powComplex, imaginary part

Percentage Accurate: 40.5% → 79.8%

Time: 10.5s

Alternatives: 21

Speedup: 1.4×

• 35.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (sqrt (+ (* x.re x.re) (* x.im x.im))))))
   (*
    (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
    (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * sin(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	return Math.exp(((t_0 * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(((t_0 * y_46_im) + (Math.atan2(x_46_im, x_46_re) * y_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	return Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(Float64(t_0 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

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

Initial Program: 40.5% 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: 79.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.im \leq 2 \cdot 10^{+207}:\\ \;\;\;\;e^{t\_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_0 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= y.im 2e+207)
     (*
      (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
      (sin (+ (* t_0 y.im) (* (atan2 x.im x.re) y.re))))
     (*
      (exp (- (* y.im (atan2 x.im x.re))))
      (sin (* 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 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_im <= 2e+207) {
		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)));
	} else {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_im <= 2e+207) {
		tmp = 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)));
	} else {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.hypot(x_46_re, x_46_im))
	tmp = 0
	if y_46_im <= 2e+207:
		tmp = 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)))
	else:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	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))
	tmp = 0.0
	if (y_46_im <= 2e+207)
		tmp = 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))));
	else
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im));
	tmp = 0.0;
	if (y_46_im <= 2e+207)
		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)));
	else
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, 2e+207], 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], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 2.0000000000000001e207

   1. Initial program 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

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

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

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

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. sqr-neg-rev N/A

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

      11. sqr-neg-rev N/A

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

      12. lower-hypot.f64 40.5

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

   3. Applied rewrites 40.5%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. sqr-neg-rev N/A

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

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

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

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

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

      7. lift-*.f64 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. sqr-neg-rev N/A

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

      11. sqr-neg-rev N/A

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

      12. lower-hypot.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if 2.0000000000000001e207 < y.im

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 40.2

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

   9. Applied rewrites 40.2%

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

Alternative 2: 64.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := \log \left(-1 \cdot x.re\right)\\ t_3 := \log \left(-x.im\right)\\ \mathbf{if}\;x.re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \mathbf{elif}\;x.re \leq 4.6 \cdot 10^{-161}:\\ \;\;\;\;e^{t\_3 \cdot y.re - t\_0} \cdot \sin \left(t\_3 \cdot y.im + t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log x.re \cdot y.im\right)\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re} - \log x.re \cdot y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* (atan2 x.im x.re) y.re))
        (t_2 (log (* -1.0 x.re)))
        (t_3 (log (- x.im))))
   (if (<= x.re -1e-310)
     (* (exp (- (* t_2 y.re) t_0)) (sin (+ (* t_2 y.im) t_1)))
     (if (<= x.re 4.6e-161)
       (* (exp (- (* t_3 y.re) t_0)) (sin (+ (* t_3 y.im) t_1)))
       (/
        (sin (fma (atan2 x.im x.re) y.re (* (log x.re) y.im)))
        (exp (- (* y.im (atan2 x.im x.re)) (* (log 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 t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = log((-1.0 * x_46_re));
	double t_3 = log(-x_46_im);
	double tmp;
	if (x_46_re <= -1e-310) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_1));
	} else if (x_46_re <= 4.6e-161) {
		tmp = exp(((t_3 * y_46_re) - t_0)) * sin(((t_3 * y_46_im) + t_1));
	} else {
		tmp = sin(fma(atan2(x_46_im, x_46_re), y_46_re, (log(x_46_re) * y_46_im))) / exp(((y_46_im * atan2(x_46_im, x_46_re)) - (log(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)
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = log(Float64(-1.0 * x_46_re))
	t_3 = log(Float64(-x_46_im))
	tmp = 0.0
	if (x_46_re <= -1e-310)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + t_1)));
	elseif (x_46_re <= 4.6e-161)
		tmp = Float64(exp(Float64(Float64(t_3 * y_46_re) - t_0)) * sin(Float64(Float64(t_3 * y_46_im) + t_1)));
	else
		tmp = Float64(sin(fma(atan(x_46_im, x_46_re), y_46_re, Float64(log(x_46_re) * y_46_im))) / exp(Float64(Float64(y_46_im * atan(x_46_im, x_46_re)) - Float64(log(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]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[(-x$46$im)], $MachinePrecision]}, If[LessEqual[x$46$re, -1e-310], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.6e-161], N[(N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$3 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] - N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -9.999999999999969e-311

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 18.5

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

   4. Applied rewrites 18.5%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 34.1

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

   7. Applied rewrites 34.1%

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

   if -9.999999999999969e-311 < x.re < 4.6e-161

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 17.8

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 31.1

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

   7. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 31.1

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

   9. Applied rewrites 31.1%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 31.1

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

   11. Applied rewrites 31.1%

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

   if 4.6e-161 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   6. Applied rewrites 32.3%

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

Alternative 3: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(-x.im\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{if}\;x.re \leq -1.4 \cdot 10^{-184}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 6.8 \cdot 10^{-307}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log x.im, y.im, t\_3\right)\right) \cdot e^{\log x.im \cdot y.re - t\_1}\\ \mathbf{elif}\;x.re \leq 4.6 \cdot 10^{-161}:\\ \;\;\;\;e^{t\_2 \cdot y.re - t\_0} \cdot \sin \left(t\_2 \cdot y.im + t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log x.re \cdot y.im\right)\right)}{e^{t\_1 - \log x.re \cdot y.re}}\\ \end{array} \end{array} \]

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.im (atan2 x.im x.re)))
        (t_2 (log (- x.im)))
        (t_3 (* (atan2 x.im x.re) y.re)))
   (if (<= x.re -1.4e-184)
     (*
      (exp (- (* (log (* -1.0 x.re)) y.re) t_0))
      (sin (* y.re (atan2 x.im x.re))))
     (if (<= x.re 6.8e-307)
       (* (sin (fma (log x.im) y.im t_3)) (exp (- (* (log x.im) y.re) t_1)))
       (if (<= x.re 4.6e-161)
         (* (exp (- (* t_2 y.re) t_0)) (sin (+ (* t_2 y.im) t_3)))
         (/
          (sin (fma (atan2 x.im x.re) y.re (* (log x.re) y.im)))
          (exp (- t_1 (* (log 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 t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = log(-x_46_im);
	double t_3 = atan2(x_46_im, x_46_re) * y_46_re;
	double tmp;
	if (x_46_re <= -1.4e-184) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (x_46_re <= 6.8e-307) {
		tmp = sin(fma(log(x_46_im), y_46_im, t_3)) * exp(((log(x_46_im) * y_46_re) - t_1));
	} else if (x_46_re <= 4.6e-161) {
		tmp = exp(((t_2 * y_46_re) - t_0)) * sin(((t_2 * y_46_im) + t_3));
	} else {
		tmp = sin(fma(atan2(x_46_im, x_46_re), y_46_re, (log(x_46_re) * y_46_im))) / exp((t_1 - (log(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)
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = log(Float64(-x_46_im))
	t_3 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	tmp = 0.0
	if (x_46_re <= -1.4e-184)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (x_46_re <= 6.8e-307)
		tmp = Float64(sin(fma(log(x_46_im), y_46_im, t_3)) * exp(Float64(Float64(log(x_46_im) * y_46_re) - t_1)));
	elseif (x_46_re <= 4.6e-161)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_0)) * sin(Float64(Float64(t_2 * y_46_im) + t_3)));
	else
		tmp = Float64(sin(fma(atan(x_46_im, x_46_re), y_46_re, Float64(log(x_46_re) * y_46_im))) / exp(Float64(t_1 - Float64(log(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]}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, If[LessEqual[x$46$re, -1.4e-184], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $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], If[LessEqual[x$46$re, 6.8e-307], N[(N[Sin[N[(N[Log[x$46$im], $MachinePrecision] * y$46$im + t$95$3), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.6e-161], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$1 - N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.3999999999999999e-184

