powComplex, imaginary part

Percentage Accurate: 40.3% → 79.8%

Time: 7.2s

Alternatives: 20

Speedup: 1.6×

• 64.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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: 40.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \log \left(\left|x.re\right|\right)\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := \frac{0 - {t\_0}^{3}}{0 + \mathsf{fma}\left(t\_0, t\_0, 0 \cdot t\_0\right)}\\ \mathbf{if}\;y.im \leq 10^{+136}:\\ \;\;\;\;e^{t\_1 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_1 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_2\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_2, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (log (fabs x.re)))
       (t_1 (log (hypot x.re x.im)))
       (t_2
        (/ (- 0.0 (pow t_0 3.0)) (+ 0.0 (fma t_0 t_0 (* 0.0 t_0))))))
  (if (<= y.im 1e+136)
    (*
     (exp (- (* t_1 y.re) (* (atan2 x.im x.re) y.im)))
     (sin (+ (* t_1 y.im) (* (atan2 x.im x.re) y.re))))
    (*
     (exp (- (* -1.0 (* y.re t_2)) (* y.im (atan2 x.im x.re))))
     (sin (fma -1.0 (* y.im t_2) (* 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(fabs(x_46_re));
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = (0.0 - pow(t_0, 3.0)) / (0.0 + fma(t_0, t_0, (0.0 * t_0)));
	double tmp;
	if (y_46_im <= 1e+136) {
		tmp = exp(((t_1 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_1 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else {
		tmp = exp(((-1.0 * (y_46_re * t_2)) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(-1.0, (y_46_im * t_2), (y_46_re * atan2(x_46_im, x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(abs(x_46_re))
	t_1 = log(hypot(x_46_re, x_46_im))
	t_2 = Float64(Float64(0.0 - (t_0 ^ 3.0)) / Float64(0.0 + fma(t_0, t_0, Float64(0.0 * t_0))))
	tmp = 0.0
	if (y_46_im <= 1e+136)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(Float64(t_1 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	else
		tmp = Float64(exp(Float64(Float64(-1.0 * Float64(y_46_re * t_2)) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(-1.0, Float64(y_46_im * t_2), Float64(y_46_re * atan(x_46_im, x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(0.0 - N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision] / N[(0.0 + N[(t$95$0 * t$95$0 + N[(0.0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, 1e+136], N[(N[Exp[N[(N[(t$95$1 * 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$1 * 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[(N[(-1.0 * N[(y$46$re * t$95$2), $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$2), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 1.0000000000000001e136

   1. Initial program 40.3%

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

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 40.3%

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

   3. Applied rewrites 40.3%

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

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

      4. lift-*.f64 N/A

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

      5. lower-hypot.f64 79.8%

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

   5. Applied rewrites 79.8%

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

   if 1.0000000000000001e136 < y.im

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lift-/.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. log-div N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. flip3-- N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. remove-sound-/ N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower--.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-pow.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       13. lower-fabs.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       14. metadata-eval N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       15. lower-+.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \left(\log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right) + 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.re\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.re\right|\right), \log \left(\left|x.re\right|\right), 0 \cdot \log \left(\left|x.re\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 79.2% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (log (hypot x.re 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(hypot(x_46_re, 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)));
}

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));
	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.hypot(x_46_re, 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(hypot(x_46_re, 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(hypot(x_46_re, 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[x$46$re ^ 2 + x$46$im ^ 2], $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]]

