powComplex, imaginary part

Percentage Accurate: 41.0% → 77.1%

Time: 27.5s

Alternatives: 16

Speedup: N/A×

• 35.4% 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: 41.0% 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: 77.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq -0.135 \lor \neg \left(y.re \leq 1.85\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_1} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{t\_0 \cdot y.re}}{e^{t\_1}} \cdot \sin \left(\mathsf{fma}\left(t\_0, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))) (t_1 (* (atan2 x.im x.re) y.im)))
   (if (or (<= y.re -0.135) (not (<= y.re 1.85)))
     (*
      (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_1))
      (sin (* y.re (atan2 x.im x.re))))
     (*
      (/ (exp (* t_0 y.re)) (exp t_1))
      (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));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if ((y_46_re <= -0.135) || !(y_46_re <= 1.85)) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_1)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = (exp((t_0 * y_46_re)) / exp(t_1)) * sin(fma(t_0, y_46_im, (atan2(x_46_im, x_46_re) * y_46_re)));
	}
	return tmp;
}

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))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if ((y_46_re <= -0.135) || !(y_46_re <= 1.85))
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_1)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(Float64(exp(Float64(t_0 * y_46_re)) / exp(t_1)) * sin(fma(t_0, y_46_im, Float64(atan(x_46_im, x_46_re) * y_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[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -0.135], N[Not[LessEqual[y$46$re, 1.85]], $MachinePrecision]], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(t$95$0 * y$46$re), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$1], $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. Split input into 2 regimes

2. if y.re < -0.13500000000000001 or 1.8500000000000001 < y.re

   1. Initial program 36.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -0.13500000000000001 < y.re < 1.8500000000000001

   1. Initial program 44.4%

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

   4. Applied rewrites 83.6%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.135 \lor \neg \left(y.re \leq 1.85\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.5% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq -0.0062 \lor \neg \left(y.re \leq 0.58\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{t\_0}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (or (<= y.re -0.0062) (not (<= y.re 0.58)))
     (*
      (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
      (sin (* y.re (atan2 x.im x.re))))
     (*
      (/ 1.0 (exp t_0))
      (sin (fma (log (hypot x.re x.im)) 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 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if ((y_46_re <= -0.0062) || !(y_46_re <= 0.58)) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = (1.0 / exp(t_0)) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if ((y_46_re <= -0.0062) || !(y_46_re <= 0.58))
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(Float64(1.0 / exp(t_0)) * sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -0.0062], N[Not[LessEqual[y$46$re, 0.58]], $MachinePrecision]], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.00619999999999999978 or 0.57999999999999996 < y.re

   1. Initial program 36.9%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -0.00619999999999999978 < y.re < 0.57999999999999996

   1. Initial program 44.4%

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

   4. Applied rewrites 83.6%

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

   5. Taylor expanded in y.re around 0

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

   7. Applied rewrites 82.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.0062 \lor \neg \left(y.re \leq 0.58\right):\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 70.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_3 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_4 := t\_3 \cdot y.im\\ \mathbf{if}\;x.im \leq -6.5 \cdot 10^{-101}:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{-1}{x.im}\right), t\_1\right)\right)\\ \mathbf{elif}\;x.im \leq 6 \cdot 10^{+154}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \mathsf{fma}\left(\sin t\_4, \cos t\_2, \cos t\_4 \cdot \sin t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.im}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* (atan2 x.im x.re) y.re))
        (t_3 (log (hypot x.re x.im)))
        (t_4 (* t_3 y.im)))
   (if (<= x.im -6.5e-101)
     (*
      (exp (- (* t_3 y.re) t_0))
      (sin (fma -1.0 (* y.im (log (/ -1.0 x.im))) t_1)))
     (if (<= x.im 6e+154)
       (*
        (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
        (fma (sin t_4) (cos t_2) (* (cos t_4) (sin t_2))))
       (*
        (/ (exp (* y.re (log x.im))) (exp (* y.im (atan2 x.im x.re))))
        (sin (fma y.im (log x.im) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_3 = log(hypot(x_46_re, x_46_im));
	double t_4 = t_3 * y_46_im;
	double tmp;
	if (x_46_im <= -6.5e-101) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(fma(-1.0, (y_46_im * log((-1.0 / x_46_im))), t_1));
	} else if (x_46_im <= 6e+154) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * fma(sin(t_4), cos(t_2), (cos(t_4) * sin(t_2)));
	} else {
		tmp = (exp((y_46_re * log(x_46_im))) / exp((y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_3 = log(hypot(x_46_re, x_46_im))
	t_4 = Float64(t_3 * y_46_im)
	tmp = 0.0
	if (x_46_im <= -6.5e-101)
		tmp = Float64(exp(Float64(Float64(t_3 * y_46_re) - t_0)) * sin(fma(-1.0, Float64(y_46_im * log(Float64(-1.0 / x_46_im))), t_1)));
	elseif (x_46_im <= 6e+154)
		tmp = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * fma(sin(t_4), cos(t_2), Float64(cos(t_4) * sin(t_2))));
	else
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_im))) / exp(Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, log(x_46_im), 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -6.5e-101], N[(N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 6e+154], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[t$95$4], $MachinePrecision] * N[Cos[t$95$2], $MachinePrecision] + N[(N[Cos[t$95$4], $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[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$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -6.4999999999999996e-101

   1. Initial program 37.7%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-atan2.f64 N/A

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

   5. Applied rewrites 40.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \left(-y.im\right) \cdot \mathsf{fma}\left(-0.5, y.im \cdot \left(\left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right)\right)\right)} \]
   6. Step-by-step derivation

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-hypot.f64 40.1

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

   7. Applied rewrites 40.1%

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

   8. Taylor expanded in x.im around -inf

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

      1. lower-sin.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift-atan2.f64 N/A

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

      7. lift-*.f64 72.0

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

   10. Applied rewrites 72.0%

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

   if -6.4999999999999996e-101 < x.im < 6.00000000000000052e154

   1. Initial program 51.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-atan2.f64 N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 67.7%

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

   if 6.00000000000000052e154 < x.im

   1. Initial program 0.0%

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

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 80.0%

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

Alternative 4: 61.5% accurate, N/A× speedup

Math

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

FPCore

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

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -inf.0

   1. Initial program 39.1%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-atan2.f64 N/A

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \left(-y.im\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, y.im \cdot \left(\left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\frac{1}{2}}\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\frac{1}{2}}\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + y.re \cdot \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \frac{-1}{6} \cdot \left(y.re \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right)\right)\right) + \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\frac{1}{2}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 60.9%

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

   if -inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 90.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-atan2.f64 N/A

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

      11. sin-sum N/A

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

      12. lower-fma.f64 N/A

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

   4. Applied rewrites 91.9%

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

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 47.4

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

   5. Applied rewrites 47.4%

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

4. Final simplification 67.1%

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

Alternative 5: 61.5% accurate, N/A× speedup

Math

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

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)))))
        (t_1 (cos (* 0.5 PI)))
        (t_2 (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im))))
        (t_3 (* t_2 (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re)))))
        (t_4 (log (pow (fma x.im x.im (* x.re x.re)) 0.5)))
        (t_5 (sin (* y.re (atan2 x.im x.re))))
        (t_6 (sin (* 0.5 PI))))
   (if (<= t_3 (- INFINITY))
     (*
      t_2
      (-
       t_5
       (*
        (* -1.0 y.im)
        (fma
         -0.5
         (* y.im (* (* t_4 t_4) t_5))
         (*
          (+
           t_6
           (*
            y.re
            (fma
             y.re
             (+
              (* -0.5 (* t_6 (pow (atan2 x.im x.re) 2.0)))
              (*
               -0.16666666666666666
               (* y.re (* t_1 (pow (atan2 x.im x.re) 3.0)))))
             (* t_1 (atan2 x.im x.re)))))
          t_4)))))
     (if (<= t_3 INFINITY) t_3 (* t_2 t_5)))))