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if -1.3999999999999999e-184 < x.re < 6.79999999999999978e-307

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.3

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

   6. Applied rewrites 32.3%

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

   if 6.79999999999999978e-307 < x.re < 4.6e-161

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 17.8

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 31.1

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

   7. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 31.1

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

   9. Applied rewrites 31.1%

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

       1. lift-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. lower-neg.f64 31.1

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

   11. Applied rewrites 31.1%

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

   if 4.6e-161 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   6. Applied rewrites 32.3%

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

Alternative 4: 61.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-x.im\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{if}\;x.re \leq -1.4 \cdot 10^{-184}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 6.8 \cdot 10^{-307}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log x.im, y.im, t\_2\right)\right) \cdot e^{\log x.im \cdot y.re - t\_0}\\ \mathbf{elif}\;x.re \leq 4.6 \cdot 10^{-161}:\\ \;\;\;\;e^{t\_1 \cdot y.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(t\_1, y.im, t\_2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log x.re \cdot y.im\right)\right)}{e^{t\_0 - \log x.re \cdot y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (log (- x.im)))
        (t_2 (* (atan2 x.im x.re) y.re)))
   (if (<= x.re -1.4e-184)
     (*
      (exp (- (* (log (* -1.0 x.re)) y.re) (* (atan2 x.im x.re) y.im)))
      (sin (* y.re (atan2 x.im x.re))))
     (if (<= x.re 6.8e-307)
       (* (sin (fma (log x.im) y.im t_2)) (exp (- (* (log x.im) y.re) t_0)))
       (if (<= x.re 4.6e-161)
         (* (exp (- (* t_1 y.re) t_0)) (sin (fma t_1 y.im t_2)))
         (/
          (sin (fma (atan2 x.im x.re) y.re (* (log x.re) y.im)))
          (exp (- t_0 (* (log 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 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = log(-x_46_im);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_re;
	double tmp;
	if (x_46_re <= -1.4e-184) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (x_46_re <= 6.8e-307) {
		tmp = sin(fma(log(x_46_im), y_46_im, t_2)) * exp(((log(x_46_im) * y_46_re) - t_0));
	} else if (x_46_re <= 4.6e-161) {
		tmp = exp(((t_1 * y_46_re) - t_0)) * sin(fma(t_1, y_46_im, t_2));
	} else {
		tmp = sin(fma(atan2(x_46_im, x_46_re), y_46_re, (log(x_46_re) * y_46_im))) / exp((t_0 - (log(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(y_46_im * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-x_46_im))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	tmp = 0.0
	if (x_46_re <= -1.4e-184)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (x_46_re <= 6.8e-307)
		tmp = Float64(sin(fma(log(x_46_im), y_46_im, t_2)) * exp(Float64(Float64(log(x_46_im) * y_46_re) - t_0)));
	elseif (x_46_re <= 4.6e-161)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_0)) * sin(fma(t_1, y_46_im, t_2)));
	else
		tmp = Float64(sin(fma(atan(x_46_im, x_46_re), y_46_re, Float64(log(x_46_re) * y_46_im))) / exp(Float64(t_0 - Float64(log(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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, If[LessEqual[x$46$re, -1.4e-184], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 6.8e-307], N[(N[Sin[N[(N[Log[x$46$im], $MachinePrecision] * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.6e-161], N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$0 - N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.3999999999999999e-184

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if -1.3999999999999999e-184 < x.re < 6.79999999999999978e-307

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.3

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

   6. Applied rewrites 32.3%

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

   if 6.79999999999999978e-307 < x.re < 4.6e-161

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 17.8

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 31.1

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

   7. Applied rewrites 31.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 31.1

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

      4. lower-neg.f64 N/A

         \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \mathsf{Rewrite=>}\left(lift-*.f64, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)} \cdot \sin \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. lower-neg.f64 N/A

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

      6. lower-neg.f64 N/A

         \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - \mathsf{Rewrite<=}\left(lift-*.f64, \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \cdot \sin \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      7. lower-neg.f64 N/A

         \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \mathsf{Rewrite=>}\left(lift-+.f64, \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. lower-neg.f64 N/A

         \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{Rewrite=>}\left(lift-*.f64, \left(\log \left(-1 \cdot x.im\right) \cdot y.im\right)\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \mathsf{Rewrite=>}\left(lift-*.f64, \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)\right) \]

      10. lower-neg.f64 N/A

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

      11. lower-neg.f64 N/A

          \[\leadsto e^{\log \left(-x.im\right) \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(-1 \cdot x.im\right) \cdot y.im + \mathsf{Rewrite<=}\left(lift-*.f64, \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right) \]

   9. Applied rewrites 31.2%

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

   if 4.6e-161 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   6. Applied rewrites 32.3%