Derivation

1. Initial program 40.3%

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

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 40.3%

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

3. Applied rewrites 40.3%

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

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

   4. lift-*.f64 N/A

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

   5. lower-hypot.f64 79.8%

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

5. Applied rewrites 79.8%

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

Alternative 3: 75.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{1}{x.re}\right)\\ t_3 := \log \left(-1 \cdot x.re\right)\\ t_4 := \log \left(\left|x.im\right|\right)\\ \mathbf{if}\;x.re \leq -1.25 \cdot 10^{+45}:\\ \;\;\;\;e^{t\_3 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_3 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \leq 1250000:\\ \;\;\;\;e^{y.re \cdot t\_4 - t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, t\_4, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_2\right) - t\_0} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_2, t\_1\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.im (atan2 x.im x.re)))
       (t_1 (* y.re (atan2 x.im x.re)))
       (t_2 (log (/ 1.0 x.re)))
       (t_3 (log (* -1.0 x.re)))
       (t_4 (log (fabs x.im))))
  (if (<= x.re -1.25e+45)
    (*
     (exp (- (* t_3 y.re) (* (atan2 x.im x.re) y.im)))
     (sin (+ (* t_3 y.im) (* (atan2 x.im x.re) y.re))))
    (if (<= x.re 1250000.0)
      (* (exp (- (* y.re t_4) t_0)) (sin (fma y.im t_4 t_1)))
      (*
       (exp (- (* -1.0 (* y.re t_2)) t_0))
       (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 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log((1.0 / x_46_re));
	double t_3 = log((-1.0 * x_46_re));
	double t_4 = log(fabs(x_46_im));
	double tmp;
	if (x_46_re <= -1.25e+45) {
		tmp = exp(((t_3 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_3 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_re <= 1250000.0) {
		tmp = exp(((y_46_re * t_4) - t_0)) * sin(fma(y_46_im, t_4, t_1));
	} else {
		tmp = exp(((-1.0 * (y_46_re * t_2)) - t_0)) * 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(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(Float64(1.0 / x_46_re))
	t_3 = log(Float64(-1.0 * x_46_re))
	t_4 = log(abs(x_46_im))
	tmp = 0.0
	if (x_46_re <= -1.25e+45)
		tmp = Float64(exp(Float64(Float64(t_3 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(Float64(t_3 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_re <= 1250000.0)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_4) - t_0)) * sin(fma(y_46_im, t_4, t_1)));
	else
		tmp = Float64(exp(Float64(Float64(-1.0 * Float64(y_46_re * t_2)) - t_0)) * 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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.25e+45], N[(N[Exp[N[(N[(t$95$3 * 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$3 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1250000.0], N[(N[Exp[N[(N[(y$46$re * t$95$4), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$4 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(-1.0 * N[(y$46$re * t$95$2), $MachinePrecision]), $MachinePrecision] - t$95$0), $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 < -1.25e45

   1. Initial program 40.3%

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

   2. 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 17.8%

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 32.9%

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

   7. Applied rewrites 32.9%

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

   if -1.25e45 < x.re < 1.25e6

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in x.re around 0

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

       1. lower-*.f64 N/A

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

   14. Applied rewrites 62.5%

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

   if 1.25e6 < x.re

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

Alternative 4: 72.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \log \left(\left|x.re\right|\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(-1 \cdot x.re\right)\\ t_3 := \log \left(\left|x.im\right|\right)\\ \mathbf{if}\;x.re \leq -1.25 \cdot 10^{+45}:\\ \;\;\;\;e^{t\_2 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_2 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \leq 2.4 \cdot 10^{+25}:\\ \;\;\;\;e^{y.re \cdot t\_3 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, t\_3, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right)} \cdot \sin \left(\mathsf{fma}\left(-1, \frac{y.im \cdot \left(t\_0 \cdot \left(-\log x.re\right)\right)}{t\_0}, t\_1\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (log (fabs x.re)))
       (t_1 (* y.re (atan2 x.im x.re)))
       (t_2 (log (* -1.0 x.re)))
       (t_3 (log (fabs x.im))))
  (if (<= x.re -1.25e+45)
    (*
     (exp (- (* t_2 y.re) (* (atan2 x.im x.re) y.im)))
     (sin (+ (* t_2 y.im) (* (atan2 x.im x.re) y.re))))
    (if (<= x.re 2.4e+25)
      (*
       (exp (- (* y.re t_3) (* y.im (atan2 x.im x.re))))
       (sin (fma y.im t_3 t_1)))
      (*
       (exp (* -1.0 (* y.re (log (/ 1.0 x.re)))))
       (sin
        (fma -1.0 (/ (* y.im (* t_0 (- (log x.re)))) 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 = log(fabs(x_46_re));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log((-1.0 * x_46_re));
	double t_3 = log(fabs(x_46_im));
	double tmp;
	if (x_46_re <= -1.25e+45) {
		tmp = exp(((t_2 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_2 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_re <= 2.4e+25) {
		tmp = exp(((y_46_re * t_3) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_3, t_1));
	} else {
		tmp = exp((-1.0 * (y_46_re * log((1.0 / x_46_re))))) * sin(fma(-1.0, ((y_46_im * (t_0 * -log(x_46_re))) / t_0), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(abs(x_46_re))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(Float64(-1.0 * x_46_re))
	t_3 = log(abs(x_46_im))
	tmp = 0.0
	if (x_46_re <= -1.25e+45)
		tmp = Float64(exp(Float64(Float64(t_2 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(Float64(t_2 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_re <= 2.4e+25)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_3) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_3, t_1)));
	else
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * log(Float64(1.0 / x_46_re))))) * sin(fma(-1.0, Float64(Float64(y_46_im * Float64(t_0 * Float64(-log(x_46_re)))) / t_0), 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[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.25e+45], N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.4e+25], N[(N[Exp[N[(N[(y$46$re * t$95$3), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$3 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(-1.0 * N[(y$46$re * N[Log[N[(1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-1.0 * N[(N[(y$46$im * N[(t$95$0 * (-N[Log[x$46$re], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.25e45

