ (pow x.im 2.0) (pow x.re 2.0))))
        (t_4 (pow t_3 y.re))
        (t_5 (* t_2 t_4))
        (t_6 (* (atan2 x.im x.re) t_4))
        (t_7 (* t_2 t_6))
        (t_8 (log t_3))
        (t_9 (* (pow (atan2 x.im x.re) 2.0) t_4))
        (t_10 (pow t_8 2.0)))
   (if (<= x.im -1.18e-306)
     (*
      (pow y.im 3.0)
      (fma
       -1.0
       (/ (* t_1 (* t_8 t_6)) y.im)
       (fma
        -1.0
        (/ t_7 (pow y.im 2.0))
        (fma
         -0.5
         (/ (* t_10 t_5) y.im)
         (fma
          -0.16666666666666666
          (* t_1 (* (pow t_8 3.0) t_4))
          (fma
           -0.16666666666666666
           (* t_2 (* (pow (atan2 x.im x.re) 3.0) t_4))
           (fma
            0.5
            (* t_1 (* t_8 t_9))
            (fma
             0.5
             (* t_10 t_7)
             (fma
              0.5
              (/ (* t_2 t_9) y.im)
              (+
               (/ (* t_1 (* t_8 t_4)) (pow y.im 2.0))
               (/ t_5 (pow y.im 3.0))))))))))))
     (*
      (exp (- (* y.re (log x.im)) (* y.im (atan2 x.im x.re))))
      (sin (fma y.im (log x.im) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0)));
	double t_4 = pow(t_3, y_46_re);
	double t_5 = t_2 * t_4;
	double t_6 = atan2(x_46_im, x_46_re) * t_4;
	double t_7 = t_2 * t_6;
	double t_8 = log(t_3);
	double t_9 = pow(atan2(x_46_im, x_46_re), 2.0) * t_4;
	double t_10 = pow(t_8, 2.0);
	double tmp;
	if (x_46_im <= -1.18e-306) {
		tmp = pow(y_46_im, 3.0) * fma(-1.0, ((t_1 * (t_8 * t_6)) / y_46_im), fma(-1.0, (t_7 / pow(y_46_im, 2.0)), fma(-0.5, ((t_10 * t_5) / y_46_im), fma(-0.16666666666666666, (t_1 * (pow(t_8, 3.0) * t_4)), fma(-0.16666666666666666, (t_2 * (pow(atan2(x_46_im, x_46_re), 3.0) * t_4)), fma(0.5, (t_1 * (t_8 * t_9)), fma(0.5, (t_10 * t_7), fma(0.5, ((t_2 * t_9) / y_46_im), (((t_1 * (t_8 * t_4)) / pow(y_46_im, 2.0)) + (t_5 / pow(y_46_im, 3.0)))))))))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0)))
	t_4 = t_3 ^ y_46_re
	t_5 = Float64(t_2 * t_4)
	t_6 = Float64(atan(x_46_im, x_46_re) * t_4)
	t_7 = Float64(t_2 * t_6)
	t_8 = log(t_3)
	t_9 = Float64((atan(x_46_im, x_46_re) ^ 2.0) * t_4)
	t_10 = t_8 ^ 2.0
	tmp = 0.0
	if (x_46_im <= -1.18e-306)
		tmp = Float64((y_46_im ^ 3.0) * fma(-1.0, Float64(Float64(t_1 * Float64(t_8 * t_6)) / y_46_im), fma(-1.0, Float64(t_7 / (y_46_im ^ 2.0)), fma(-0.5, Float64(Float64(t_10 * t_5) / y_46_im), fma(-0.16666666666666666, Float64(t_1 * Float64((t_8 ^ 3.0) * t_4)), fma(-0.16666666666666666, Float64(t_2 * Float64((atan(x_46_im, x_46_re) ^ 3.0) * t_4)), fma(0.5, Float64(t_1 * Float64(t_8 * t_9)), fma(0.5, Float64(t_10 * t_7), fma(0.5, Float64(Float64(t_2 * t_9) / y_46_im), Float64(Float64(Float64(t_1 * Float64(t_8 * t_4)) / (y_46_im ^ 2.0)) + Float64(t_5 / (y_46_im ^ 3.0))))))))))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[x$46$im, 2.0], $MachinePrecision] + N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$3, y$46$re], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$7 = N[(t$95$2 * t$95$6), $MachinePrecision]}, Block[{t$95$8 = N[Log[t$95$3], $MachinePrecision]}, Block[{t$95$9 = N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision] * t$95$4), $MachinePrecision]}, Block[{t$95$10 = N[Power[t$95$8, 2.0], $MachinePrecision]}, If[LessEqual[x$46$im, -1.18e-306], N[(N[Power[y$46$im, 3.0], $MachinePrecision] * N[(-1.0 * N[(N[(t$95$1 * N[(t$95$8 * t$95$6), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(-1.0 * N[(t$95$7 / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(t$95$10 * t$95$5), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$1 * N[(N[Power[t$95$8, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$2 * N[(N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 * N[(t$95$8 * t$95$9), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$10 * t$95$7), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 * t$95$9), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(N[(N[(t$95$1 * N[(t$95$8 * t$95$4), $MachinePrecision]), $MachinePrecision] / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$5 / N[Power[y$46$im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.17999999999999999e-306