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

Alternative 5: 61.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq -1.4 \cdot 10^{-184}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log x.im, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\log x.im \cdot y.re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, \log x.re \cdot y.im\right)\right)}{e^{t\_0 - \log x.re \cdot y.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re))))
   (if (<= x.re -1.4e-184)
     (*
      (exp (- (* (log (* -1.0 x.re)) y.re) (* (atan2 x.im x.re) y.im)))
      (sin (* y.re (atan2 x.im x.re))))
     (if (<= x.re -1e-310)
       (*
        (sin (fma (log x.im) y.im (* (atan2 x.im x.re) y.re)))
        (exp (- (* (log x.im) y.re) t_0)))
       (/
        (sin (fma (atan2 x.im x.re) y.re (* (log x.re) y.im)))
        (exp (- t_0 (* (log 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 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -1.4e-184) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (x_46_re <= -1e-310) {
		tmp = sin(fma(log(x_46_im), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re))) * exp(((log(x_46_im) * y_46_re) - t_0));
	} else {
		tmp = sin(fma(atan2(x_46_im, x_46_re), y_46_re, (log(x_46_re) * y_46_im))) / exp((t_0 - (log(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(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= -1.4e-184)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (x_46_re <= -1e-310)
		tmp = Float64(sin(fma(log(x_46_im), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))) * exp(Float64(Float64(log(x_46_im) * y_46_re) - t_0)));
	else
		tmp = Float64(sin(fma(atan(x_46_im, x_46_re), y_46_re, Float64(log(x_46_re) * y_46_im))) / exp(Float64(t_0 - Float64(log(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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1.4e-184], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -1e-310], N[(N[Sin[N[(N[Log[x$46$im], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[(N[Log[x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(t$95$0 - N[(N[Log[x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.3999999999999999e-184

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if -1.3999999999999999e-184 < x.re < -9.999999999999969e-311

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.3

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

   6. Applied rewrites 32.3%

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

   if -9.999999999999969e-311 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   6. Applied rewrites 32.3%

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

Alternative 6: 59.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\frac{1}{x.re}\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \sin t\_2\\ \mathbf{if}\;x.re \leq -1.4 \cdot 10^{-184}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot t\_3\\ \mathbf{elif}\;x.re \leq -2.9 \cdot 10^{-295}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log x.im, y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot e^{\log x.im \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)}\right) \cdot y.re - t\_0} \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_1\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, t\_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (/ 1.0 x.re)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (sin t_2)))
   (if (<= x.re -1.4e-184)
     (* (exp (- (* (log (* -1.0 x.re)) y.re) t_0)) t_3)
     (if (<= x.re -2.9e-295)
       (*
        (sin (fma (log x.im) y.im (* (atan2 x.im x.re) y.re)))
        (exp (- (* (log x.im) y.re) (* y.im (atan2 x.im x.re)))))
       (if (<= x.re 7.5e-10)
         (*
          (exp (- (* (log (sqrt (fma x.re x.re (* x.im x.im)))) y.re) t_0))
          t_3)
         (* (exp (* -1.0 (* y.re t_1))) (sin (fma -1.0 (* y.im 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 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log((1.0 / x_46_re));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = sin(t_2);
	double tmp;
	if (x_46_re <= -1.4e-184) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * t_3;
	} else if (x_46_re <= -2.9e-295) {
		tmp = sin(fma(log(x_46_im), y_46_im, (atan2(x_46_im, x_46_re) * y_46_re))) * exp(((log(x_46_im) * y_46_re) - (y_46_im * atan2(x_46_im, x_46_re))));
	} else if (x_46_re <= 7.5e-10) {
		tmp = exp(((log(sqrt(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))) * y_46_re) - t_0)) * t_3;
	} else {
		tmp = exp((-1.0 * (y_46_re * t_1))) * sin(fma(-1.0, (y_46_im * t_1), t_2));
	}
	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 = log(Float64(1.0 / x_46_re))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = sin(t_2)
	tmp = 0.0
	if (x_46_re <= -1.4e-184)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * t_3);
	elseif (x_46_re <= -2.9e-295)
		tmp = Float64(sin(fma(log(x_46_im), y_46_im, Float64(atan(x_46_im, x_46_re) * y_46_re))) * exp(Float64(Float64(log(x_46_im) * y_46_re) - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	elseif (x_46_re <= 7.5e-10)
		tmp = Float64(exp(Float64(Float64(log(sqrt(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * t_3);
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_1))) * sin(fma(-1.0, Float64(y_46_im * 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[t$95$2], $MachinePrecision]}, If[LessEqual[x$46$re, -1.4e-184], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[x$46$re, -2.9e-295], N[(N[Sin[N[(N[Log[x$46$im], $MachinePrecision] * y$46$im + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[x$46$im], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7.5e-10], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.3999999999999999e-184

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if -1.3999999999999999e-184 < x.re < -2.90000000000000015e-295

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 32.3

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

   6. Applied rewrites 32.3%

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

   if -2.90000000000000015e-295 < x.re < 7.49999999999999995e-10

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   if 7.49999999999999995e-10 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 24.7

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

   7. Applied rewrites 24.7%

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

Alternative 7: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t\_1\\ t_3 := \log \left(\sqrt{\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)}\right) \cdot y.re\\ t_4 := \log \left(\frac{1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -1.1 \cdot 10^{-180}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot t\_2\\ \mathbf{elif}\;x.re \leq 5.8 \cdot 10^{-132}:\\ \;\;\;\;\sin \left(\left(-\tan^{-1}_* \frac{x.im}{x.re}\right) \cdot y.re\right) \cdot e^{t\_3 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;e^{t\_3 - t\_0} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_4\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_4, t\_1\right)\right)\\ \end{array} \end{array} \]

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 (sin t_1))
        (t_3 (* (log (sqrt (fma x.re x.re (* x.im x.im)))) y.re))
        (t_4 (log (/ 1.0 x.re))))
   (if (<= x.re -1.1e-180)
     (* (exp (- (* (log (* -1.0 x.re)) y.re) t_0)) t_2)
     (if (<= x.re 5.8e-132)
       (*
        (sin (* (- (atan2 x.im x.re)) y.re))
        (exp (- t_3 (* y.im (atan2 x.im x.re)))))
       (if (<= x.re 7.5e-10)
         (* (exp (- t_3 t_0)) t_2)
         (* (exp (* -1.0 (* y.re t_4))) (sin (fma -1.0 (* y.im t_4) 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 = sin(t_1);
	double t_3 = log(sqrt(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))) * y_46_re;
	double t_4 = log((1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -1.1e-180) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * t_2;
	} else if (x_46_re <= 5.8e-132) {
		tmp = sin((-atan2(x_46_im, x_46_re) * y_46_re)) * exp((t_3 - (y_46_im * atan2(x_46_im, x_46_re))));
	} else if (x_46_re <= 7.5e-10) {
		tmp = exp((t_3 - t_0)) * t_2;
	} else {
		tmp = exp((-1.0 * (y_46_re * t_4))) * sin(fma(-1.0, (y_46_im * t_4), 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 = sin(t_1)
	t_3 = Float64(log(sqrt(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))) * y_46_re)
	t_4 = log(Float64(1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -1.1e-180)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * t_2);
	elseif (x_46_re <= 5.8e-132)
		tmp = Float64(sin(Float64(Float64(-atan(x_46_im, x_46_re)) * y_46_re)) * exp(Float64(t_3 - Float64(y_46_im * atan(x_46_im, x_46_re)))));
	elseif (x_46_re <= 7.5e-10)
		tmp = Float64(exp(Float64(t_3 - t_0)) * t_2);
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_4))) * sin(fma(-1.0, Float64(y_46_im * t_4), 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[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[(N[Log[N[Sqrt[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$4 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.1e-180], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[x$46$re, 5.8e-132], N[(N[Sin[N[((-N[ArcTan[x$46$im / x$46$re], $MachinePrecision]) * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(t$95$3 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7.5e-10], N[(N[Exp[N[(t$95$3 - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * t$95$4), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$4), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.re < -1.10000000000000007e-180