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

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))))
	t_1 = cos(Float64(0.5 * pi))
	t_2 = exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_3 = Float64(t_2 * sin(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
	t_4 = log((fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ 0.5))
	t_5 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_6 = sin(Float64(0.5 * pi))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(t_2 * Float64(t_5 - Float64(Float64(-1.0 * y_46_im) * fma(-0.5, Float64(y_46_im * Float64(Float64(t_4 * t_4) * t_5)), Float64(Float64(t_6 + Float64(y_46_re * fma(y_46_re, Float64(Float64(-0.5 * Float64(t_6 * (atan(x_46_im, x_46_re) ^ 2.0))) + Float64(-0.16666666666666666 * Float64(y_46_re * Float64(t_1 * (atan(x_46_im, x_46_re) ^ 3.0))))), Float64(t_1 * atan(x_46_im, x_46_re))))) * t_4)))));
	elseif (t_3 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(t_2 * t_5);
	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[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = 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]}, Block[{t$95$3 = N[(t$95$2 * 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]}, Block[{t$95$4 = N[Log[N[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[Sin[N[(0.5 * Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(t$95$2 * N[(t$95$5 - N[(N[(-1.0 * y$46$im), $MachinePrecision] * N[(-0.5 * N[(y$46$im * N[(N[(t$95$4 * t$95$4), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$6 + N[(y$46$re * N[(y$46$re * N[(N[(-0.5 * N[(t$95$6 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(y$46$re * N[(t$95$1 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$3, N[(t$95$2 * t$95$5), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -inf.0

   1. Initial program 39.1%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-atan2.f64 N/A

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \left(-y.im\right) \cdot \mathsf{fma}\left(\frac{-1}{2}, y.im \cdot \left(\left(\log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\frac{1}{2}}\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\frac{1}{2}}\right)\right) \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right), \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) + y.re \cdot \left(y.re \cdot \left(\frac{-1}{2} \cdot \left(\sin \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \frac{-1}{6} \cdot \left(y.re \cdot \left(\cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right)\right)\right) + \cos \left(\frac{1}{2} \cdot \mathsf{PI}\left(\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{\frac{1}{2}}\right)\right)\right) \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lower-*.f64 N/A

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

   8. Applied rewrites 60.9%

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

   if -inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 90.0%

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

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 0.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 47.4

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

   5. Applied rewrites 47.4%

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

4. Final simplification 66.3%

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

Alternative 6: 61.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_1 := e^{t\_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ t_2 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;t\_1 \cdot \sin \left(t\_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \leq 0.5:\\ \;\;\;\;t\_1 \cdot \left(t\_2 - \left(-1 \cdot y.im\right) \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot t\_2\\ \end{array} \end{array} \]

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)))))
        (t_1 (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im))))
        (t_2 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= (* t_1 (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re)))) 0.5)
     (*
      t_1
      (-
       t_2
       (*
        (* -1.0 y.im)
        (*
         (sin (fma y.re (atan2 x.im x.re) (/ PI 2.0)))
         (log (pow (fma x.im x.im (* x.re x.re)) 0.5))))))
     (* t_1 t_2))))

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))));
	double t_1 = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double t_2 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if ((t_1 * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)))) <= 0.5) {
		tmp = t_1 * (t_2 - ((-1.0 * y_46_im) * (sin(fma(y_46_re, atan2(x_46_im, x_46_re), (((double) M_PI) / 2.0))) * log(pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), 0.5)))));
	} else {
		tmp = t_1 * t_2;
	}
	return tmp;
}

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))))
	t_1 = exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	t_2 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (Float64(t_1 * sin(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re)))) <= 0.5)
		tmp = Float64(t_1 * Float64(t_2 - Float64(Float64(-1.0 * y_46_im) * Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(pi / 2.0))) * log((fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ 0.5))))));
	else
		tmp = Float64(t_1 * t_2);
	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[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = 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]}, Block[{t$95$2 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$1 * 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], 0.5], N[(t$95$1 * N[(t$95$2 - N[(N[(-1.0 * y$46$im), $MachinePrecision] * N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * t$95$2), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 0.5

   1. Initial program 84.3%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-atan2.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 85.8%

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

   if 0.5 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 7.0%

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

   3. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lift-atan2.f64 46.8

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

   5. Applied rewrites 46.8%

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

4. Final simplification 63.8%

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

Alternative 7: 55.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq 1.1 \cdot 10^{+16}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin t\_0 - \left(-1 \cdot y.im\right) \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.im}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_0\right)\right)\\ \end{array} \end{array} \]

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 1.1e+16)
     (*
      (exp
       (-
        (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
        (* (atan2 x.im x.re) y.im)))
      (-
       (sin t_0)
       (*
        (* -1.0 y.im)
        (*
         (sin (fma y.re (atan2 x.im x.re) (/ PI 2.0)))
         (log (pow (fma x.im x.im (* x.re x.re)) 0.5))))))
     (*
      (/ (exp (* y.re (log x.im))) (exp (* 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 <= 1.1e+16) {
		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) - ((-1.0 * y_46_im) * (sin(fma(y_46_re, atan2(x_46_im, x_46_re), (((double) M_PI) / 2.0))) * log(pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), 0.5)))));
	} else {
		tmp = (exp((y_46_re * log(x_46_im))) / exp((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 <= 1.1e+16)
		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(Float64(-1.0 * y_46_im) * Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(pi / 2.0))) * log((fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ 0.5))))));
	else
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_im))) / exp(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, 1.1e+16], 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[(N[(-1.0 * y$46$im), $MachinePrecision] * N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Log[N[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[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 < 1.1e16

   1. Initial program 44.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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. fp-cancel-sign-sub-inv N/A

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

      2. lower--.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-atan2.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 N/A