   1. Initial program 40.3%

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

   2. 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 17.8%

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 32.9%

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

   7. Applied rewrites 32.9%

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

   if -1.25e45 < x.re < 2.4e25

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in x.re around 0

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

       1. lower-*.f64 N/A

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

   14. Applied rewrites 62.5%

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

   if 2.4e25 < x.re

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in y.im around 0

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

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

   10. Applied rewrites 24.7%

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

       1. lift-*.f64 N/A

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

       2. lift-log.f64 N/A

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

       3. lift-/.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. log-div N/A

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

       5. metadata-eval N/A

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

       6. metadata-eval N/A

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

       7. lift-fabs.f64 N/A

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

       8. lift-log.f64 N/A

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

       9. flip-- N/A

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

       10. metadata-eval N/A

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

       11. lift-*.f64 N/A

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

       12. lift--.f64 N/A

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

       13. +-lft-identity N/A

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

       14. remove-sound-/ N/A

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

       15. associate-*r/ N/A

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

       16. lower-/.f64 N/A

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

   12. Applied rewrites 24.8%

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

Alternative 5: 72.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-1 \cdot x.re\right)\\ t_2 := \log \left(\left|x.im\right|\right)\\ t_3 := \log \left(\left|x.re\right|\right)\\ \mathbf{if}\;x.re \leq -1.25 \cdot 10^{+45}:\\ \;\;\;\;e^{t\_1 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t\_1 \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;e^{y.re \cdot t\_2 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, t\_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left|x.re\right|\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - t\_3 \cdot t\_3}{0 + t\_3}, t\_0\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (log (* -1.0 x.re)))
       (t_2 (log (fabs x.im)))
       (t_3 (log (fabs x.re))))
  (if (<= x.re -1.25e+45)
    (*
     (exp (- (* t_1 y.re) (* (atan2 x.im x.re) y.im)))
     (sin (+ (* t_1 y.im) (* (atan2 x.im x.re) y.re))))
    (if (<= x.re 1.35e+31)
      (*
       (exp (- (* y.re t_2) (* y.im (atan2 x.im x.re))))
       (sin (fma y.im t_2 t_0)))
      (*
       (pow (fabs x.re) y.re)
       (sin
        (fma
         -1.0
         (* y.im (/ (- 0.0 (* t_3 t_3)) (+ 0.0 t_3)))
         t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log((-1.0 * x_46_re));
	double t_2 = log(fabs(x_46_im));
	double t_3 = log(fabs(x_46_re));
	double tmp;
	if (x_46_re <= -1.25e+45) {
		tmp = exp(((t_1 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_1 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
	} else if (x_46_re <= 1.35e+31) {
		tmp = exp(((y_46_re * t_2) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_2, t_0));
	} else {
		tmp = pow(fabs(x_46_re), y_46_re) * sin(fma(-1.0, (y_46_im * ((0.0 - (t_3 * t_3)) / (0.0 + t_3))), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-1.0 * x_46_re))
	t_2 = log(abs(x_46_im))
	t_3 = log(abs(x_46_re))
	tmp = 0.0
	if (x_46_re <= -1.25e+45)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(Float64(t_1 * y_46_im) + Float64(atan(x_46_im, x_46_re) * y_46_re))));
	elseif (x_46_re <= 1.35e+31)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_2) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_2, t_0)));
	else
		tmp = Float64((abs(x_46_re) ^ y_46_re) * sin(fma(-1.0, Float64(y_46_im * Float64(Float64(0.0 - Float64(t_3 * t_3)) / Float64(0.0 + t_3))), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(-1.0 * x$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.25e+45], N[(N[Exp[N[(N[(t$95$1 * 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$1 * y$46$im), $MachinePrecision] + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.35e+31], N[(N[Exp[N[(N[(y$46$re * t$95$2), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$2 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Abs[x$46$re], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * N[(N[(0.0 - N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] / N[(0.0 + t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.25e45