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if -1.10000000000000007e-180 < x.re < 5.79999999999999967e-132

   1. Initial program 40.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in y.re around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 47.1

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

   6. Applied rewrites 47.1%

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

   8. Applied rewrites 47.1%

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

   if 5.79999999999999967e-132 < x.re < 7.49999999999999995e-10

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   if 7.49999999999999995e-10 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 24.7

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

   7. Applied rewrites 24.7%

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

Alternative 8: 59.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t\_1\\ t_3 := \log \left(\frac{1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -2.9 \cdot 10^{-142}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot t\_2\\ \mathbf{elif}\;x.re \leq 7.5 \cdot 10^{-10}:\\ \;\;\;\;e^{\log \left(\sqrt{\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)}\right) \cdot y.re - t\_0} \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_3\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_3, t\_1\right)\right)\\ \end{array} \end{array} \]

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 (sin t_1))
        (t_3 (log (/ 1.0 x.re))))
   (if (<= x.re -2.9e-142)
     (* (exp (- (* (log (* -1.0 x.re)) y.re) t_0)) t_2)
     (if (<= x.re 7.5e-10)
       (*
        (exp (- (* (log (sqrt (fma x.re x.re (* x.im x.im)))) y.re) t_0))
        t_2)
       (* (exp (* -1.0 (* y.re t_3))) (sin (fma -1.0 (* y.im t_3) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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 = sin(t_1);
	double t_3 = log((1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -2.9e-142) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * t_2;
	} else if (x_46_re <= 7.5e-10) {
		tmp = exp(((log(sqrt(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))) * y_46_re) - t_0)) * t_2;
	} else {
		tmp = exp((-1.0 * (y_46_re * t_3))) * sin(fma(-1.0, (y_46_im * t_3), 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 = sin(t_1)
	t_3 = log(Float64(1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -2.9e-142)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * t_2);
	elseif (x_46_re <= 7.5e-10)
		tmp = Float64(exp(Float64(Float64(log(sqrt(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))) * y_46_re) - t_0)) * t_2);
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_3))) * sin(fma(-1.0, Float64(y_46_im * t_3), 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[Sin[t$95$1], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -2.9e-142], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[x$46$re, 7.5e-10], N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$3), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.8999999999999999e-142

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if -2.8999999999999999e-142 < x.re < 7.49999999999999995e-10

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   if 7.49999999999999995e-10 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 24.7

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

   7. Applied rewrites 24.7%

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

Alternative 9: 59.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{-148}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - t\_0} \cdot \sin t\_1\\ \mathbf{elif}\;x.re \leq 3.9 \cdot 10^{-118}:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - t\_0} \cdot \sin \pi\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_2\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_2, t\_1\right)\right)\\ \end{array} \end{array} \]

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 (log (/ 1.0 x.re))))
   (if (<= x.re -2e-148)
     (* (exp (- (* (log (* -1.0 x.re)) y.re) t_0)) (sin t_1))
     (if (<= x.re 3.9e-118)
       (*
        (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) t_0))
        (sin PI))
       (* (exp (* -1.0 (* y.re t_2))) (sin (fma -1.0 (* y.im t_2) 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 = log((1.0 / x_46_re));
	double tmp;
	if (x_46_re <= -2e-148) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(t_1);
	} else if (x_46_re <= 3.9e-118) {
		tmp = exp(((log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - t_0)) * sin(((double) M_PI));
	} else {
		tmp = exp((-1.0 * (y_46_re * t_2))) * sin(fma(-1.0, (y_46_im * t_2), 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 = log(Float64(1.0 / x_46_re))
	tmp = 0.0
	if (x_46_re <= -2e-148)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_re)) * y_46_re) - t_0)) * sin(t_1));
	elseif (x_46_re <= 3.9e-118)
		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(pi));
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_2))) * sin(fma(-1.0, Float64(y_46_im * t_2), 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[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -2e-148], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 3.9e-118], 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[Pi], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.99999999999999987e-148

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 31.0

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

   9. Applied rewrites 31.0%

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

   if -1.99999999999999987e-148 < x.re < 3.90000000000000001e-118

   1. Initial program 40.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-PI.f64 49.3

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

   9. Applied rewrites 49.3%

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

   if 3.90000000000000001e-118 < x.re

   1. Initial program 40.5%

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

   2. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 24.7

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

   7. Applied rewrites 24.7%

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

Alternative 10: 58.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \pi\\ \mathbf{if}\;y.re \leq -135000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.58 \cdot 10^{-18}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (exp
           (-
            (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re)
            (* (atan2 x.im x.re) y.im)))
          (sin PI))))
   (if (<= y.re -135000.0)
     t_0
     (if (<= y.re 1.58e-18)
       (*
        (exp (- (* y.im (atan2 x.im x.re))))
        (sin (* y.re (atan2 x.im x.re))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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(((double) M_PI));
	double tmp;
	if (y_46_re <= -135000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.58e-18) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(((Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(Math.PI);
	double tmp;
	if (y_46_re <= -135000.0) {
		tmp = t_0;
	} else if (y_46_re <= 1.58e-18) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(((math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im)))) * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin(math.pi)
	tmp = 0
	if y_46_re <= -135000.0:
		tmp = t_0
	elif y_46_re <= 1.58e-18:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(pi))
	tmp = 0.0
	if (y_46_re <= -135000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.58e-18)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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(pi);
	tmp = 0.0;
	if (y_46_re <= -135000.0)
		tmp = t_0;
	elseif (y_46_re <= 1.58e-18)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -135000.0], t$95$0, If[LessEqual[y$46$re, 1.58e-18], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -135000 or 1.5800000000000001e-18 < y.re