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

   5. Applied rewrites 50.8%

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

   if 1.1e16 < x.im

   1. Initial program 27.6%

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

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 76.8%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.1 \cdot 10^{+16}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - \left(-1 \cdot y.im\right) \cdot \left(\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) \cdot \log \left({\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.im}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.3% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)}^{0.5}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t\_1\\ t_3 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_4 := \frac{e^{y.re \cdot \log x.im}}{t\_3} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_1\right)\right)\\ t_5 := {t\_0}^{y.re}\\ \mathbf{if}\;x.re \leq -3.2 \cdot 10^{+84}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;x.re \leq -3.4 \cdot 10^{-93}:\\ \;\;\;\;\mathsf{fma}\left(y.im, \mathsf{fma}\left(-1, t\_2 \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot t\_5\right), \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \frac{\pi}{2}\right)\right) \cdot \left(\log t\_0 \cdot t\_5\right)\right), t\_2 \cdot t\_5\right)\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{-225}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.re}}{t\_3} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (fma x.im x.im (* x.re x.re)) 0.5))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (sin t_1))
        (t_3 (exp (* y.im (atan2 x.im x.re))))
        (t_4
         (* (/ (exp (* y.re (log x.im))) t_3) (sin (fma y.im (log x.im) t_1))))
        (t_5 (pow t_0 y.re)))
   (if (<= x.re -3.2e+84)
     t_4
     (if (<= x.re -3.4e-93)
       (fma
        y.im
        (fma
         -1.0
         (* t_2 (* (atan2 x.im x.re) t_5))
         (* (sin (fma y.re (atan2 x.im x.re) (/ PI 2.0))) (* (log t_0) t_5)))
        (* t_2 t_5))
       (if (<= x.re 6.5e-225)
         t_4
         (*
          (/ (exp (* y.re (log x.re))) t_3)
          (sin (fma y.im (log x.re) t_1))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(fma(x_46_im, x_46_im, (x_46_re * x_46_re)), 0.5);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(t_1);
	double t_3 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double t_4 = (exp((y_46_re * log(x_46_im))) / t_3) * sin(fma(y_46_im, log(x_46_im), t_1));
	double t_5 = pow(t_0, y_46_re);
	double tmp;
	if (x_46_re <= -3.2e+84) {
		tmp = t_4;
	} else if (x_46_re <= -3.4e-93) {
		tmp = fma(y_46_im, fma(-1.0, (t_2 * (atan2(x_46_im, x_46_re) * t_5)), (sin(fma(y_46_re, atan2(x_46_im, x_46_re), (((double) M_PI) / 2.0))) * (log(t_0) * t_5))), (t_2 * t_5));
	} else if (x_46_re <= 6.5e-225) {
		tmp = t_4;
	} else {
		tmp = (exp((y_46_re * log(x_46_re))) / t_3) * sin(fma(y_46_im, log(x_46_re), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)) ^ 0.5
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = sin(t_1)
	t_3 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_4 = Float64(Float64(exp(Float64(y_46_re * log(x_46_im))) / t_3) * sin(fma(y_46_im, log(x_46_im), t_1)))
	t_5 = t_0 ^ y_46_re
	tmp = 0.0
	if (x_46_re <= -3.2e+84)
		tmp = t_4;
	elseif (x_46_re <= -3.4e-93)
		tmp = fma(y_46_im, fma(-1.0, Float64(t_2 * Float64(atan(x_46_im, x_46_re) * t_5)), Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(pi / 2.0))) * Float64(log(t_0) * t_5))), Float64(t_2 * t_5));
	elseif (x_46_re <= 6.5e-225)
		tmp = t_4;
	else
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_re))) / t_3) * sin(fma(y_46_im, log(x_46_re), 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[Power[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[t$95$0, y$46$re], $MachinePrecision]}, If[LessEqual[x$46$re, -3.2e+84], t$95$4, If[LessEqual[x$46$re, -3.4e-93], N[(y$46$im * N[(-1.0 * N[(t$95$2 * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision] + N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(Pi / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Log[t$95$0], $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * t$95$5), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 6.5e-225], t$95$4, N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -3.2000000000000001e84 or -3.40000000000000001e-93 < x.re < 6.5000000000000005e-225

   1. Initial program 26.8%

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

   3. Taylor expanded in 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. exp-diff N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 39.0%

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

   if -3.2000000000000001e84 < x.re < -3.40000000000000001e-93

   1. Initial program 63.9%

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

   3. Taylor expanded in y.im around 0

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

      1. lower-fma.f64 N/A

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

   5. Applied rewrites 40.8%

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

   if 6.5000000000000005e-225 < x.re

   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 x.im around 0

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

      1. lower-*.f64 N/A

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

      2. exp-diff N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 63.2%

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

Alternative 9: 45.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ t_2 := \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\\ t_3 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.re \leq -5.5 \cdot 10^{+151}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.im}}{t\_3} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_0\right)\right)\\ \mathbf{elif}\;x.re \leq 6.5 \cdot 10^{-225}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(t\_1 - {y.im}^{2} \cdot \mathsf{fma}\left(-1, \frac{t\_2 \cdot \sin \left(\mathsf{fma}\left(0.5, \pi, t\_0\right)\right)}{y.im}, 0.5 \cdot \left({t\_2}^{2} \cdot t\_1\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.re}}{t\_3} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (log (sqrt (+ (pow x.im 2.0) (pow x.re 2.0)))))
        (t_3 (exp (* y.im (atan2 x.im x.re)))))
   (if (<= x.re -5.5e+151)
     (* (/ (exp (* y.re (log x.im))) t_3) (sin (fma y.im (log x.im) t_0)))
     (if (<= x.re 6.5e-225)
       (*
        (exp
         (-
          (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
          (* (atan2 x.im x.re) y.im)))
        (-
         t_1
         (*
          (pow y.im 2.0)
          (fma
           -1.0
           (/ (* t_2 (sin (fma 0.5 PI t_0))) y.im)
           (* 0.5 (* (pow t_2 2.0) t_1))))))
       (*
        (/ (exp (* y.re (log x.re))) t_3)
        (sin (fma y.im (log x.re) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = log(sqrt((pow(x_46_im, 2.0) + pow(x_46_re, 2.0))));
	double t_3 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -5.5e+151) {
		tmp = (exp((y_46_re * log(x_46_im))) / t_3) * sin(fma(y_46_im, log(x_46_im), t_0));
	} else if (x_46_re <= 6.5e-225) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * (t_1 - (pow(y_46_im, 2.0) * fma(-1.0, ((t_2 * sin(fma(0.5, ((double) M_PI), t_0))) / y_46_im), (0.5 * (pow(t_2, 2.0) * t_1)))));
	} else {
		tmp = (exp((y_46_re * log(x_46_re))) / t_3) * sin(fma(y_46_im, log(x_46_re), 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 = sin(t_0)
	t_2 = log(sqrt(Float64((x_46_im ^ 2.0) + (x_46_re ^ 2.0))))
	t_3 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_re <= -5.5e+151)
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_im))) / t_3) * sin(fma(y_46_im, log(x_46_im), t_0)));
	elseif (x_46_re <= 6.5e-225)
		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(t_1 - Float64((y_46_im ^ 2.0) * fma(-1.0, Float64(Float64(t_2 * sin(fma(0.5, pi, t_0))) / y_46_im), Float64(0.5 * Float64((t_2 ^ 2.0) * t_1))))));
	else
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_re))) / t_3) * sin(fma(y_46_im, log(x_46_re), 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[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[Power[x$46$im, 2.0], $MachinePrecision] + N[Power[x$46$re, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -5.5e+151], N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 6.5e-225], 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[(t$95$1 - N[(N[Power[y$46$im, 2.0], $MachinePrecision] * N[(-1.0 * N[(N[(t$95$2 * N[Sin[N[(0.5 * Pi + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] + N[(0.5 * N[(N[Power[t$95$2, 2.0], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -5.4999999999999994e151

   1. Initial program 0.0%

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

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 27.2%

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

   if -5.4999999999999994e151 < x.re < 6.5000000000000005e-225

   1. Initial program 51.3%

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

   3. Taylor expanded in y.im around 0

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

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

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

      2. lower--.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift-atan2.f64 N/A

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

   5. Applied rewrites 46.6%

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

   6. Taylor expanded in y.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 N/A

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

   8. Applied rewrites 42.3%

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

   if 6.5000000000000005e-225 < x.re

   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 x.im around 0

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

      1. lower-*.f64 N/A

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

      2. exp-diff N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 63.2%

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

Alternative 10: 44.2% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;x.re \leq 6.5 \cdot 10^{-225}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.im}}{t\_1} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.re}}{t\_1} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_0\right)\right)\\ \end{array} \end{array} \]

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 (* y.im (atan2 x.im x.re)))))
   (if (<= x.re 6.5e-225)
     (* (/ (exp (* y.re (log x.im))) t_1) (sin (fma y.im (log x.im) t_0)))
     (* (/ (exp (* y.re (log x.re))) t_1) (sin (fma y.im (log x.re) t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= 6.5e-225) {
		tmp = (exp((y_46_re * log(x_46_im))) / t_1) * sin(fma(y_46_im, log(x_46_im), t_0));
	} else {
		tmp = (exp((y_46_re * log(x_46_re))) / t_1) * sin(fma(y_46_im, log(x_46_re), 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(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_re <= 6.5e-225)
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_im))) / t_1) * sin(fma(y_46_im, log(x_46_im), t_0)));
	else
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_re))) / t_1) * sin(fma(y_46_im, log(x_46_re), 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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, 6.5e-225], N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 6.5000000000000005e-225