   1. Initial program 41.5%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(y.im \cdot \left(-1 \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)\right) + \left(\frac{1}{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + y.im \cdot \left(\frac{-1}{6} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{-1}{6} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \frac{1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \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. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), \mathsf{fma}\left(-0.5, {\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), y.im \cdot \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), 0.5 \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   6. Taylor expanded in y.im around inf

      \[\leadsto {y.im}^{3} \cdot \color{blue}{\left(-1 \cdot \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}{y.im} + \left(-1 \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}} + \left(\frac{-1}{2} \cdot \frac{{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)}{y.im} + \left(\frac{-1}{6} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{-1}{6} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{1}{2} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{y.im} + \left(\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}} + \frac{\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}}{{y.im}^{3}}\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 13.2%

      \[\leadsto {y.im}^{3} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}{y.im}, \mathsf{fma}\left(-1, \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)}{y.im}, \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \mathsf{fma}\left(0.5, {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \mathsf{fma}\left(0.5, \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{y.im}, \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}} + \frac{\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}}{{y.im}^{3}}\right)\right)\right)\right)\right)\right)\right)\right)} \]

   if -1.17999999999999999e-306 < x.im

   1. Initial program 35.0%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 63.3

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

   5. Applied rewrites 63.3%

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

Alternative 10: 26.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log x.im\\ t_1 := \sin t\_0\\ t_2 := {\log x.im}^{2}\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := \cos t\_3\\ t_5 := \sin t\_3\\ t_6 := {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\\ t_7 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_8 := \sqrt{{x.im}^{2} + {x.re}^{2}}\\ t_9 := {t\_8}^{y.re}\\ t_10 := t\_5 \cdot t\_9\\ t_11 := \tan^{-1}_* \frac{x.im}{x.re} \cdot t\_9\\ t_12 := t\_5 \cdot t\_11\\ t_13 := \log t\_8\\ t_14 := {t\_13}^{2}\\ t_15 := {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\\ t_16 := t\_1 \cdot t\_15\\ t_17 := t\_15 \cdot t\_9\\ t_18 := \cos t\_0\\ \mathbf{if}\;x.im \leq -1.18 \cdot 10^{-306}:\\ \;\;\;\;{y.im}^{3} \cdot \mathsf{fma}\left(-1, \frac{t\_4 \cdot \left(t\_13 \cdot t\_11\right)}{y.im}, \mathsf{fma}\left(-1, \frac{t\_12}{{y.im}^{2}}, \mathsf{fma}\left(-0.5, \frac{t\_14 \cdot t\_10}{y.im}, \mathsf{fma}\left(-0.16666666666666666, t\_4 \cdot \left({t\_13}^{3} \cdot t\_9\right), \mathsf{fma}\left(-0.16666666666666666, t\_5 \cdot \left(t\_6 \cdot t\_9\right), \mathsf{fma}\left(0.5, t\_4 \cdot \left(t\_13 \cdot t\_17\right), \mathsf{fma}\left(0.5, t\_14 \cdot t\_12, \mathsf{fma}\left(0.5, \frac{t\_5 \cdot t\_17}{y.im}, \frac{t\_4 \cdot \left(t\_13 \cdot t\_9\right)}{{y.im}^{2}} + \frac{t\_10}{{y.im}^{3}}\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, t\_7 \cdot t\_16, \mathsf{fma}\left(0.5, t\_7 \cdot \left(t\_2 \cdot t\_1\right), \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, t\_7 \cdot \left(\log x.im \cdot t\_16\right), \mathsf{fma}\left(-0.16666666666666666, t\_18 \cdot \left(t\_7 \cdot t\_6\right), \mathsf{fma}\left(0.16666666666666666, t\_7 \cdot \left({\log x.im}^{3} \cdot t\_1\right), 0.5 \cdot \left(t\_18 \cdot \left(t\_7 \cdot \left(t\_2 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right), t\_18 \cdot \left(t\_7 \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{fma}\left(t\_18, t\_7 \cdot \tan^{-1}_* \frac{x.im}{x.re}, t\_7 \cdot \left(\log x.im \cdot t\_1\right)\right)\right), t\_7 \cdot t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log x.im)))