   1. Initial program 40.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. sub-negate-rev N/A

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

      6. sin-neg N/A

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

      7. sin-+PI-rev N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-+.f64 N/A

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

   3. Applied rewrites 29.0%

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

   4. Taylor expanded in y.im around 0

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

      1. lower--.f64 N/A

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

      2. lower-PI.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 49.7

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-PI.f64 49.3

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

   9. Applied rewrites 49.3%

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

   if -135000 < y.re < 1.5800000000000001e-18

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 40.2

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

   9. Applied rewrites 40.2%

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

Alternative 11: 49.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+64}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{+44}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= y.re -8e+64)
     (* (pow x.im y.re) t_0)
     (if (<= y.re 6e+44)
       (* (exp (- (* y.im (atan2 x.im x.re)))) t_0)
       (log (pow (sqrt (fma x.im x.im (* x.re x.re))) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_re <= -8e+64) {
		tmp = pow(x_46_im, y_46_re) * t_0;
	} else if (y_46_re <= 6e+44) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_0;
	} else {
		tmp = log(pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_re <= -8e+64)
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	elseif (y_46_re <= 6e+44)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_0);
	else
		tmp = log((sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -8e+64], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 6e+44], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * t$95$0), $MachinePrecision], N[Log[N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.00000000000000017e64

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 19.2

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   10. Applied rewrites 31.5%

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

   if -8.00000000000000017e64 < y.re < 5.99999999999999974e44

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 53.6

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

   4. Applied rewrites 53.6%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. sqr-neg-rev N/A

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

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

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

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

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

      8. lift-*.f64 N/A

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

      9. remove-double-neg N/A

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

      10. sqr-neg-rev N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 53.6

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

   6. Applied rewrites 53.6%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 40.2

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

   9. Applied rewrites 40.2%

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

   if 5.99999999999999974e44 < y.re

   1. Initial program 40.5%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 22.6

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

   4. Applied rewrites 22.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       2. lift-log.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       3. log-pow-rev N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       5. pow1/2 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       6. lift-+.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       7. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       8. pow2 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       9. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       10. pow2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       11. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       13. pow1/2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       14. +-commutative N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       15. lift-+.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       18. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

       19. lift-sqrt.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

   11. Applied rewrites 23.0%

       \[\leadsto \log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 40.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_1 := {x.im}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := \frac{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\right)}{t\_0}\\ \mathbf{if}\;x.im \leq -9 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x.im \leq -1.2 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.im \leq -9.5 \cdot 10^{-202}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;x.im \leq 1.45 \cdot 10^{-252}:\\ \;\;\;\;\log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right)\\ \mathbf{elif}\;x.im \leq 1.65 \cdot 10^{+218}:\\ \;\;\;\;\frac{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* y.im (atan2 x.im x.re))))
        (t_1 (* (pow x.im y.re) (sin (* y.re (atan2 x.im x.re)))))
        (t_2 (/ (sin (* -1.0 (* y.im (log (/ -1.0 x.im))))) t_0)))
   (if (<= x.im -9e+62)
     t_2
     (if (<= x.im -1.2e+16)
       t_1
       (if (<= x.im -9.5e-202)
         t_2
         (if (<= x.im 1.45e-252)
           (log (pow (sqrt (fma x.im x.im (* x.re x.re))) y.im))
           (if (<= x.im 1.65e+218)
             (/ (sin (* -1.0 (* y.im (log (/ 1.0 x.im))))) t_0)
             t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((y_46_im * atan2(x_46_im, x_46_re)));
	double t_1 = pow(x_46_im, y_46_re) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = sin((-1.0 * (y_46_im * log((-1.0 / x_46_im))))) / t_0;
	double tmp;
	if (x_46_im <= -9e+62) {
		tmp = t_2;
	} else if (x_46_im <= -1.2e+16) {
		tmp = t_1;
	} else if (x_46_im <= -9.5e-202) {
		tmp = t_2;
	} else if (x_46_im <= 1.45e-252) {
		tmp = log(pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_im));
	} else if (x_46_im <= 1.65e+218) {
		tmp = sin((-1.0 * (y_46_im * log((1.0 / x_46_im))))) / t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(y_46_im * atan(x_46_im, x_46_re)))
	t_1 = Float64((x_46_im ^ y_46_re) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))))
	t_2 = Float64(sin(Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))))) / t_0)
	tmp = 0.0
	if (x_46_im <= -9e+62)
		tmp = t_2;
	elseif (x_46_im <= -1.2e+16)
		tmp = t_1;
	elseif (x_46_im <= -9.5e-202)
		tmp = t_2;
	elseif (x_46_im <= 1.45e-252)
		tmp = log((sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_im));
	elseif (x_46_im <= 1.65e+218)
		tmp = Float64(sin(Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_im))))) / t_0);
	else
		tmp = 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[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[x$46$im, -9e+62], t$95$2, If[LessEqual[x$46$im, -1.2e+16], t$95$1, If[LessEqual[x$46$im, -9.5e-202], t$95$2, If[LessEqual[x$46$im, 1.45e-252], N[Log[N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]], $MachinePrecision], If[LessEqual[x$46$im, 1.65e+218], N[(N[Sin[N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -8.99999999999999997e62 or -1.2e16 < x.im < -9.5000000000000001e-202

   1. Initial program 40.5%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 22.6

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

   4. Applied rewrites 22.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 17.8

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

   9. Applied rewrites 17.8%

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

   if -8.99999999999999997e62 < x.im < -1.2e16 or 1.64999999999999999e218 < x.im

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 19.2

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   10. Applied rewrites 31.5%

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

   if -9.5000000000000001e-202 < x.im < 1.45e-252

   1. Initial program 40.5%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 22.6

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

   4. Applied rewrites 22.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       2. lift-log.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       3. log-pow-rev N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       5. pow1/2 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       6. lift-+.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       7. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       8. pow2 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       9. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       10. pow2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       11. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       13. pow1/2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       14. +-commutative N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       15. lift-+.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       18. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

       19. lift-sqrt.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

   11. Applied rewrites 23.0%

       \[\leadsto \log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right) \]

   if 1.45e-252 < x.im < 1.64999999999999999e218

   1. Initial program 40.5%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 22.6

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

   4. Applied rewrites 22.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. lower-/.f64 18.9

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

   9. Applied rewrites 18.9%

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

Alternative 13: 37.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ \mathbf{if}\;y.re \leq -220000:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \mathbf{elif}\;y.re \leq -7.6 \cdot 10^{-109}:\\ \;\;\;\;1 \cdot t\_0\\ \mathbf{elif}\;y.re \leq 130000000000:\\ \;\;\;\;\frac{\sin \left(\log t\_1 \cdot y.im\right)}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({t\_1}^{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (sqrt (fma x.im x.im (* x.re x.re)))))
   (if (<= y.re -220000.0)
     (* (pow x.im y.re) t_0)
     (if (<= y.re -7.6e-109)
       (* 1.0 t_0)
       (if (<= y.re 130000000000.0)
         (/ (sin (* (log t_1) y.im)) (exp (* y.im (atan2 x.im x.re))))
         (log (pow t_1 y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double tmp;
	if (y_46_re <= -220000.0) {
		tmp = pow(x_46_im, y_46_re) * t_0;
	} else if (y_46_re <= -7.6e-109) {
		tmp = 1.0 * t_0;
	} else if (y_46_re <= 130000000000.0) {
		tmp = sin((log(t_1) * y_46_im)) / exp((y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = log(pow(t_1, y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	tmp = 0.0
	if (y_46_re <= -220000.0)
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	elseif (y_46_re <= -7.6e-109)
		tmp = Float64(1.0 * t_0);
	elseif (y_46_re <= 130000000000.0)
		tmp = Float64(sin(Float64(log(t_1) * y_46_im)) / exp(Float64(y_46_im * atan(x_46_im, x_46_re))));
	else
		tmp = log((t_1 ^ y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -220000.0], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, -7.6e-109], N[(1.0 * t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 130000000000.0], N[(N[Sin[N[(N[Log[t$95$1], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[N[Power[t$95$1, y$46$im], $MachinePrecision]], $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.2e5