   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. exp-diff N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 31.6%

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

   if 6.5000000000000005e-225 < x.re

   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 x.im around 0

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

      1. lower-*.f64 N/A

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

      2. exp-diff N/A

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 63.2%

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

Alternative 11: 38.0% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \log \left(\frac{1}{x.re}\right)\\ t_2 := \sin \left(\left(-1 \cdot y.im\right) \cdot t\_1\right)\\ t_3 := t\_2 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\\ t_4 := {t\_1}^{2}\\ t_5 := \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, 0.5 \cdot \pi\right)\right)\\ t_6 := t\_5 \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \mathsf{fma}\left(-1, \frac{t\_1 \cdot t\_2}{t\_0}, \mathsf{fma}\left(y.re, \mathsf{fma}\left(-1, \frac{t\_1 \cdot t\_6}{t\_0}, \mathsf{fma}\left(-0.5, \frac{t\_3}{t\_0}, \mathsf{fma}\left(0.5, \frac{t\_4 \cdot t\_2}{t\_0}, y.re \cdot \left(-0.16666666666666666 \cdot \frac{t\_5 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}}{t\_0} + \left(-0.16666666666666666 \cdot \frac{{t\_1}^{3} \cdot t\_2}{t\_0} + \mathsf{fma}\left(0.5, \frac{t\_1 \cdot t\_3}{t\_0}, 0.5 \cdot \frac{t\_4 \cdot t\_6}{t\_0}\right)\right)\right)\right)\right)\right), \frac{t\_6}{t\_0}\right)\right), \frac{t\_2}{t\_0}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.im}}{t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

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 (log (/ 1.0 x.re)))
        (t_2 (sin (* (* -1.0 y.im) t_1)))
        (t_3 (* t_2 (pow (atan2 x.im x.re) 2.0)))
        (t_4 (pow t_1 2.0))
        (t_5 (sin (fma -1.0 (* y.im t_1) (* 0.5 PI))))
        (t_6 (* t_5 (atan2 x.im x.re))))
   (if (<= x.im -5e-310)
     (fma
      y.re
      (fma
       -1.0
       (/ (* t_1 t_2) t_0)
       (fma
        y.re
        (fma
         -1.0
         (/ (* t_1 t_6) t_0)
         (fma
          -0.5
          (/ t_3 t_0)
          (fma
           0.5
           (/ (* t_4 t_2) t_0)
           (*
            y.re
            (+
             (*
              -0.16666666666666666
              (/ (* t_5 (pow (atan2 x.im x.re) 3.0)) t_0))
             (+
              (* -0.16666666666666666 (/ (* (pow t_1 3.0) t_2) t_0))
              (fma 0.5 (/ (* t_1 t_3) t_0) (* 0.5 (/ (* t_4 t_6) t_0)))))))))
        (/ t_6 t_0)))
      (/ t_2 t_0))
     (*
      (/ (exp (* y.re (log x.im))) t_0)
      (sin (fma y.im (log x.im) (* 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((y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = log((1.0 / x_46_re));
	double t_2 = sin(((-1.0 * y_46_im) * t_1));
	double t_3 = t_2 * pow(atan2(x_46_im, x_46_re), 2.0);
	double t_4 = pow(t_1, 2.0);
	double t_5 = sin(fma(-1.0, (y_46_im * t_1), (0.5 * ((double) M_PI))));
	double t_6 = t_5 * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= -5e-310) {
		tmp = fma(y_46_re, fma(-1.0, ((t_1 * t_2) / t_0), fma(y_46_re, fma(-1.0, ((t_1 * t_6) / t_0), fma(-0.5, (t_3 / t_0), fma(0.5, ((t_4 * t_2) / t_0), (y_46_re * ((-0.16666666666666666 * ((t_5 * pow(atan2(x_46_im, x_46_re), 3.0)) / t_0)) + ((-0.16666666666666666 * ((pow(t_1, 3.0) * t_2) / t_0)) + fma(0.5, ((t_1 * t_3) / t_0), (0.5 * ((t_4 * t_6) / t_0))))))))), (t_6 / t_0))), (t_2 / t_0));
	} else {
		tmp = (exp((y_46_re * log(x_46_im))) / t_0) * sin(fma(y_46_im, log(x_46_im), (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(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_1 = log(Float64(1.0 / x_46_re))
	t_2 = sin(Float64(Float64(-1.0 * y_46_im) * t_1))
	t_3 = Float64(t_2 * (atan(x_46_im, x_46_re) ^ 2.0))
	t_4 = t_1 ^ 2.0
	t_5 = sin(fma(-1.0, Float64(y_46_im * t_1), Float64(0.5 * pi)))
	t_6 = Float64(t_5 * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= -5e-310)
		tmp = fma(y_46_re, fma(-1.0, Float64(Float64(t_1 * t_2) / t_0), fma(y_46_re, fma(-1.0, Float64(Float64(t_1 * t_6) / t_0), fma(-0.5, Float64(t_3 / t_0), fma(0.5, Float64(Float64(t_4 * t_2) / t_0), Float64(y_46_re * Float64(Float64(-0.16666666666666666 * Float64(Float64(t_5 * (atan(x_46_im, x_46_re) ^ 3.0)) / t_0)) + Float64(Float64(-0.16666666666666666 * Float64(Float64((t_1 ^ 3.0) * t_2) / t_0)) + fma(0.5, Float64(Float64(t_1 * t_3) / t_0), Float64(0.5 * Float64(Float64(t_4 * t_6) / t_0))))))))), Float64(t_6 / t_0))), Float64(t_2 / t_0));
	else
		tmp = Float64(Float64(exp(Float64(y_46_re * log(x_46_im))) / t_0) * sin(fma(y_46_im, log(x_46_im), 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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(-1.0 * y$46$im), $MachinePrecision] * t$95$1), $MachinePrecision]], $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[Power[t$95$1, 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -5e-310], N[(y$46$re * N[(-1.0 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(y$46$re * N[(-1.0 * N[(N[(t$95$1 * t$95$6), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(-0.5 * N[(t$95$3 / t$95$0), $MachinePrecision] + N[(0.5 * N[(N[(t$95$4 * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(y$46$re * N[(N[(-0.16666666666666666 * N[(N[(t$95$5 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(0.5 * N[(N[(t$95$4 * t$95$6), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$6 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Exp[N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -4.999999999999985e-310

   1. Initial program 39.2%

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

   3. Taylor expanded in y.re around 0

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

   5. Applied rewrites 8.7%

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

   6. Taylor expanded in x.re around inf

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

   8. Applied rewrites 16.5%

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

   if -4.999999999999985e-310 < x.im

   1. Initial program 42.4%

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

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

      3. lower-/.f64 N/A

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

      4. lower-exp.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. lower-exp.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift-atan2.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 60.1%