        (t_1 (sin t_0))
        (t_2 (pow (log x.im) 2.0))
        (t_3 (* y.re (atan2 x.im x.re)))
        (t_4 (cos t_3))
        (t_5 (sin t_3))
        (t_6 (pow (atan2 x.im x.re) 3.0))
        (t_7 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_8 (sqrt (+ (pow x.im 2.0) (pow x.re 2.0))))
        (t_9 (pow t_8 y.re))
        (t_10 (* t_5 t_9))
        (t_11 (* (atan2 x.im x.re) t_9))
        (t_12 (* t_5 t_11))
        (t_13 (log t_8))
        (t_14 (pow t_13 2.0))
        (t_15 (pow (atan2 x.im x.re) 2.0))
        (t_16 (* t_1 t_15))
        (t_17 (* t_15 t_9))
        (t_18 (cos t_0)))
   (if (<= x.im -1.18e-306)
     (*
      (pow y.im 3.0)
      (fma
       -1.0
       (/ (* t_4 (* t_13 t_11)) y.im)
       (fma
        -1.0
        (/ t_12 (pow y.im 2.0))
        (fma
         -0.5
         (/ (* t_14 t_10) y.im)
         (fma
          -0.16666666666666666
          (* t_4 (* (pow t_13 3.0) t_9))
          (fma
           -0.16666666666666666
           (* t_5 (* t_6 t_9))
           (fma
            0.5
            (* t_4 (* t_13 t_17))
            (fma
             0.5
             (* t_14 t_12)
             (fma
              0.5
              (/ (* t_5 t_17) y.im)
              (+
               (/ (* t_4 (* t_13 t_9)) (pow y.im 2.0))
               (/ t_10 (pow y.im 3.0))))))))))))
     (fma
      y.re
      (fma
       y.re
       (fma
        -0.5
        (* t_7 t_16)
        (fma
         0.5
         (* t_7 (* t_2 t_1))
         (fma
          y.re
          (fma
           -0.5
           (* t_7 (* (log x.im) t_16))
           (fma
            -0.16666666666666666
            (* t_18 (* t_7 t_6))
            (fma
             0.16666666666666666
             (* t_7 (* (pow (log x.im) 3.0) t_1))
             (* 0.5 (* t_18 (* t_7 (* t_2 (atan2 x.im x.re))))))))
          (* t_18 (* t_7 (* (log x.im) (atan2 x.im x.re)))))))
       (fma t_18 (* t_7 (atan2 x.im x.re)) (* t_7 (* (log x.im) t_1))))
      (* t_7 t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * log(x_46_im);
	double t_1 = sin(t_0);
	double t_2 = pow(log(x_46_im), 2.0);
	double t_3 = y_46_re * atan2(x_46_im, x_46_re);
	double t_4 = cos(t_3);
	double t_5 = sin(t_3);
	double t_6 = pow(atan2(x_46_im, x_46_re), 3.0);
	double t_7 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_8 = sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0)));
	double t_9 = pow(t_8, y_46_re);
	double t_10 = t_5 * t_9;
	double t_11 = atan2(x_46_im, x_46_re) * t_9;
	double t_12 = t_5 * t_11;
	double t_13 = log(t_8);
	double t_14 = pow(t_13, 2.0);
	double t_15 = pow(atan2(x_46_im, x_46_re), 2.0);
	double t_16 = t_1 * t_15;
	double t_17 = t_15 * t_9;
	double t_18 = cos(t_0);
	double tmp;
	if (x_46_im <= -1.18e-306) {
		tmp = pow(y_46_im, 3.0) * fma(-1.0, ((t_4 * (t_13 * t_11)) / y_46_im), fma(-1.0, (t_12 / pow(y_46_im, 2.0)), fma(-0.5, ((t_14 * t_10) / y_46_im), fma(-0.16666666666666666, (t_4 * (pow(t_13, 3.0) * t_9)), fma(-0.16666666666666666, (t_5 * (t_6 * t_9)), fma(0.5, (t_4 * (t_13 * t_17)), fma(0.5, (t_14 * t_12), fma(0.5, ((t_5 * t_17) / y_46_im), (((t_4 * (t_13 * t_9)) / pow(y_46_im, 2.0)) + (t_10 / pow(y_46_im, 3.0)))))))))));
	} else {
		tmp = fma(y_46_re, fma(y_46_re, fma(-0.5, (t_7 * t_16), fma(0.5, (t_7 * (t_2 * t_1)), fma(y_46_re, fma(-0.5, (t_7 * (log(x_46_im) * t_16)), fma(-0.16666666666666666, (t_18 * (t_7 * t_6)), fma(0.16666666666666666, (t_7 * (pow(log(x_46_im), 3.0) * t_1)), (0.5 * (t_18 * (t_7 * (t_2 * atan2(x_46_im, x_46_re)))))))), (t_18 * (t_7 * (log(x_46_im) * atan2(x_46_im, x_46_re))))))), fma(t_18, (t_7 * atan2(x_46_im, x_46_re)), (t_7 * (log(x_46_im) * t_1)))), (t_7 * t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(x_46_im))
	t_1 = sin(t_0)
	t_2 = log(x_46_im) ^ 2.0
	t_3 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_4 = cos(t_3)
	t_5 = sin(t_3)
	t_6 = atan(x_46_im, x_46_re) ^ 3.0
	t_7 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_8 = sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0)))
	t_9 = t_8 ^ y_46_re
	t_10 = Float64(t_5 * t_9)