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 19.2

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   10. Applied rewrites 31.5%

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

   if -2.2e5 < y.re < -7.60000000000000003e-109

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 19.2

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 13.8%

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

   if -7.60000000000000003e-109 < y.re < 1.3e11

   1. Initial program 40.5%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 22.6

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

   4. Applied rewrites 22.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

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

   if 1.3e11 < y.re

   1. Initial program 40.5%

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

   2. Taylor expanded in y.re around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. lower-+.f64 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 22.6

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

   4. Applied rewrites 22.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift-neg.f64 N/A

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

      5. exp-neg N/A

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

      6. mult-flip-rev N/A

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

      7. lower-/.f64 N/A

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

   6. Applied rewrites 22.6%

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

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       2. lift-log.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       3. log-pow-rev N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       5. pow1/2 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       6. lift-+.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       7. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       8. pow2 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       9. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       10. pow2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       11. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       13. pow1/2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       14. +-commutative N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       15. lift-+.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       18. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

       19. lift-sqrt.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

   11. Applied rewrites 23.0%

       \[\leadsto \log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 36.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := 1 \cdot t\_1\\ \mathbf{if}\;y.re \leq -220000:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_1\\ \mathbf{elif}\;y.re \leq -2.8 \cdot 10^{-109}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-168}:\\ \;\;\;\;\log t\_0 \cdot y.im\\ \mathbf{elif}\;y.re \leq 1.58 \cdot 10^{-18}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\log \left({t\_0}^{y.im}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_1 (sin (* y.re (atan2 x.im x.re))))
        (t_2 (* 1.0 t_1)))
   (if (<= y.re -220000.0)
     (* (pow x.im y.re) t_1)
     (if (<= y.re -2.8e-109)
       t_2
       (if (<= y.re 7e-168)
         (* (log t_0) y.im)
         (if (<= y.re 1.58e-18) t_2 (log (pow t_0 y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = 1.0 * t_1;
	double tmp;
	if (y_46_re <= -220000.0) {
		tmp = pow(x_46_im, y_46_re) * t_1;
	} else if (y_46_re <= -2.8e-109) {
		tmp = t_2;
	} else if (y_46_re <= 7e-168) {
		tmp = log(t_0) * y_46_im;
	} else if (y_46_re <= 1.58e-18) {
		tmp = t_2;
	} else {
		tmp = log(pow(t_0, y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_2 = Float64(1.0 * t_1)
	tmp = 0.0
	if (y_46_re <= -220000.0)
		tmp = Float64((x_46_im ^ y_46_re) * t_1);
	elseif (y_46_re <= -2.8e-109)
		tmp = t_2;
	elseif (y_46_re <= 7e-168)
		tmp = Float64(log(t_0) * y_46_im);
	elseif (y_46_re <= 1.58e-18)
		tmp = t_2;
	else
		tmp = log((t_0 ^ y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 * t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -220000.0], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, -2.8e-109], t$95$2, If[LessEqual[y$46$re, 7e-168], N[(N[Log[t$95$0], $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.58e-18], t$95$2, N[Log[N[Power[t$95$0, y$46$im], $MachinePrecision]], $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.2e5

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-atan2.f64 19.2

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

   7. Applied rewrites 19.2%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-pow.f64 31.5

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

   10. Applied rewrites 31.5%

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

   if -2.2e5 < y.re < -2.79999999999999979e-109 or 6.99999999999999964e-168 < y.re < 1.5800000000000001e-18

   1. Initial program 40.5%

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

   2. Taylor expanded in x.im around inf

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

      1. lower-*.f64 N/A

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

   4. Applied rewrites 32.3%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-exp.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lower-atan2.f64 19.2

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 19.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.8%

       \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -2.79999999999999979e-109 < y.re < 6.99999999999999964e-168

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   11. Applied rewrites 18.3%

       \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im} \]

   if 1.5800000000000001e-18 < y.re

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       2. lift-log.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       3. log-pow-rev N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       5. pow1/2 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       6. lift-+.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       7. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       8. pow2 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       9. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       10. pow2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       11. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       13. pow1/2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       14. +-commutative N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       15. lift-+.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       18. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

       19. lift-sqrt.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

   11. Applied rewrites 23.0%

       \[\leadsto \log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 29.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_2 := \log \left({t\_1}^{y.im}\right)\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-49}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -2.8 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-168}:\\ \;\;\;\;\log t\_1 \cdot y.im\\ \mathbf{elif}\;y.re \leq 1.58 \cdot 10^{-18}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* 1.0 (sin (* y.re (atan2 x.im x.re)))))
        (t_1 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_2 (log (pow t_1 y.im))))
   (if (<= y.re -1.15e-49)
     t_2
     (if (<= y.re -2.8e-109)
       t_0
       (if (<= y.re 7e-168)
         (* (log t_1) y.im)
         (if (<= y.re 1.58e-18) t_0 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 * sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_2 = log(pow(t_1, y_46_im));
	double tmp;
	if (y_46_re <= -1.15e-49) {
		tmp = t_2;
	} else if (y_46_re <= -2.8e-109) {
		tmp = t_0;
	} else if (y_46_re <= 7e-168) {
		tmp = log(t_1) * y_46_im;
	} else if (y_46_re <= 1.58e-18) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 * sin(Float64(y_46_re * atan(x_46_im, x_46_re))))
	t_1 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_2 = log((t_1 ^ y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.15e-49)
		tmp = t_2;
	elseif (y_46_re <= -2.8e-109)
		tmp = t_0;
	elseif (y_46_re <= 7e-168)
		tmp = Float64(log(t_1) * y_46_im);
	elseif (y_46_re <= 1.58e-18)
		tmp = t_0;
	else
		tmp = 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[(1.0 * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Power[t$95$1, y$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e-49], t$95$2, If[LessEqual[y$46$re, -2.8e-109], t$95$0, If[LessEqual[y$46$re, 7e-168], N[(N[Log[t$95$1], $MachinePrecision] * y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.58e-18], t$95$0, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.15e-49 or 1.5800000000000001e-18 < y.re