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

4. Final simplification 38.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \mathsf{fma}\left(-1, \frac{\log \left(\frac{1}{x.re}\right) \cdot \sin \left(\left(-1 \cdot y.im\right) \cdot \log \left(\frac{1}{x.re}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, \mathsf{fma}\left(y.re, \mathsf{fma}\left(-1, \frac{\log \left(\frac{1}{x.re}\right) \cdot \left(\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), 0.5 \cdot \pi\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, \mathsf{fma}\left(-0.5, \frac{\sin \left(\left(-1 \cdot y.im\right) \cdot \log \left(\frac{1}{x.re}\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, \mathsf{fma}\left(0.5, \frac{{\log \left(\frac{1}{x.re}\right)}^{2} \cdot \sin \left(\left(-1 \cdot y.im\right) \cdot \log \left(\frac{1}{x.re}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, y.re \cdot \left(-0.16666666666666666 \cdot \frac{\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), 0.5 \cdot \pi\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} + \left(-0.16666666666666666 \cdot \frac{{\log \left(\frac{1}{x.re}\right)}^{3} \cdot \sin \left(\left(-1 \cdot y.im\right) \cdot \log \left(\frac{1}{x.re}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} + \mathsf{fma}\left(0.5, \frac{\log \left(\frac{1}{x.re}\right) \cdot \left(\sin \left(\left(-1 \cdot y.im\right) \cdot \log \left(\frac{1}{x.re}\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, 0.5 \cdot \frac{{\log \left(\frac{1}{x.re}\right)}^{2} \cdot \left(\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), 0.5 \cdot \pi\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right)\right)\right)\right), \frac{\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), 0.5 \cdot \pi\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right), \frac{\sin \left(\left(-1 \cdot y.im\right) \cdot \log \left(\frac{1}{x.re}\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{y.re \cdot \log x.im}}{e^{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 12: 19.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \log \left(\frac{1}{x.re}\right)\\ t_2 := \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, 0.5 \cdot \pi\right)\right)\\ t_3 := \sin \left(\left(-1 \cdot y.im\right) \cdot t\_1\right)\\ t_4 := t\_3 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\\ t_5 := t\_2 \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_6 := {t\_1}^{2}\\ \mathsf{fma}\left(y.re, \mathsf{fma}\left(-1, \frac{t\_1 \cdot t\_3}{t\_0}, \mathsf{fma}\left(y.re, \mathsf{fma}\left(-1, \frac{t\_1 \cdot t\_5}{t\_0}, \mathsf{fma}\left(-0.5, \frac{t\_4}{t\_0}, \mathsf{fma}\left(0.5, \frac{t\_6 \cdot t\_3}{t\_0}, y.re \cdot \left(-0.16666666666666666 \cdot \frac{t\_2 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}}{t\_0} + \left(-0.16666666666666666 \cdot \frac{{t\_1}^{3} \cdot t\_3}{t\_0} + \mathsf{fma}\left(0.5, \frac{t\_1 \cdot t\_4}{t\_0}, 0.5 \cdot \frac{t\_6 \cdot t\_5}{t\_0}\right)\right)\right)\right)\right)\right), \frac{t\_5}{t\_0}\right)\right), \frac{t\_3}{t\_0}\right) \end{array} \end{array} \]

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 (log (/ 1.0 x.re)))
        (t_2 (sin (fma -1.0 (* y.im t_1) (* 0.5 PI))))
        (t_3 (sin (* (* -1.0 y.im) t_1)))
        (t_4 (* t_3 (pow (atan2 x.im x.re) 2.0)))
        (t_5 (* t_2 (atan2 x.im x.re)))
        (t_6 (pow t_1 2.0)))
   (fma
    y.re
    (fma
     -1.0
     (/ (* t_1 t_3) t_0)
     (fma
      y.re
      (fma
       -1.0
       (/ (* t_1 t_5) t_0)
       (fma
        -0.5
        (/ t_4 t_0)
        (fma
         0.5
         (/ (* t_6 t_3) t_0)
         (*
          y.re
          (+
           (* -0.16666666666666666 (/ (* t_2 (pow (atan2 x.im x.re) 3.0)) t_0))
           (+
            (* -0.16666666666666666 (/ (* (pow t_1 3.0) t_3) t_0))
            (fma 0.5 (/ (* t_1 t_4) t_0) (* 0.5 (/ (* t_6 t_5) t_0)))))))))
      (/ t_5 t_0)))
    (/ t_3 t_0))))

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 = log((1.0 / x_46_re));
	double t_2 = sin(fma(-1.0, (y_46_im * t_1), (0.5 * ((double) M_PI))));
	double t_3 = sin(((-1.0 * y_46_im) * t_1));
	double t_4 = t_3 * pow(atan2(x_46_im, x_46_re), 2.0);
	double t_5 = t_2 * atan2(x_46_im, x_46_re);
	double t_6 = pow(t_1, 2.0);
	return fma(y_46_re, fma(-1.0, ((t_1 * t_3) / t_0), fma(y_46_re, fma(-1.0, ((t_1 * t_5) / t_0), fma(-0.5, (t_4 / t_0), fma(0.5, ((t_6 * t_3) / t_0), (y_46_re * ((-0.16666666666666666 * ((t_2 * pow(atan2(x_46_im, x_46_re), 3.0)) / t_0)) + ((-0.16666666666666666 * ((pow(t_1, 3.0) * t_3) / t_0)) + fma(0.5, ((t_1 * t_4) / t_0), (0.5 * ((t_6 * t_5) / t_0))))))))), (t_5 / t_0))), (t_3 / t_0));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_1 = log(Float64(1.0 / x_46_re))
	t_2 = sin(fma(-1.0, Float64(y_46_im * t_1), Float64(0.5 * pi)))
	t_3 = sin(Float64(Float64(-1.0 * y_46_im) * t_1))
	t_4 = Float64(t_3 * (atan(x_46_im, x_46_re) ^ 2.0))
	t_5 = Float64(t_2 * atan(x_46_im, x_46_re))
	t_6 = t_1 ^ 2.0
	return fma(y_46_re, fma(-1.0, Float64(Float64(t_1 * t_3) / t_0), fma(y_46_re, fma(-1.0, Float64(Float64(t_1 * t_5) / t_0), fma(-0.5, Float64(t_4 / t_0), fma(0.5, Float64(Float64(t_6 * t_3) / t_0), Float64(y_46_re * Float64(Float64(-0.16666666666666666 * Float64(Float64(t_2 * (atan(x_46_im, x_46_re) ^ 3.0)) / t_0)) + Float64(Float64(-0.16666666666666666 * Float64(Float64((t_1 ^ 3.0) * t_3) / t_0)) + fma(0.5, Float64(Float64(t_1 * t_4) / t_0), Float64(0.5 * Float64(Float64(t_6 * t_5) / t_0))))))))), Float64(t_5 / t_0))), Float64(t_3 / t_0))
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[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(-1.0 * y$46$im), $MachinePrecision] * t$95$1), $MachinePrecision]], $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[(t$95$2 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Power[t$95$1, 2.0], $MachinePrecision]}, N[(y$46$re * N[(-1.0 * N[(N[(t$95$1 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(y$46$re * N[(-1.0 * N[(N[(t$95$1 * t$95$5), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(-0.5 * N[(t$95$4 / t$95$0), $MachinePrecision] + N[(0.5 * N[(N[(t$95$6 * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(y$46$re * N[(N[(-0.16666666666666666 * N[(N[(t$95$2 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.16666666666666666 * N[(N[(N[Power[t$95$1, 3.0], $MachinePrecision] * t$95$3), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$1 * t$95$4), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(0.5 * N[(N[(t$95$6 * t$95$5), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$5 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t$95$3 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in y.re around 0