   1. Initial program 40.3%

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

   2. 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 17.8%

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

   4. Applied rewrites 17.8%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 32.9%

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

   7. Applied rewrites 32.9%

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

   if -1.25e45 < x.re < 1.3499999999999999e31

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in x.re around 0

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

       1. lower-*.f64 N/A

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

   14. Applied rewrites 62.5%

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

   if 1.3499999999999999e31 < x.re

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-- N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-fabs.f64 32.7%

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

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. remove-sound-/ N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-fabs.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-+.f64 N/A

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

       17. lower-log.f64 N/A

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

       18. lower-fabs.f64 65.6%

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in y.im around 0

       \[\leadsto {\left(\left|x.re\right|\right)}^{y.re} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

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

       2. lower-fabs.f64 46.6%

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

   14. Applied rewrites 46.6%

       \[\leadsto {\left(\left|x.re\right|\right)}^{y.re} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 70.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\left|x.re\right|\right)\\ t_2 := \log \left(\left|x.im\right|\right)\\ \mathbf{if}\;x.re \leq -1.8 \cdot 10^{+45}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin t\_0\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{+31}:\\ \;\;\;\;e^{y.re \cdot t\_2 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, t\_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\left|x.re\right|\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - t\_1 \cdot t\_1}{0 + t\_1}, t\_0\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (log (fabs x.re)))
       (t_2 (log (fabs x.im))))
  (if (<= x.re -1.8e+45)
    (*
     (exp (- (* (log (* -1.0 x.re)) y.re) (* (atan2 x.im x.re) y.im)))
     (sin t_0))
    (if (<= x.re 1.35e+31)
      (*
       (exp (- (* y.re t_2) (* y.im (atan2 x.im x.re))))
       (sin (fma y.im t_2 t_0)))
      (*
       (pow (fabs x.re) y.re)
       (sin
        (fma
         -1.0
         (* y.im (/ (- 0.0 (* t_1 t_1)) (+ 0.0 t_1)))
         t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(fabs(x_46_re));
	double t_2 = log(fabs(x_46_im));
	double tmp;
	if (x_46_re <= -1.8e+45) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(t_0);
	} else if (x_46_re <= 1.35e+31) {
		tmp = exp(((y_46_re * t_2) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_2, t_0));
	} else {
		tmp = pow(fabs(x_46_re), y_46_re) * sin(fma(-1.0, (y_46_im * ((0.0 - (t_1 * t_1)) / (0.0 + t_1))), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(abs(x_46_re))
	t_2 = log(abs(x_46_im))
	tmp = 0.0
	if (x_46_re <= -1.8e+45)
		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(t_0));
	elseif (x_46_re <= 1.35e+31)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_2) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_2, t_0)));
	else
		tmp = Float64((abs(x_46_re) ^ y_46_re) * sin(fma(-1.0, Float64(y_46_im * Float64(Float64(0.0 - Float64(t_1 * t_1)) / Float64(0.0 + t_1))), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.8e+45], 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[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.35e+31], N[(N[Exp[N[(N[(y$46$re * t$95$2), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$2 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Abs[x$46$re], $MachinePrecision], y$46$re], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * N[(N[(0.0 - N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] / N[(0.0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.8e45

   1. Initial program 40.3%

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

   2. 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 52.6%

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 29.4%

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

   7. Applied rewrites 29.4%

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

   if -1.8e45 < x.re < 1.3499999999999999e31

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in x.re around 0

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

       1. lower-*.f64 N/A

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

   14. Applied rewrites 62.5%

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

   if 1.3499999999999999e31 < x.re

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-- N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-fabs.f64 32.7%

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

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. remove-sound-/ N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-fabs.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-+.f64 N/A

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

       17. lower-log.f64 N/A

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

       18. lower-fabs.f64 65.6%

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in y.im around 0

       \[\leadsto {\left(\left|x.re\right|\right)}^{y.re} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   13. Step-by-step derivation