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       2. lift-log.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       3. log-pow-rev N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       5. pow1/2 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       6. lift-+.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       7. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       8. pow2 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       9. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       10. pow2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       11. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       13. pow1/2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       14. +-commutative N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       15. lift-+.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       18. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

       19. lift-sqrt.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

   11. Applied rewrites 23.0%

       \[\leadsto \log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right) \]

   if -1.15e-49 < y.re < -2.79999999999999979e-109 or 6.99999999999999964e-168 < y.re < 1.5800000000000001e-18

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in x.im around inf

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   4. Applied rewrites 32.3%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. lower-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      9. lower-atan2.f64 19.2

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 19.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   8. Taylor expanded in y.re around 0

      \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 13.8%

       \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -2.79999999999999979e-109 < y.re < 6.99999999999999964e-168

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   11. Applied rewrites 18.3%

       \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 26.8% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_1 := \log \left({t\_0}^{y.im}\right)\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-28}:\\ \;\;\;\;\log t\_0 \cdot y.im\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sqrt (fma x.im x.im (* x.re x.re)))) (t_1 (log (pow t_0 y.im))))
   (if (<= y.re -2.15e-78) t_1 (if (<= y.re 7e-28) (* (log t_0) y.im) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_1 = log(pow(t_0, y_46_im));
	double tmp;
	if (y_46_re <= -2.15e-78) {
		tmp = t_1;
	} else if (y_46_re <= 7e-28) {
		tmp = log(t_0) * y_46_im;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_1 = log((t_0 ^ y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.15e-78)
		tmp = t_1;
	elseif (y_46_re <= 7e-28)
		tmp = Float64(log(t_0) * y_46_im);
	else
		tmp = 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[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Power[t$95$0, y$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -2.15e-78], t$95$1, If[LessEqual[y$46$re, 7e-28], N[(N[Log[t$95$0], $MachinePrecision] * y$46$im), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.14999999999999997e-78 or 6.9999999999999999e-28 < y.re

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       2. lift-log.f64 N/A

          \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

       3. log-pow-rev N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \log \left({\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.im}\right) \]

       5. pow1/2 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       6. lift-+.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       7. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left({x.im}^{2} + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       8. pow2 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       9. lift-pow.f64 N/A

          \[\leadsto \log \left({\left({\left(x.im \cdot x.im + {x.re}^{2}\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       10. pow2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       11. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.im}\right) \]

       13. pow1/2 N/A

           \[\leadsto \log \left({\left({\left(x.im \cdot x.im + x.re \cdot x.re\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       14. +-commutative N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       15. lift-+.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       16. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto \log \left({\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)}^{y.im}\right) \]

       18. pow1/2 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

       19. lift-sqrt.f64 N/A

           \[\leadsto \log \left({\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.im}\right) \]

   11. Applied rewrites 23.0%

       \[\leadsto \log \left({\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.im}\right) \]

   if -2.14999999999999997e-78 < y.re < 6.9999999999999999e-28

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   11. Applied rewrites 18.3%

       \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 18.3% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (log (sqrt (fma x.im x.im (* x.re x.re)))) y.im))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) * y_46_im;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) * y_46_im)
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]

Derivation

1. Initial program 40.5%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

2. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   2. lower-exp.f64 N/A

      \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   3. lower-neg.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   5. lower-atan2.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   6. lower-sin.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   8. lower-log.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   11. lower-pow.f64 N/A

       \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   12. lower-pow.f64 22.6

       \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

4. Applied rewrites 22.6%

   \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   4. lift-neg.f64 N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   5. exp-neg N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. mult-flip-rev N/A

      \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

6. Applied rewrites 22.6%

   \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

7. Taylor expanded in y.im around 0

   \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   2. lower-log.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   6. lower-pow.f64 18.3

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

9. Applied rewrites 18.3%

   \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

11. Applied rewrites 18.3%

    \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im} \]
12. Add Preprocessing

Alternative 18: 11.7% accurate, 5.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -4.3 \cdot 10^{-141}:\\ \;\;\;\;-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{elif}\;x.re \leq 7.8 \cdot 10^{-91}:\\ \;\;\;\;-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -4.3e-141)
   (* -1.0 (* y.im (log (/ -1.0 x.re))))
   (if (<= x.re 7.8e-91)
     (* -1.0 (* y.im (log (/ -1.0 x.im))))
     (* -1.0 (* y.im (log (/ 1.0 x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -4.3e-141) {
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_re)));
	} else if (x_46_re <= 7.8e-91) {
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_im)));
	} else {
		tmp = -1.0 * (y_46_im * log((1.0 / x_46_re)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-4.3d-141)) then
        tmp = (-1.0d0) * (y_46im * log(((-1.0d0) / x_46re)))
    else if (x_46re <= 7.8d-91) then
        tmp = (-1.0d0) * (y_46im * log(((-1.0d0) / x_46im)))
    else
        tmp = (-1.0d0) * (y_46im * log((1.0d0 / x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -4.3e-141) {
		tmp = -1.0 * (y_46_im * Math.log((-1.0 / x_46_re)));
	} else if (x_46_re <= 7.8e-91) {
		tmp = -1.0 * (y_46_im * Math.log((-1.0 / x_46_im)));
	} else {
		tmp = -1.0 * (y_46_im * Math.log((1.0 / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -4.3e-141:
		tmp = -1.0 * (y_46_im * math.log((-1.0 / x_46_re)))
	elif x_46_re <= 7.8e-91:
		tmp = -1.0 * (y_46_im * math.log((-1.0 / x_46_im)))
	else:
		tmp = -1.0 * (y_46_im * math.log((1.0 / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -4.3e-141)
		tmp = Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_re))));
	elseif (x_46_re <= 7.8e-91)
		tmp = Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))));
	else
		tmp = Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -4.3e-141)
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_re)));
	elseif (x_46_re <= 7.8e-91)
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_im)));
	else
		tmp = -1.0 * (y_46_im * log((1.0 / x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -4.3e-141], N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 7.8e-91], N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -4.29999999999999974e-141

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   10. Taylor expanded in x.re around -inf

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.re}\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

       4. lower-/.f64 4.7

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

   12. Applied rewrites 4.7%

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.re}\right)}\right) \]

   if -4.29999999999999974e-141 < x.re < 7.79999999999999987e-91

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   10. Taylor expanded in x.im around -inf