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

5. Applied rewrites 7.4%

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

6. Taylor expanded in x.re around inf

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

8. Applied rewrites 19.9%

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

9. Final simplification 19.9%

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

Alternative 13: 18.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := \log \left(\frac{1}{x.re}\right)\\ t_2 := \sin \left(\left(-1 \cdot y.im\right) \cdot t\_1\right)\\ \mathsf{fma}\left(y.re, \mathsf{fma}\left(-1, \frac{t\_1 \cdot t\_2}{t\_0}, \frac{\sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, 0.5 \cdot \pi\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}{t\_0}\right), \frac{t\_2}{t\_0}\right) \end{array} \end{array} \]

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 (log (/ 1.0 x.re)))
        (t_2 (sin (* (* -1.0 y.im) t_1))))
   (fma
    y.re
    (fma
     -1.0
     (/ (* t_1 t_2) t_0)
     (/ (* (sin (fma -1.0 (* y.im t_1) (* 0.5 PI))) (atan2 x.im x.re)) t_0))
    (/ t_2 t_0))))

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 = log((1.0 / x_46_re));
	double t_2 = sin(((-1.0 * y_46_im) * t_1));
	return fma(y_46_re, fma(-1.0, ((t_1 * t_2) / t_0), ((sin(fma(-1.0, (y_46_im * t_1), (0.5 * ((double) M_PI)))) * atan2(x_46_im, x_46_re)) / t_0)), (t_2 / t_0));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_1 = log(Float64(1.0 / x_46_re))
	t_2 = sin(Float64(Float64(-1.0 * y_46_im) * t_1))
	return fma(y_46_re, fma(-1.0, Float64(Float64(t_1 * t_2) / t_0), Float64(Float64(sin(fma(-1.0, Float64(y_46_im * t_1), Float64(0.5 * pi))) * atan(x_46_im, x_46_re)) / t_0)), Float64(t_2 / t_0))
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[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(-1.0 * y$46$im), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision]}, N[(y$46$re * N[(-1.0 * N[(N[(t$95$1 * t$95$2), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(N[(N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / t$95$0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in y.re around 0

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

5. Applied rewrites 7.4%

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

6. Taylor expanded in x.re around inf

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

8. Applied rewrites 19.9%

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

9. Taylor expanded in y.re around 0

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

11. Applied rewrites 19.6%

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

12. Final simplification 19.6%

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

Alternative 14: 12.4% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{1}{x.re}\right)\\ t_1 := \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_0, 0.5 \cdot \pi\right)\right)\\ t_2 := {t\_0}^{2}\\ t_3 := t\_1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := \sin \left(\left(-1 \cdot y.im\right) \cdot t\_0\right)\\ t_5 := t\_4 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\\ t_6 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \left(-1 \cdot {y.re}^{3}\right) \cdot \left(\left(\left(--0.16666666666666666\right) \cdot \frac{t\_1 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}}{t\_6} - \left(-0.16666666666666666 \cdot \frac{{t\_0}^{3} \cdot t\_4}{t\_6} + \mathsf{fma}\left(0.5, \frac{t\_0 \cdot t\_5}{t\_6}, 0.5 \cdot \frac{t\_2 \cdot t\_3}{t\_6}\right)\right)\right) - \frac{\mathsf{fma}\left(-1, \frac{t\_0 \cdot t\_3}{t\_6}, \mathsf{fma}\left(-1, \frac{\mathsf{fma}\left(-1, \mathsf{fma}\left(-1, \frac{t\_0 \cdot t\_4}{t\_6}, \frac{t\_3}{t\_6}\right), \frac{t\_4}{\left(-1 \cdot y.re\right) \cdot t\_6}\right)}{y.re}, \mathsf{fma}\left(-0.5, \frac{t\_5}{t\_6}, 0.5 \cdot \frac{t\_2 \cdot t\_4}{t\_6}\right)\right)\right)}{y.re}\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (/ 1.0 x.re)))
        (t_1 (sin (fma -1.0 (* y.im t_0) (* 0.5 PI))))
        (t_2 (pow t_0 2.0))
        (t_3 (* t_1 (atan2 x.im x.re)))
        (t_4 (sin (* (* -1.0 y.im) t_0)))
        (t_5 (* t_4 (pow (atan2 x.im x.re) 2.0)))
        (t_6 (exp (* y.im (atan2 x.im x.re)))))
   (*
    (* -1.0 (pow y.re 3.0))
    (-
     (-
      (* (- -0.16666666666666666) (/ (* t_1 (pow (atan2 x.im x.re) 3.0)) t_6))
      (+
       (* -0.16666666666666666 (/ (* (pow t_0 3.0) t_4) t_6))
       (fma 0.5 (/ (* t_0 t_5) t_6) (* 0.5 (/ (* t_2 t_3) t_6)))))
     (/
      (fma
       -1.0
       (/ (* t_0 t_3) t_6)
       (fma
        -1.0
        (/
         (fma
          -1.0
          (fma -1.0 (/ (* t_0 t_4) t_6) (/ t_3 t_6))
          (/ t_4 (* (* -1.0 y.re) t_6)))
         y.re)
        (fma -0.5 (/ t_5 t_6) (* 0.5 (/ (* t_2 t_4) t_6)))))
      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((1.0 / x_46_re));
	double t_1 = sin(fma(-1.0, (y_46_im * t_0), (0.5 * ((double) M_PI))));
	double t_2 = pow(t_0, 2.0);
	double t_3 = t_1 * atan2(x_46_im, x_46_re);
	double t_4 = sin(((-1.0 * y_46_im) * t_0));
	double t_5 = t_4 * pow(atan2(x_46_im, x_46_re), 2.0);
	double t_6 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	return (-1.0 * pow(y_46_re, 3.0)) * (((-(-0.16666666666666666) * ((t_1 * pow(atan2(x_46_im, x_46_re), 3.0)) / t_6)) - ((-0.16666666666666666 * ((pow(t_0, 3.0) * t_4) / t_6)) + fma(0.5, ((t_0 * t_5) / t_6), (0.5 * ((t_2 * t_3) / t_6))))) - (fma(-1.0, ((t_0 * t_3) / t_6), fma(-1.0, (fma(-1.0, fma(-1.0, ((t_0 * t_4) / t_6), (t_3 / t_6)), (t_4 / ((-1.0 * y_46_re) * t_6))) / y_46_re), fma(-0.5, (t_5 / t_6), (0.5 * ((t_2 * t_4) / t_6))))) / y_46_re));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(Float64(1.0 / x_46_re))
	t_1 = sin(fma(-1.0, Float64(y_46_im * t_0), Float64(0.5 * pi)))
	t_2 = t_0 ^ 2.0
	t_3 = Float64(t_1 * atan(x_46_im, x_46_re))
	t_4 = sin(Float64(Float64(-1.0 * y_46_im) * t_0))
	t_5 = Float64(t_4 * (atan(x_46_im, x_46_re) ^ 2.0))
	t_6 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	return Float64(Float64(-1.0 * (y_46_re ^ 3.0)) * Float64(Float64(Float64(Float64(-(-0.16666666666666666)) * Float64(Float64(t_1 * (atan(x_46_im, x_46_re) ^ 3.0)) / t_6)) - Float64(Float64(-0.16666666666666666 * Float64(Float64((t_0 ^ 3.0) * t_4) / t_6)) + fma(0.5, Float64(Float64(t_0 * t_5) / t_6), Float64(0.5 * Float64(Float64(t_2 * t_3) / t_6))))) - Float64(fma(-1.0, Float64(Float64(t_0 * t_3) / t_6), fma(-1.0, Float64(fma(-1.0, fma(-1.0, Float64(Float64(t_0 * t_4) / t_6), Float64(t_3 / t_6)), Float64(t_4 / Float64(Float64(-1.0 * y_46_re) * t_6))) / y_46_re), fma(-0.5, Float64(t_5 / t_6), Float64(0.5 * Float64(Float64(t_2 * t_4) / t_6))))) / 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[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-1.0 * N[(y$46$im * t$95$0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(-1.0 * y$46$im), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(-1.0 * N[Power[y$46$re, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[((--0.16666666666666666) * N[(N[(t$95$1 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.16666666666666666 * N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$0 * t$95$5), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(-1.0 * N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(-1.0 * N[(N[(-1.0 * N[(-1.0 * N[(N[(t$95$0 * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(t$95$3 / t$95$6), $MachinePrecision]), $MachinePrecision] + N[(t$95$4 / N[(N[(-1.0 * y$46$re), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision] + N[(-0.5 * N[(t$95$5 / t$95$6), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in y.re around 0