       1. lower-pow.f64 N/A

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

       2. lower-fabs.f64 46.6%

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

   14. Applied rewrites 46.6%

       \[\leadsto {\left(\left|x.re\right|\right)}^{y.re} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := -\log x.re\\ t_2 := \log \left(\left|x.im\right|\right)\\ \mathbf{if}\;x.re \leq -1.8 \cdot 10^{+45}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.re\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin t\_0\\ \mathbf{elif}\;x.re \leq 1.48 \cdot 10^{+38}:\\ \;\;\;\;e^{y.re \cdot t\_2 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, t\_2, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - t\_1 \cdot y.im\right) \cdot e^{-t\_1 \cdot y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (- (log x.re)))
       (t_2 (log (fabs x.im))))
  (if (<= x.re -1.8e+45)
    (*
     (exp (- (* (log (* -1.0 x.re)) y.re) (* (atan2 x.im x.re) y.im)))
     (sin t_0))
    (if (<= x.re 1.48e+38)
      (*
       (exp (- (* y.re t_2) (* y.im (atan2 x.im x.re))))
       (sin (fma y.im t_2 t_0)))
      (*
       (sin (- (* (atan2 x.im x.re) y.re) (* t_1 y.im)))
       (exp (- (* t_1 y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = -log(x_46_re);
	double t_2 = log(fabs(x_46_im));
	double tmp;
	if (x_46_re <= -1.8e+45) {
		tmp = exp(((log((-1.0 * x_46_re)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(t_0);
	} else if (x_46_re <= 1.48e+38) {
		tmp = exp(((y_46_re * t_2) - (y_46_im * atan2(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_2, t_0));
	} else {
		tmp = sin(((atan2(x_46_im, x_46_re) * y_46_re) - (t_1 * y_46_im))) * exp(-(t_1 * y_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(-log(x_46_re))
	t_2 = log(abs(x_46_im))
	tmp = 0.0
	if (x_46_re <= -1.8e+45)
		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(t_0));
	elseif (x_46_re <= 1.48e+38)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_2) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(fma(y_46_im, t_2, t_0)));
	else
		tmp = Float64(sin(Float64(Float64(atan(x_46_im, x_46_re) * y_46_re) - Float64(t_1 * y_46_im))) * exp(Float64(-Float64(t_1 * 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$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Log[x$46$re], $MachinePrecision])}, Block[{t$95$2 = N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.8e+45], 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[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.48e+38], N[(N[Exp[N[(N[(y$46$re * t$95$2), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$2 + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(t$95$1 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[(-N[(t$95$1 * y$46$re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.8e45

   1. Initial program 40.3%

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

   2. 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 52.6%

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in x.re around -inf

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

      1. lower-*.f64 29.4%

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

   7. Applied rewrites 29.4%

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

   if -1.8e45 < x.re < 1.48e38

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in x.re around 0

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

       1. lower-*.f64 N/A

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

   14. Applied rewrites 62.5%

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

   if 1.48e38 < x.re

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in y.im around 0

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

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

   10. Applied rewrites 24.7%

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

   12. Applied rewrites 24.7%

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

Alternative 8: 59.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\frac{1}{x.im}\right)\\ t_2 := -1 \cdot \left(y.re \cdot t\_1\right)\\ t_3 := -\log \left(\left|x.re\right|\right)\\ \mathbf{if}\;x.im \leq -3.9 \cdot 10^{-112}:\\ \;\;\;\;e^{\log \left(-1 \cdot x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin t\_0\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{-161}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_3\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_3, t\_0\right)\right)\\ \mathbf{elif}\;x.im \leq 2.3 \cdot 10^{+190}:\\ \;\;\;\;e^{t\_2 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t\_2} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, t\_0\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (log (/ 1.0 x.im)))
       (t_2 (* -1.0 (* y.re t_1)))
       (t_3 (- (log (fabs x.re)))))
  (if (<= x.im -3.9e-112)
    (*
     (exp (- (* (log (* -1.0 x.im)) y.re) (* (atan2 x.im x.re) y.im)))
     (sin t_0))
    (if (<= x.im 9e-161)
      (*
       (exp (* -1.0 (* y.re t_3)))
       (sin (fma -1.0 (* y.im t_3) t_0)))
      (if (<= x.im 2.3e+190)
        (*
         (exp (- t_2 (* y.im (atan2 x.im x.re))))
         (sin (* y.im (log x.im))))
        (* (exp t_2) (sin (fma -1.0 (* y.im t_1) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log((1.0 / x_46_im));
	double t_2 = -1.0 * (y_46_re * t_1);
	double t_3 = -log(fabs(x_46_re));
	double tmp;
	if (x_46_im <= -3.9e-112) {
		tmp = exp(((log((-1.0 * x_46_im)) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(t_0);
	} else if (x_46_im <= 9e-161) {
		tmp = exp((-1.0 * (y_46_re * t_3))) * sin(fma(-1.0, (y_46_im * t_3), t_0));
	} else if (x_46_im <= 2.3e+190) {
		tmp = exp((t_2 - (y_46_im * atan2(x_46_im, x_46_re)))) * sin((y_46_im * log(x_46_im)));
	} else {
		tmp = exp(t_2) * sin(fma(-1.0, (y_46_im * t_1), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(1.0 / x_46_im))
	t_2 = Float64(-1.0 * Float64(y_46_re * t_1))
	t_3 = Float64(-log(abs(x_46_re)))
	tmp = 0.0
	if (x_46_im <= -3.9e-112)
		tmp = Float64(exp(Float64(Float64(log(Float64(-1.0 * x_46_im)) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(t_0));
	elseif (x_46_im <= 9e-161)
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_3))) * sin(fma(-1.0, Float64(y_46_im * t_3), t_0)));
	elseif (x_46_im <= 2.3e+190)
		tmp = Float64(exp(Float64(t_2 - Float64(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_im * log(x_46_im))));
	else
		tmp = Float64(exp(t_2) * sin(fma(-1.0, Float64(y_46_im * t_1), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(y$46$re * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision])}, If[LessEqual[x$46$im, -3.9e-112], N[(N[Exp[N[(N[(N[Log[N[(-1.0 * x$46$im), $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 9e-161], 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$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.3e+190], N[(N[Exp[N[(t$95$2 - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$2], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -3.9000000000000001e-112