	t_11 = Float64(atan(x_46_im, x_46_re) * t_9)
	t_12 = Float64(t_5 * t_11)
	t_13 = log(t_8)
	t_14 = t_13 ^ 2.0
	t_15 = atan(x_46_im, x_46_re) ^ 2.0
	t_16 = Float64(t_1 * t_15)
	t_17 = Float64(t_15 * t_9)
	t_18 = cos(t_0)
	tmp = 0.0
	if (x_46_im <= -1.18e-306)
		tmp = Float64((y_46_im ^ 3.0) * fma(-1.0, Float64(Float64(t_4 * Float64(t_13 * t_11)) / y_46_im), fma(-1.0, Float64(t_12 / (y_46_im ^ 2.0)), fma(-0.5, Float64(Float64(t_14 * t_10) / y_46_im), fma(-0.16666666666666666, Float64(t_4 * Float64((t_13 ^ 3.0) * t_9)), fma(-0.16666666666666666, Float64(t_5 * Float64(t_6 * t_9)), fma(0.5, Float64(t_4 * Float64(t_13 * t_17)), fma(0.5, Float64(t_14 * t_12), fma(0.5, Float64(Float64(t_5 * t_17) / y_46_im), Float64(Float64(Float64(t_4 * Float64(t_13 * t_9)) / (y_46_im ^ 2.0)) + Float64(t_10 / (y_46_im ^ 3.0))))))))))));
	else
		tmp = fma(y_46_re, fma(y_46_re, fma(-0.5, Float64(t_7 * t_16), fma(0.5, Float64(t_7 * Float64(t_2 * t_1)), fma(y_46_re, fma(-0.5, Float64(t_7 * Float64(log(x_46_im) * t_16)), fma(-0.16666666666666666, Float64(t_18 * Float64(t_7 * t_6)), fma(0.16666666666666666, Float64(t_7 * Float64((log(x_46_im) ^ 3.0) * t_1)), Float64(0.5 * Float64(t_18 * Float64(t_7 * Float64(t_2 * atan(x_46_im, x_46_re)))))))), Float64(t_18 * Float64(t_7 * Float64(log(x_46_im) * atan(x_46_im, x_46_re))))))), fma(t_18, Float64(t_7 * atan(x_46_im, x_46_re)), Float64(t_7 * Float64(log(x_46_im) * t_1)))), Float64(t_7 * t_1));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Log[x$46$im], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[t$95$3], $MachinePrecision]}, Block[{t$95$5 = N[Sin[t$95$3], $MachinePrecision]}, Block[{t$95$6 = N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$7 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[N[(N[Power[x$46$im, 2.0], $MachinePrecision] + N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$9 = N[Power[t$95$8, y$46$re], $MachinePrecision]}, Block[{t$95$10 = N[(t$95$5 * t$95$9), $MachinePrecision]}, Block[{t$95$11 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$9), $MachinePrecision]}, Block[{t$95$12 = N[(t$95$5 * t$95$11), $MachinePrecision]}, Block[{t$95$13 = N[Log[t$95$8], $MachinePrecision]}, Block[{t$95$14 = N[Power[t$95$13, 2.0], $MachinePrecision]}, Block[{t$95$15 = N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$16 = N[(t$95$1 * t$95$15), $MachinePrecision]}, Block[{t$95$17 = N[(t$95$15 * t$95$9), $MachinePrecision]}, Block[{t$95$18 = N[Cos[t$95$0], $MachinePrecision]}, If[LessEqual[x$46$im, -1.18e-306], N[(N[Power[y$46$im, 3.0], $MachinePrecision] * N[(-1.0 * N[(N[(t$95$4 * N[(t$95$13 * t$95$11), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(-1.0 * N[(t$95$12 / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(N[(t$95$14 * t$95$10), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$4 * N[(N[Power[t$95$13, 3.0], $MachinePrecision] * t$95$9), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$5 * N[(t$95$6 * t$95$9), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$4 * N[(t$95$13 * t$95$17), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$14 * t$95$12), $MachinePrecision] + N[(0.5 * N[(N[(t$95$5 * t$95$17), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(N[(N[(t$95$4 * N[(t$95$13 * t$95$9), $MachinePrecision]), $MachinePrecision] / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision] + N[(t$95$10 / N[Power[y$46$im, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(y$46$re * N[(-0.5 * N[(t$95$7 * t$95$16), $MachinePrecision] + N[(0.5 * N[(t$95$7 * N[(t$95$2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[(-0.5 * N[(t$95$7 * N[(N[Log[x$46$im], $MachinePrecision] * t$95$16), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$18 * N[(t$95$7 * t$95$6), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(t$95$7 * N[(N[Power[N[Log[x$46$im], $MachinePrecision], 3.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$18 * N[(t$95$7 * N[(t$95$2 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$18 * N[(t$95$7 * N[(N[Log[x$46$im], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$18 * N[(t$95$7 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[(t$95$7 * N[(N[Log[x$46$im], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$7 * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.17999999999999999e-306