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

5. Applied rewrites 7.4%

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

6. Taylor expanded in x.re around inf

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

8. Applied rewrites 19.9%

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

9. Taylor expanded in y.re around -inf

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

11. Applied rewrites 10.7%

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

12. Final simplification 10.7%

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

Alternative 15: 10.9% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\frac{1}{x.re}\right)\\ t_1 := \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_0, 0.5 \cdot \pi\right)\right)\\ t_2 := {t\_0}^{2}\\ t_3 := t\_1 \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := \sin \left(\left(-1 \cdot y.im\right) \cdot t\_0\right)\\ t_5 := t\_4 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\\ t_6 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \left(-1 \cdot {y.re}^{3}\right) \cdot \left(\left(\left(--0.16666666666666666\right) \cdot \frac{t\_1 \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}}{t\_6} - \left(-0.16666666666666666 \cdot \frac{{t\_0}^{3} \cdot t\_4}{t\_6} + \mathsf{fma}\left(0.5, \frac{t\_0 \cdot t\_5}{t\_6}, 0.5 \cdot \frac{t\_2 \cdot t\_3}{t\_6}\right)\right)\right) - \frac{\frac{t\_0 \cdot t\_4}{\left(-1 \cdot y.re\right) \cdot t\_6} + \mathsf{fma}\left(-1, \frac{t\_0 \cdot t\_3}{t\_6}, \mathsf{fma}\left(-0.5, \frac{t\_5}{t\_6}, \mathsf{fma}\left(0.5, \frac{t\_2 \cdot t\_4}{t\_6}, \frac{t\_3}{y.re \cdot t\_6}\right)\right)\right)}{y.re}\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (/ 1.0 x.re)))
        (t_1 (sin (fma -1.0 (* y.im t_0) (* 0.5 PI))))
        (t_2 (pow t_0 2.0))
        (t_3 (* t_1 (atan2 x.im x.re)))
        (t_4 (sin (* (* -1.0 y.im) t_0)))
        (t_5 (* t_4 (pow (atan2 x.im x.re) 2.0)))
        (t_6 (exp (* y.im (atan2 x.im x.re)))))
   (*
    (* -1.0 (pow y.re 3.0))
    (-
     (-
      (* (- -0.16666666666666666) (/ (* t_1 (pow (atan2 x.im x.re) 3.0)) t_6))
      (+
       (* -0.16666666666666666 (/ (* (pow t_0 3.0) t_4) t_6))
       (fma 0.5 (/ (* t_0 t_5) t_6) (* 0.5 (/ (* t_2 t_3) t_6)))))
     (/
      (+
       (/ (* t_0 t_4) (* (* -1.0 y.re) t_6))
       (fma
        -1.0
        (/ (* t_0 t_3) t_6)
        (fma
         -0.5
         (/ t_5 t_6)
         (fma 0.5 (/ (* t_2 t_4) t_6) (/ t_3 (* y.re t_6))))))
      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((1.0 / x_46_re));
	double t_1 = sin(fma(-1.0, (y_46_im * t_0), (0.5 * ((double) M_PI))));
	double t_2 = pow(t_0, 2.0);
	double t_3 = t_1 * atan2(x_46_im, x_46_re);
	double t_4 = sin(((-1.0 * y_46_im) * t_0));
	double t_5 = t_4 * pow(atan2(x_46_im, x_46_re), 2.0);
	double t_6 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	return (-1.0 * pow(y_46_re, 3.0)) * (((-(-0.16666666666666666) * ((t_1 * pow(atan2(x_46_im, x_46_re), 3.0)) / t_6)) - ((-0.16666666666666666 * ((pow(t_0, 3.0) * t_4) / t_6)) + fma(0.5, ((t_0 * t_5) / t_6), (0.5 * ((t_2 * t_3) / t_6))))) - ((((t_0 * t_4) / ((-1.0 * y_46_re) * t_6)) + fma(-1.0, ((t_0 * t_3) / t_6), fma(-0.5, (t_5 / t_6), fma(0.5, ((t_2 * t_4) / t_6), (t_3 / (y_46_re * t_6)))))) / y_46_re));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(Float64(1.0 / x_46_re))
	t_1 = sin(fma(-1.0, Float64(y_46_im * t_0), Float64(0.5 * pi)))
	t_2 = t_0 ^ 2.0
	t_3 = Float64(t_1 * atan(x_46_im, x_46_re))
	t_4 = sin(Float64(Float64(-1.0 * y_46_im) * t_0))
	t_5 = Float64(t_4 * (atan(x_46_im, x_46_re) ^ 2.0))
	t_6 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	return Float64(Float64(-1.0 * (y_46_re ^ 3.0)) * Float64(Float64(Float64(Float64(-(-0.16666666666666666)) * Float64(Float64(t_1 * (atan(x_46_im, x_46_re) ^ 3.0)) / t_6)) - Float64(Float64(-0.16666666666666666 * Float64(Float64((t_0 ^ 3.0) * t_4) / t_6)) + fma(0.5, Float64(Float64(t_0 * t_5) / t_6), Float64(0.5 * Float64(Float64(t_2 * t_3) / t_6))))) - Float64(Float64(Float64(Float64(t_0 * t_4) / Float64(Float64(-1.0 * y_46_re) * t_6)) + fma(-1.0, Float64(Float64(t_0 * t_3) / t_6), fma(-0.5, Float64(t_5 / t_6), fma(0.5, Float64(Float64(t_2 * t_4) / t_6), Float64(t_3 / Float64(y_46_re * t_6)))))) / 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[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-1.0 * N[(y$46$im * t$95$0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(-1.0 * y$46$im), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, N[(N[(-1.0 * N[Power[y$46$re, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[((--0.16666666666666666) * N[(N[(t$95$1 * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.16666666666666666 * N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$0 * t$95$5), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(t$95$0 * t$95$4), $MachinePrecision] / N[(N[(-1.0 * y$46$re), $MachinePrecision] * t$95$6), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(-0.5 * N[(t$95$5 / t$95$6), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(t$95$3 / N[(y$46$re * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in y.re around 0