   1. Initial program 40.3%

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

   2. 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 52.6%

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in x.im around -inf

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

      1. lower-*.f64 27.4%

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

   7. Applied rewrites 27.4%

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

   if -3.9000000000000001e-112 < x.im < 8.9999999999999993e-161

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in y.im around 0

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

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

   10. Applied rewrites 24.7%

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. lift-fabs.f64 N/A

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

       6. diff-log N/A

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

       7. neg-log N/A

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

       8. lift-log.f64 N/A

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

       9. lower-neg.f64 24.7%

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

   12. Applied rewrites 24.7%

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. lift-fabs.f64 N/A

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

       6. diff-log N/A

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

       7. neg-log N/A

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

       8. lift-log.f64 N/A

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

       9. lower-neg.f64 46.6%

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

   14. Applied rewrites 46.6%

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

   if 8.9999999999999993e-161 < x.im < 2.2999999999999999e190

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 27.1%

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

   9. Applied rewrites 27.1%

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

   if 2.2999999999999999e190 < x.im

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 22.0%

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

   7. Applied rewrites 22.0%

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

Alternative 9: 53.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\frac{1}{x.im}\right)\\ t_2 := -1 \cdot \left(y.re \cdot t\_1\right)\\ t_3 := -\log \left(\left|x.re\right|\right)\\ t_4 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq -1.2 \cdot 10^{-98}:\\ \;\;\;\;e^{-t\_4} \cdot \sin t\_0\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{-161}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_3\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_3, t\_0\right)\right)\\ \mathbf{elif}\;x.im \leq 2.3 \cdot 10^{+190}:\\ \;\;\;\;e^{t\_2 - t\_4} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \mathbf{else}:\\ \;\;\;\;e^{t\_2} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, t\_0\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (log (/ 1.0 x.im)))
       (t_2 (* -1.0 (* y.re t_1)))
       (t_3 (- (log (fabs x.re))))
       (t_4 (* y.im (atan2 x.im x.re))))
  (if (<= x.im -1.2e-98)
    (* (exp (- t_4)) (sin t_0))
    (if (<= x.im 9e-161)
      (*
       (exp (* -1.0 (* y.re t_3)))
       (sin (fma -1.0 (* y.im t_3) t_0)))
      (if (<= x.im 2.3e+190)
        (* (exp (- t_2 t_4)) (sin (* y.im (log x.im))))
        (* (exp t_2) (sin (fma -1.0 (* y.im t_1) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log((1.0 / x_46_im));
	double t_2 = -1.0 * (y_46_re * t_1);
	double t_3 = -log(fabs(x_46_re));
	double t_4 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= -1.2e-98) {
		tmp = exp(-t_4) * sin(t_0);
	} else if (x_46_im <= 9e-161) {
		tmp = exp((-1.0 * (y_46_re * t_3))) * sin(fma(-1.0, (y_46_im * t_3), t_0));
	} else if (x_46_im <= 2.3e+190) {
		tmp = exp((t_2 - t_4)) * sin((y_46_im * log(x_46_im)));
	} else {
		tmp = exp(t_2) * sin(fma(-1.0, (y_46_im * t_1), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(1.0 / x_46_im))
	t_2 = Float64(-1.0 * Float64(y_46_re * t_1))
	t_3 = Float64(-log(abs(x_46_re)))
	t_4 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= -1.2e-98)
		tmp = Float64(exp(Float64(-t_4)) * sin(t_0));
	elseif (x_46_im <= 9e-161)
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_3))) * sin(fma(-1.0, Float64(y_46_im * t_3), t_0)));
	elseif (x_46_im <= 2.3e+190)
		tmp = Float64(exp(Float64(t_2 - t_4)) * sin(Float64(y_46_im * log(x_46_im))));
	else
		tmp = Float64(exp(t_2) * sin(fma(-1.0, Float64(y_46_im * t_1), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(y$46$re * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = (-N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision])}, Block[{t$95$4 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.2e-98], N[(N[Exp[(-t$95$4)], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 9e-161], 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$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.3e+190], N[(N[Exp[N[(t$95$2 - t$95$4), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[t$95$2], $MachinePrecision] * N[Sin[N[(-1.0 * N[(y$46$im * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -1.2e-98