   1. Initial program 41.5%

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

   3. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(-1 \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(y.im \cdot \left(-1 \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{-1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)\right) + \left(\frac{1}{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + y.im \cdot \left(\frac{-1}{6} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{-1}{6} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \frac{1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right) + \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \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. Applied rewrites 54.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), \mathsf{fma}\left(-0.5, {\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), y.im \cdot \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right), 0.5 \cdot \left({\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right)\right)\right)\right)\right)\right)\right), \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)\right)\right), \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]

   6. Taylor expanded in y.im around inf

      \[\leadsto {y.im}^{3} \cdot \color{blue}{\left(-1 \cdot \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}{y.im} + \left(-1 \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}} + \left(\frac{-1}{2} \cdot \frac{{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)}{y.im} + \left(\frac{-1}{6} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{-1}{6} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right) + \left(\frac{1}{2} \cdot \left(\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{1}{2} \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)\right) + \left(\frac{1}{2} \cdot \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{y.im} + \left(\frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}} + \frac{\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}}{{y.im}^{3}}\right)\right)\right)\right)\right)\right)\right)\right)\right)} \]

   8. Applied rewrites 13.2%

      \[\leadsto {y.im}^{3} \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}{y.im}, \mathsf{fma}\left(-1, \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)}{y.im}, \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \mathsf{fma}\left(0.5, {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \mathsf{fma}\left(0.5, \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{y.im}, \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}} + \frac{\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}}{{y.im}^{3}}\right)\right)\right)\right)\right)\right)\right)\right)} \]

   if -1.17999999999999999e-306 < x.im

   1. Initial program 35.0%

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

   3. Taylor expanded in x.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-atan2.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-log.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-atan2.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto y.re \cdot \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) + \left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \frac{1}{2} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right) + \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) + \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right) + \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log x.im\right)} \]