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       4. lower-/.f64 4.5

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

   12. Applied rewrites 4.5%

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]

   if 7.79999999999999987e-91 < x.re

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   10. Taylor expanded in x.re around inf

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.re}\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) \]

       4. lower-/.f64 6.8

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) \]

   12. Applied rewrites 6.8%

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.re}\right)}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 9.0% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.im -4e-310)
   (* -1.0 (* y.im (log (/ -1.0 x.im))))
   (* -1.0 (* y.im (log (/ 1.0 x.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -4e-310) {
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_im)));
	} else {
		tmp = -1.0 * (y_46_im * log((1.0 / x_46_im)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46im <= (-4d-310)) then
        tmp = (-1.0d0) * (y_46im * log(((-1.0d0) / x_46im)))
    else
        tmp = (-1.0d0) * (y_46im * log((1.0d0 / x_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_im <= -4e-310) {
		tmp = -1.0 * (y_46_im * Math.log((-1.0 / x_46_im)));
	} else {
		tmp = -1.0 * (y_46_im * Math.log((1.0 / x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_im <= -4e-310:
		tmp = -1.0 * (y_46_im * math.log((-1.0 / x_46_im)))
	else:
		tmp = -1.0 * (y_46_im * math.log((1.0 / x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= -4e-310)
		tmp = Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))));
	else
		tmp = Float64(-1.0 * Float64(y_46_im * log(Float64(1.0 / x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_im <= -4e-310)
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_im)));
	else
		tmp = -1.0 * (y_46_im * log((1.0 / x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, -4e-310], N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(y$46$im * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.999999999999988e-310

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   10. Taylor expanded in x.im around -inf

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       4. lower-/.f64 4.5

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

   12. Applied rewrites 4.5%

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]

   if -3.999999999999988e-310 < x.im

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   10. Taylor expanded in x.im around inf

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.im}\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) \]

       4. lower-/.f64 4.4

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) \]

   12. Applied rewrites 4.4%

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{1}{x.im}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 7.2% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x.re \leq -4.3 \cdot 10^{-141}:\\ \;\;\;\;-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= x.re -4.3e-141)
   (* -1.0 (* y.im (log (/ -1.0 x.re))))
   (* -1.0 (* y.im (log (/ -1.0 x.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -4.3e-141) {
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_re)));
	} else {
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_im)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (x_46re <= (-4.3d-141)) then
        tmp = (-1.0d0) * (y_46im * log(((-1.0d0) / x_46re)))
    else
        tmp = (-1.0d0) * (y_46im * log(((-1.0d0) / x_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (x_46_re <= -4.3e-141) {
		tmp = -1.0 * (y_46_im * Math.log((-1.0 / x_46_re)));
	} else {
		tmp = -1.0 * (y_46_im * Math.log((-1.0 / x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if x_46_re <= -4.3e-141:
		tmp = -1.0 * (y_46_im * math.log((-1.0 / x_46_re)))
	else:
		tmp = -1.0 * (y_46_im * math.log((-1.0 / x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_re <= -4.3e-141)
		tmp = Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_re))));
	else
		tmp = Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (x_46_re <= -4.3e-141)
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_re)));
	else
		tmp = -1.0 * (y_46_im * log((-1.0 / x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$re, -4.3e-141], N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.29999999999999974e-141

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   10. Taylor expanded in x.re around -inf

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.re}\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

       4. lower-/.f64 4.7

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.re}\right)\right) \]

   12. Applied rewrites 4.7%

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.re}\right)}\right) \]

   if -4.29999999999999974e-141 < x.re

   1. Initial program 40.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      8. lower-log.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      9. lower-sqrt.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      10. lower-+.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

      12. lower-pow.f64 22.6

          \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. Applied rewrites 22.6%

      \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

      3. lift-exp.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      4. lift-neg.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      5. exp-neg N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      6. mult-flip-rev N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. Applied rewrites 22.6%

      \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. Taylor expanded in y.im around 0

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      2. lower-log.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      3. lower-sqrt.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      4. lower-+.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      5. lower-pow.f64 N/A

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

      6. lower-pow.f64 18.3

         \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

   10. Taylor expanded in x.im around -inf

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       3. lower-log.f64 N/A

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

       4. lower-/.f64 4.5

          \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

   12. Applied rewrites 4.5%

       \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 4.5% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* -1.0 (* y.im (log (/ -1.0 x.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return -1.0 * (y_46_im * log((-1.0 / x_46_im)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = (-1.0d0) * (y_46im * log(((-1.0d0) / x_46im)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return -1.0 * (y_46_im * Math.log((-1.0 / x_46_im)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return -1.0 * (y_46_im * math.log((-1.0 / x_46_im)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(-1.0 * Float64(y_46_im * log(Float64(-1.0 / x_46_im))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = -1.0 * (y_46_im * log((-1.0 / x_46_im)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(-1.0 * N[(y$46$im * N[Log[N[(-1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.5%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

2. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   2. lower-exp.f64 N/A

      \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \color{blue}{\left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   3. lower-neg.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\color{blue}{y.im} \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   4. lower-*.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   5. lower-atan2.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   6. lower-sin.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   7. lower-*.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   8. lower-log.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   9. lower-sqrt.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   10. lower-+.f64 N/A

       \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   11. lower-pow.f64 N/A

       \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

   12. lower-pow.f64 22.6

       \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \]

4. Applied rewrites 22.6%

   \[\leadsto \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)} \]

   2. *-commutative N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \color{blue}{e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   4. lift-neg.f64 N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   5. exp-neg N/A

      \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right) \cdot \frac{1}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   6. mult-flip-rev N/A

      \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

   7. lower-/.f64 N/A

      \[\leadsto \frac{\sin \left(y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

6. Applied rewrites 22.6%

   \[\leadsto \frac{\sin \left(\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot y.im\right)}{\color{blue}{e^{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}} \]

7. Taylor expanded in y.im around 0

   \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   2. lower-log.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   3. lower-sqrt.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   4. lower-+.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   5. lower-pow.f64 N/A

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

   6. lower-pow.f64 18.3

      \[\leadsto y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right) \]

9. Applied rewrites 18.3%

   \[\leadsto y.im \cdot \color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)} \]

10. Taylor expanded in x.im around -inf

    \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]
11. Step-by-step derivation

    1. lower-*.f64 N/A

       \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

    2. lower-*.f64 N/A

       \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

    3. lower-log.f64 N/A

       \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

    4. lower-/.f64 4.5

       \[\leadsto -1 \cdot \left(y.im \cdot \log \left(\frac{-1}{x.im}\right)\right) \]

12. Applied rewrites 4.5%

    \[\leadsto -1 \cdot \left(y.im \cdot \color{blue}{\log \left(\frac{-1}{x.im}\right)}\right) \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(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)))))