   1. Initial program 40.3%

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

   2. 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 52.6%

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 39.7%

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

   7. Applied rewrites 39.7%

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

   if -1.2e-98 < x.im < 8.9999999999999993e-161

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in y.im around 0

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

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

   10. Applied rewrites 24.7%

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. lift-fabs.f64 N/A

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

       6. diff-log N/A

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

       7. neg-log N/A

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

       8. lift-log.f64 N/A

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

       9. lower-neg.f64 24.7%

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

   12. Applied rewrites 24.7%

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. lift-fabs.f64 N/A

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

       6. diff-log N/A

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

       7. neg-log N/A

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

       8. lift-log.f64 N/A

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

       9. lower-neg.f64 46.6%

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

   14. Applied rewrites 46.6%

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

   if 8.9999999999999993e-161 < x.im < 2.2999999999999999e190

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 27.1%

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

   9. Applied rewrites 27.1%

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

   if 2.2999999999999999e190 < x.im

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-/.f64 22.0%

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

   7. Applied rewrites 22.0%

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

Alternative 10: 52.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := -\log \left(\left|x.re\right|\right)\\ t_2 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq -1.2 \cdot 10^{-98}:\\ \;\;\;\;e^{-t\_2} \cdot \sin t\_0\\ \mathbf{elif}\;x.im \leq 9 \cdot 10^{-161}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot t\_1\right)} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot t\_1, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - t\_2} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (* y.re (atan2 x.im x.re)))
       (t_1 (- (log (fabs x.re))))
       (t_2 (* y.im (atan2 x.im x.re))))
  (if (<= x.im -1.2e-98)
    (* (exp (- t_2)) (sin t_0))
    (if (<= x.im 9e-161)
      (*
       (exp (* -1.0 (* y.re t_1)))
       (sin (fma -1.0 (* y.im t_1) t_0)))
      (*
       (exp (- (* -1.0 (* y.re (log (/ 1.0 x.im)))) t_2))
       (sin (* y.im (log x.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = -log(fabs(x_46_re));
	double t_2 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= -1.2e-98) {
		tmp = exp(-t_2) * sin(t_0);
	} else if (x_46_im <= 9e-161) {
		tmp = exp((-1.0 * (y_46_re * t_1))) * sin(fma(-1.0, (y_46_im * t_1), t_0));
	} else {
		tmp = exp(((-1.0 * (y_46_re * log((1.0 / x_46_im)))) - t_2)) * sin((y_46_im * log(x_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(-log(abs(x_46_re)))
	t_2 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= -1.2e-98)
		tmp = Float64(exp(Float64(-t_2)) * sin(t_0));
	elseif (x_46_im <= 9e-161)
		tmp = Float64(exp(Float64(-1.0 * Float64(y_46_re * t_1))) * sin(fma(-1.0, Float64(y_46_im * t_1), t_0)));
	else
		tmp = Float64(exp(Float64(Float64(-1.0 * Float64(y_46_re * log(Float64(1.0 / x_46_im)))) - t_2)) * sin(Float64(y_46_im * log(x_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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision])}, Block[{t$95$2 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.2e-98], N[(N[Exp[(-t$95$2)], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 9e-161], 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$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(-1.0 * N[(y$46$re * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.2e-98