   8. Applied rewrites 39.0%

      \[\leadsto \mathsf{fma}\left(y.re, \color{blue}{\mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \mathsf{fma}\left(0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right), \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right), 0.5 \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right), \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{fma}\left(\cos \left(y.im \cdot \log x.im\right), e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \tan^{-1}_* \frac{x.im}{x.re}, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right)}, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.18 \cdot 10^{-306}:\\ \;\;\;\;{y.im}^{3} \cdot \mathsf{fma}\left(-1, \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right)}{y.im}, \mathsf{fma}\left(-1, \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}}, \mathsf{fma}\left(-0.5, \frac{{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\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}\right)}{y.im}, \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \mathsf{fma}\left(-0.16666666666666666, \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{3} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right), \mathsf{fma}\left(0.5, \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \mathsf{fma}\left(0.5, {\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{2} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)\right), \mathsf{fma}\left(0.5, \frac{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left({\tan^{-1}_* \frac{x.im}{x.re}}^{2} \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{y.im}, \frac{\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \left(\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \cdot {\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}\right)}{{y.im}^{2}} + \frac{\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}}{{y.im}^{3}}\right)\right)\right)\right)\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \mathsf{fma}\left(0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right), \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right), 0.5 \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right), \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{fma}\left(\cos \left(y.im \cdot \log x.im\right), e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \tan^{-1}_* \frac{x.im}{x.re}, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right), e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 20.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := {\log x.im}^{2}\\ t_2 := y.im \cdot \log x.im\\ t_3 := \sin t\_2\\ t_4 := t\_3 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\\ t_5 := \cos t\_2\\ \mathsf{fma}\left(y.re, \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, t\_0 \cdot t\_4, \mathsf{fma}\left(0.5, t\_0 \cdot \left(t\_1 \cdot t\_3\right), \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, t\_0 \cdot \left(\log x.im \cdot t\_4\right), \mathsf{fma}\left(-0.16666666666666666, t\_5 \cdot \left(t\_0 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, t\_0 \cdot \left({\log x.im}^{3} \cdot t\_3\right), 0.5 \cdot \left(t\_5 \cdot \left(t\_0 \cdot \left(t\_1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right), t\_5 \cdot \left(t\_0 \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{fma}\left(t\_5, t\_0 \cdot \tan^{-1}_* \frac{x.im}{x.re}, t\_0 \cdot \left(\log x.im \cdot t\_3\right)\right)\right), t\_0 \cdot t\_3\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_1 (pow (log x.im) 2.0))
        (t_2 (* y.im (log x.im)))
        (t_3 (sin t_2))
        (t_4 (* t_3 (pow (atan2 x.im x.re) 2.0)))
        (t_5 (cos t_2)))
   (fma
    y.re
    (fma
     y.re
     (fma
      -0.5
      (* t_0 t_4)
      (fma
       0.5
       (* t_0 (* t_1 t_3))
       (fma
        y.re
        (fma
         -0.5
         (* t_0 (* (log x.im) t_4))
         (fma
          -0.16666666666666666
          (* t_5 (* t_0 (pow (atan2 x.im x.re) 3.0)))
          (fma
           0.16666666666666666
           (* t_0 (* (pow (log x.im) 3.0) t_3))
           (* 0.5 (* t_5 (* t_0 (* t_1 (atan2 x.im x.re))))))))
        (* t_5 (* t_0 (* (log x.im) (atan2 x.im x.re)))))))
     (fma t_5 (* t_0 (atan2 x.im x.re)) (* t_0 (* (log x.im) t_3))))
    (* t_0 t_3))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = pow(log(x_46_im), 2.0);
	double t_2 = y_46_im * log(x_46_im);
	double t_3 = sin(t_2);
	double t_4 = t_3 * pow(atan2(x_46_im, x_46_re), 2.0);
	double t_5 = cos(t_2);
	return fma(y_46_re, fma(y_46_re, fma(-0.5, (t_0 * t_4), fma(0.5, (t_0 * (t_1 * t_3)), fma(y_46_re, fma(-0.5, (t_0 * (log(x_46_im) * t_4)), fma(-0.16666666666666666, (t_5 * (t_0 * pow(atan2(x_46_im, x_46_re), 3.0))), fma(0.16666666666666666, (t_0 * (pow(log(x_46_im), 3.0) * t_3)), (0.5 * (t_5 * (t_0 * (t_1 * atan2(x_46_im, x_46_re)))))))), (t_5 * (t_0 * (log(x_46_im) * atan2(x_46_im, x_46_re))))))), fma(t_5, (t_0 * atan2(x_46_im, x_46_re)), (t_0 * (log(x_46_im) * t_3)))), (t_0 * t_3));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_1 = log(x_46_im) ^ 2.0
	t_2 = Float64(y_46_im * log(x_46_im))
	t_3 = sin(t_2)
	t_4 = Float64(t_3 * (atan(x_46_im, x_46_re) ^ 2.0))
	t_5 = cos(t_2)
	return fma(y_46_re, fma(y_46_re, fma(-0.5, Float64(t_0 * t_4), fma(0.5, Float64(t_0 * Float64(t_1 * t_3)), fma(y_46_re, fma(-0.5, Float64(t_0 * Float64(log(x_46_im) * t_4)), fma(-0.16666666666666666, Float64(t_5 * Float64(t_0 * (atan(x_46_im, x_46_re) ^ 3.0))), fma(0.16666666666666666, Float64(t_0 * Float64((log(x_46_im) ^ 3.0) * t_3)), Float64(0.5 * Float64(t_5 * Float64(t_0 * Float64(t_1 * atan(x_46_im, x_46_re)))))))), Float64(t_5 * Float64(t_0 * Float64(log(x_46_im) * atan(x_46_im, x_46_re))))))), fma(t_5, Float64(t_0 * atan(x_46_im, x_46_re)), Float64(t_0 * Float64(log(x_46_im) * t_3)))), Float64(t_0 * t_3))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[x$46$im], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Cos[t$95$2], $MachinePrecision]}, N[(y$46$re * N[(y$46$re * N[(-0.5 * N[(t$95$0 * t$95$4), $MachinePrecision] + N[(0.5 * N[(t$95$0 * N[(t$95$1 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[(-0.5 * N[(t$95$0 * N[(N[Log[x$46$im], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$5 * N[(t$95$0 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(t$95$0 * N[(N[Power[N[Log[x$46$im], $MachinePrecision], 3.0], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$5 * N[(t$95$0 * N[(t$95$1 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * N[(t$95$0 * N[(N[Log[x$46$im], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 * N[(t$95$0 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(N[Log[x$46$im], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 38.0%

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

3. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-atan2.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-atan2.f64 34.1

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

5. Applied rewrites 34.1%

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

6. Taylor expanded in y.re around 0

   \[\leadsto y.re \cdot \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) + \left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \frac{1}{2} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right) + \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) + \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right) + \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log x.im\right)} \]