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

5. Applied rewrites 7.4%

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

6. Taylor expanded in x.re around inf

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

8. Applied rewrites 19.9%

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

9. Taylor expanded in y.re around -inf

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

11. Applied rewrites 9.4%

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

12. Final simplification 9.4%

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

Alternative 16: 9.6% accurate, N/A× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (/ 1.0 x.re)))
        (t_1 (sin (fma -1.0 (* y.im t_0) (* 0.5 PI))))
        (t_2 (pow t_0 2.0))
        (t_3 (* t_1 (atan2 x.im x.re)))
        (t_4 (sin (* (* -1.0 y.im) t_0)))
        (t_5 (pow (atan2 x.im x.re) 2.0))
        (t_6 (exp (* y.im (atan2 x.im x.re))))
        (t_7 (pow (atan2 x.im x.re) 3.0))
        (t_8 (* t_4 t_5)))
   (*
    (* -1.0 (pow y.re 3.0))
    (-
     (-
      (* (- -0.16666666666666666) (/ (* t_1 t_7) t_6))
      (+
       (* -0.16666666666666666 (/ (* (pow t_0 3.0) t_4) t_6))
       (fma 0.5 (/ (* t_0 t_8) t_6) (* 0.5 (/ (* t_2 t_3) t_6)))))
     (/
      (+
       (/
        (* t_0 t_4)
        (*
         (* -1.0 y.re)
         (+
          1.0
          (*
           y.im
           (fma
            y.im
            (fma 0.16666666666666666 (* y.im t_7) (* 0.5 t_5))
            (atan2 x.im x.re))))))
       (fma
        -1.0
        (/ (* t_0 t_3) t_6)
        (fma
         -0.5
         (/ t_8 t_6)
         (fma 0.5 (/ (* t_2 t_4) t_6) (/ t_3 (* y.re t_6))))))
      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((1.0 / x_46_re));
	double t_1 = sin(fma(-1.0, (y_46_im * t_0), (0.5 * ((double) M_PI))));
	double t_2 = pow(t_0, 2.0);
	double t_3 = t_1 * atan2(x_46_im, x_46_re);
	double t_4 = sin(((-1.0 * y_46_im) * t_0));
	double t_5 = pow(atan2(x_46_im, x_46_re), 2.0);
	double t_6 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double t_7 = pow(atan2(x_46_im, x_46_re), 3.0);
	double t_8 = t_4 * t_5;
	return (-1.0 * pow(y_46_re, 3.0)) * (((-(-0.16666666666666666) * ((t_1 * t_7) / t_6)) - ((-0.16666666666666666 * ((pow(t_0, 3.0) * t_4) / t_6)) + fma(0.5, ((t_0 * t_8) / t_6), (0.5 * ((t_2 * t_3) / t_6))))) - ((((t_0 * t_4) / ((-1.0 * y_46_re) * (1.0 + (y_46_im * fma(y_46_im, fma(0.16666666666666666, (y_46_im * t_7), (0.5 * t_5)), atan2(x_46_im, x_46_re)))))) + fma(-1.0, ((t_0 * t_3) / t_6), fma(-0.5, (t_8 / t_6), fma(0.5, ((t_2 * t_4) / t_6), (t_3 / (y_46_re * t_6)))))) / y_46_re));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(Float64(1.0 / x_46_re))
	t_1 = sin(fma(-1.0, Float64(y_46_im * t_0), Float64(0.5 * pi)))
	t_2 = t_0 ^ 2.0
	t_3 = Float64(t_1 * atan(x_46_im, x_46_re))
	t_4 = sin(Float64(Float64(-1.0 * y_46_im) * t_0))
	t_5 = atan(x_46_im, x_46_re) ^ 2.0
	t_6 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_7 = atan(x_46_im, x_46_re) ^ 3.0
	t_8 = Float64(t_4 * t_5)
	return Float64(Float64(-1.0 * (y_46_re ^ 3.0)) * Float64(Float64(Float64(Float64(-(-0.16666666666666666)) * Float64(Float64(t_1 * t_7) / t_6)) - Float64(Float64(-0.16666666666666666 * Float64(Float64((t_0 ^ 3.0) * t_4) / t_6)) + fma(0.5, Float64(Float64(t_0 * t_8) / t_6), Float64(0.5 * Float64(Float64(t_2 * t_3) / t_6))))) - Float64(Float64(Float64(Float64(t_0 * t_4) / Float64(Float64(-1.0 * y_46_re) * Float64(1.0 + Float64(y_46_im * fma(y_46_im, fma(0.16666666666666666, Float64(y_46_im * t_7), Float64(0.5 * t_5)), atan(x_46_im, x_46_re)))))) + fma(-1.0, Float64(Float64(t_0 * t_3) / t_6), fma(-0.5, Float64(t_8 / t_6), fma(0.5, Float64(Float64(t_2 * t_4) / t_6), Float64(t_3 / Float64(y_46_re * t_6)))))) / 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[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(-1.0 * N[(y$46$im * t$95$0), $MachinePrecision] + N[(0.5 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[t$95$0, 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(-1.0 * y$46$im), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]}, Block[{t$95$8 = N[(t$95$4 * t$95$5), $MachinePrecision]}, N[(N[(-1.0 * N[Power[y$46$re, 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[(N[((--0.16666666666666666) * N[(N[(t$95$1 * t$95$7), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.16666666666666666 * N[(N[(N[Power[t$95$0, 3.0], $MachinePrecision] * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(N[(t$95$0 * t$95$8), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(t$95$0 * t$95$4), $MachinePrecision] / N[(N[(-1.0 * y$46$re), $MachinePrecision] * N[(1.0 + N[(y$46$im * N[(y$46$im * N[(0.16666666666666666 * N[(y$46$im * t$95$7), $MachinePrecision] + N[(0.5 * t$95$5), $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(N[(t$95$0 * t$95$3), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(-0.5 * N[(t$95$8 / t$95$6), $MachinePrecision] + N[(0.5 * N[(N[(t$95$2 * t$95$4), $MachinePrecision] / t$95$6), $MachinePrecision] + N[(t$95$3 / N[(y$46$re * t$95$6), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Initial program 40.8%

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

3. Taylor expanded in y.re around 0

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

5. Applied rewrites 7.4%

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

6. Taylor expanded in x.re around inf

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

8. Applied rewrites 19.9%

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

9. Taylor expanded in y.re around -inf

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

11. Applied rewrites 9.4%

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

12. Taylor expanded in y.im around 0

    \[\leadsto -1 \cdot \left({y.re}^{3} \cdot \left(-1 \cdot \left(\frac{-1}{6} \cdot \frac{\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), \frac{1}{2} \cdot \pi\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} + \left(\frac{-1}{6} \cdot \frac{{\log \left(\frac{1}{x.re}\right)}^{3} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} + \mathsf{fma}\left(\frac{1}{2}, \frac{\log \left(\frac{1}{x.re}\right) \cdot \left(\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, \frac{1}{2} \cdot \frac{{\log \left(\frac{1}{x.re}\right)}^{2} \cdot \left(\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), \frac{1}{2} \cdot \pi\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right) + -1 \cdot \frac{-1 \cdot \frac{\log \left(\frac{1}{x.re}\right) \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)}{y.re \cdot \left(1 + y.im \cdot \left(y.im \cdot \left(\frac{1}{6} \cdot \left(y.im \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \frac{1}{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} + \mathsf{fma}\left(-1, \frac{\log \left(\frac{1}{x.re}\right) \cdot \left(\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), \frac{1}{2} \cdot \pi\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, \mathsf{fma}\left(\frac{-1}{2}, \frac{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{2}}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, \mathsf{fma}\left(\frac{1}{2}, \frac{{\log \left(\frac{1}{x.re}\right)}^{2} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}, \frac{\sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), \frac{1}{2} \cdot \pi\right)\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.re \cdot e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\right)\right)}{y.re}\right)\right) \]
13. Step-by-step derivation

    1. lower-+.f64 N/A

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

    2. lower-*.f64 N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lift-pow.f64 N/A

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

    7. lift-atan2.f64 N/A

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

    8. lower-*.f64 N/A

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

    9. lift-pow.f64 N/A

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

    10. lift-atan2.f64 N/A

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

    11. lift-atan2.f64 7.5

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

14. Applied rewrites 7.5%

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

15. Final simplification 7.5%

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

Reproduce

herbie shell --seed 2025065 
(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)))))