8. Applied rewrites 21.0%

   \[\leadsto \mathsf{fma}\left(y.re, \color{blue}{\mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \mathsf{fma}\left(0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right), \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right), 0.5 \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right), \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{fma}\left(\cos \left(y.im \cdot \log x.im\right), e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \tan^{-1}_* \frac{x.im}{x.re}, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right)}, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right) \]

9. Final simplification 21.0%

   \[\leadsto \mathsf{fma}\left(y.re, \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \mathsf{fma}\left(0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right), \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right), 0.5 \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right), \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{fma}\left(\cos \left(y.im \cdot \log x.im\right), e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \tan^{-1}_* \frac{x.im}{x.re}, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right), e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right) \]
10. Add Preprocessing

Alternative 12: 15.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \log x.im\\ t_1 := \cos t\_0\\ t_2 := \sin t\_0\\ t_3 := t\_2 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\\ t_4 := e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_5 := {\log x.im}^{2}\\ \mathsf{fma}\left(y.re, {y.re}^{2} \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-0.5, t\_4 \cdot t\_3, \mathsf{fma}\left(0.5, t\_4 \cdot \left(t\_5 \cdot t\_2\right), t\_1 \cdot \left(t\_4 \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \frac{\mathsf{fma}\left(t\_1, t\_4 \cdot \tan^{-1}_* \frac{x.im}{x.re}, t\_4 \cdot \left(\log x.im \cdot t\_2\right)\right)}{-y.re}\right)}{y.re}, \mathsf{fma}\left(-0.5, t\_4 \cdot \left(\log x.im \cdot t\_3\right), \mathsf{fma}\left(-0.16666666666666666, t\_1 \cdot \left(t\_4 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, t\_4 \cdot \left({\log x.im}^{3} \cdot t\_2\right), 0.5 \cdot \left(t\_1 \cdot \left(t\_4 \cdot \left(t\_5 \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right)\right), t\_4 \cdot t\_2\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (log x.im)))
        (t_1 (cos t_0))
        (t_2 (sin t_0))
        (t_3 (* t_2 (pow (atan2 x.im x.re) 2.0)))
        (t_4 (exp (* (- y.im) (atan2 x.im x.re))))
        (t_5 (pow (log x.im) 2.0)))
   (fma
    y.re
    (*
     (pow y.re 2.0)
     (fma
      -1.0
      (/
       (fma
        -1.0
        (fma
         -0.5
         (* t_4 t_3)
         (fma
          0.5
          (* t_4 (* t_5 t_2))
          (* t_1 (* t_4 (* (log x.im) (atan2 x.im x.re))))))
        (/
         (fma t_1 (* t_4 (atan2 x.im x.re)) (* t_4 (* (log x.im) t_2)))
         (- y.re)))
       y.re)
      (fma
       -0.5
       (* t_4 (* (log x.im) t_3))
       (fma
        -0.16666666666666666
        (* t_1 (* t_4 (pow (atan2 x.im x.re) 3.0)))
        (fma
         0.16666666666666666
         (* t_4 (* (pow (log x.im) 3.0) t_2))
         (* 0.5 (* t_1 (* t_4 (* t_5 (atan2 x.im x.re))))))))))
    (* t_4 t_2))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * log(x_46_im);
	double t_1 = cos(t_0);
	double t_2 = sin(t_0);
	double t_3 = t_2 * pow(atan2(x_46_im, x_46_re), 2.0);
	double t_4 = exp((-y_46_im * atan2(x_46_im, x_46_re)));
	double t_5 = pow(log(x_46_im), 2.0);
	return fma(y_46_re, (pow(y_46_re, 2.0) * fma(-1.0, (fma(-1.0, fma(-0.5, (t_4 * t_3), fma(0.5, (t_4 * (t_5 * t_2)), (t_1 * (t_4 * (log(x_46_im) * atan2(x_46_im, x_46_re)))))), (fma(t_1, (t_4 * atan2(x_46_im, x_46_re)), (t_4 * (log(x_46_im) * t_2))) / -y_46_re)) / y_46_re), fma(-0.5, (t_4 * (log(x_46_im) * t_3)), fma(-0.16666666666666666, (t_1 * (t_4 * pow(atan2(x_46_im, x_46_re), 3.0))), fma(0.16666666666666666, (t_4 * (pow(log(x_46_im), 3.0) * t_2)), (0.5 * (t_1 * (t_4 * (t_5 * atan2(x_46_im, x_46_re)))))))))), (t_4 * t_2));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * log(x_46_im))
	t_1 = cos(t_0)
	t_2 = sin(t_0)
	t_3 = Float64(t_2 * (atan(x_46_im, x_46_re) ^ 2.0))
	t_4 = exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re)))
	t_5 = log(x_46_im) ^ 2.0
	return fma(y_46_re, Float64((y_46_re ^ 2.0) * fma(-1.0, Float64(fma(-1.0, fma(-0.5, Float64(t_4 * t_3), fma(0.5, Float64(t_4 * Float64(t_5 * t_2)), Float64(t_1 * Float64(t_4 * Float64(log(x_46_im) * atan(x_46_im, x_46_re)))))), Float64(fma(t_1, Float64(t_4 * atan(x_46_im, x_46_re)), Float64(t_4 * Float64(log(x_46_im) * t_2))) / Float64(-y_46_re))) / y_46_re), fma(-0.5, Float64(t_4 * Float64(log(x_46_im) * t_3)), fma(-0.16666666666666666, Float64(t_1 * Float64(t_4 * (atan(x_46_im, x_46_re) ^ 3.0))), fma(0.16666666666666666, Float64(t_4 * Float64((log(x_46_im) ^ 3.0) * t_2)), Float64(0.5 * Float64(t_1 * Float64(t_4 * Float64(t_5 * atan(x_46_im, x_46_re)))))))))), Float64(t_4 * t_2))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[Log[x$46$im], $MachinePrecision], 2.0], $MachinePrecision]}, N[(y$46$re * N[(N[Power[y$46$re, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[(-1.0 * N[(-0.5 * N[(t$95$4 * t$95$3), $MachinePrecision] + N[(0.5 * N[(t$95$4 * N[(t$95$5 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(t$95$4 * N[(N[Log[x$46$im], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$1 * N[(t$95$4 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * N[(N[Log[x$46$im], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision] + N[(-0.5 * N[(t$95$4 * N[(N[Log[x$46$im], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(t$95$1 * N[(t$95$4 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.16666666666666666 * N[(t$95$4 * N[(N[Power[N[Log[x$46$im], $MachinePrecision], 3.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(t$95$1 * N[(t$95$4 * N[(t$95$5 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Initial program 38.0%

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

3. Taylor expanded in x.re around 0

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

   1. lower-*.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-atan2.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-log.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lift-atan2.f64 34.1

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

5. Applied rewrites 34.1%

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

6. Taylor expanded in y.re around 0

   \[\leadsto y.re \cdot \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) + \left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \frac{1}{2} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right) + \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right) + \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right) + \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log x.im\right)} \]

8. Applied rewrites 21.0%

   \[\leadsto \mathsf{fma}\left(y.re, \color{blue}{\mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \mathsf{fma}\left(0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right), \mathsf{fma}\left(y.re, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right), 0.5 \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right), \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right), \mathsf{fma}\left(\cos \left(y.im \cdot \log x.im\right), e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \tan^{-1}_* \frac{x.im}{x.re}, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)\right)}, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right) \]

9. Taylor expanded in y.re around -inf

   \[\leadsto \mathsf{fma}\left(y.re, {y.re}^{2} \cdot \left(-1 \cdot \frac{-1 \cdot \left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right) + \left(\frac{1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right) + -1 \cdot \frac{\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) + e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)}{y.re}}{y.re} + \color{blue}{\left(\frac{-1}{2} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right)\right) + \left(\frac{-1}{6} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right)\right) + \left(\frac{1}{6} \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right) + \frac{1}{2} \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right)}\right), e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right) \]

11. Applied rewrites 13.5%

    \[\leadsto \mathsf{fma}\left(y.re, {y.re}^{2} \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \mathsf{fma}\left(0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right), \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), -1 \cdot \frac{\mathsf{fma}\left(\cos \left(y.im \cdot \log x.im\right), e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \tan^{-1}_* \frac{x.im}{x.re}, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)}{y.re}\right)}{y.re}}, \mathsf{fma}\left(-0.5, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right), 0.5 \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right)\right), e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right) \]

12. Final simplification 13.5%

    \[\leadsto \mathsf{fma}\left(y.re, {y.re}^{2} \cdot \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right), \mathsf{fma}\left(0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \sin \left(y.im \cdot \log x.im\right)\right), \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right), \frac{\mathsf{fma}\left(\cos \left(y.im \cdot \log x.im\right), e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \tan^{-1}_* \frac{x.im}{x.re}, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \sin \left(y.im \cdot \log x.im\right)\right)\right)}{-y.re}\right)}{y.re}, \mathsf{fma}\left(-0.5, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left(\log x.im \cdot \left(\sin \left(y.im \cdot \log x.im\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)\right), \mathsf{fma}\left(-0.16666666666666666, \cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right), \mathsf{fma}\left(0.16666666666666666, e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{3} \cdot \sin \left(y.im \cdot \log x.im\right)\right), 0.5 \cdot \left(\cos \left(y.im \cdot \log x.im\right) \cdot \left(e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \left({\log x.im}^{2} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right)\right)\right)\right)\right), e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025057 
(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)))))