   1. Initial program 40.3%

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

   2. 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 52.6%

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 39.7%

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

   7. Applied rewrites 39.7%

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

   if -1.2e-98 < x.im < 8.9999999999999993e-161

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in y.im around 0

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

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

   10. Applied rewrites 24.7%

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. lift-fabs.f64 N/A

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

       6. diff-log N/A

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

       7. neg-log N/A

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

       8. lift-log.f64 N/A

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

       9. lower-neg.f64 24.7%

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

   12. Applied rewrites 24.7%

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

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. lift-fabs.f64 N/A

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

       6. diff-log N/A

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

       7. neg-log N/A

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

       8. lift-log.f64 N/A

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

       9. lower-neg.f64 46.6%

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

   14. Applied rewrites 46.6%

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

   if 8.9999999999999993e-161 < x.im

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 27.1%

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

   9. Applied rewrites 27.1%

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

Alternative 11: 51.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := -\log x.re\\ t_1 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{-t\_1}\\ t_3 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -2.8 \cdot 10^{+94}:\\ \;\;\;\;t\_2 \cdot t\_3\\ \mathbf{elif}\;x.im \leq -9.2 \cdot 10^{-146}:\\ \;\;\;\;t\_2 \cdot \sin \left(y.im \cdot \log \left(\left|x.re\right|\right)\right)\\ \mathbf{elif}\;x.im \leq -9.5 \cdot 10^{-298}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - t\_0 \cdot y.im\right) \cdot e^{-t\_0 \cdot y.re}\\ \mathbf{elif}\;x.im \leq 10^{-161}:\\ \;\;\;\;t\_3 \cdot {\left(\left|x.re\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - t\_1} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (- (log x.re)))
       (t_1 (* y.im (atan2 x.im x.re)))
       (t_2 (exp (- t_1)))
       (t_3 (sin (* y.re (atan2 x.im x.re)))))
  (if (<= x.im -2.8e+94)
    (* t_2 t_3)
    (if (<= x.im -9.2e-146)
      (* t_2 (sin (* y.im (log (fabs x.re)))))
      (if (<= x.im -9.5e-298)
        (*
         (sin (- (* (atan2 x.im x.re) y.re) (* t_0 y.im)))
         (exp (- (* t_0 y.re))))
        (if (<= x.im 1e-161)
          (* t_3 (pow (fabs x.re) y.re))
          (*
           (exp (- (* -1.0 (* y.re (log (/ 1.0 x.im)))) t_1))
           (sin (* y.im (log x.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -log(x_46_re);
	double t_1 = y_46_im * atan2(x_46_im, x_46_re);
	double t_2 = exp(-t_1);
	double t_3 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -2.8e+94) {
		tmp = t_2 * t_3;
	} else if (x_46_im <= -9.2e-146) {
		tmp = t_2 * sin((y_46_im * log(fabs(x_46_re))));
	} else if (x_46_im <= -9.5e-298) {
		tmp = sin(((atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * exp(-(t_0 * y_46_re));
	} else if (x_46_im <= 1e-161) {
		tmp = t_3 * pow(fabs(x_46_re), y_46_re);
	} else {
		tmp = exp(((-1.0 * (y_46_re * log((1.0 / x_46_im)))) - t_1)) * sin((y_46_im * log(x_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = -log(x_46re)
    t_1 = y_46im * atan2(x_46im, x_46re)
    t_2 = exp(-t_1)
    t_3 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46im <= (-2.8d+94)) then
        tmp = t_2 * t_3
    else if (x_46im <= (-9.2d-146)) then
        tmp = t_2 * sin((y_46im * log(abs(x_46re))))
    else if (x_46im <= (-9.5d-298)) then
        tmp = sin(((atan2(x_46im, x_46re) * y_46re) - (t_0 * y_46im))) * exp(-(t_0 * y_46re))
    else if (x_46im <= 1d-161) then
        tmp = t_3 * (abs(x_46re) ** y_46re)
    else
        tmp = exp((((-1.0d0) * (y_46re * log((1.0d0 / x_46im)))) - t_1)) * sin((y_46im * log(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 t_0 = -Math.log(x_46_re);
	double t_1 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.exp(-t_1);
	double t_3 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -2.8e+94) {
		tmp = t_2 * t_3;
	} else if (x_46_im <= -9.2e-146) {
		tmp = t_2 * Math.sin((y_46_im * Math.log(Math.abs(x_46_re))));
	} else if (x_46_im <= -9.5e-298) {
		tmp = Math.sin(((Math.atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * Math.exp(-(t_0 * y_46_re));
	} else if (x_46_im <= 1e-161) {
		tmp = t_3 * Math.pow(Math.abs(x_46_re), y_46_re);
	} else {
		tmp = Math.exp(((-1.0 * (y_46_re * Math.log((1.0 / x_46_im)))) - t_1)) * Math.sin((y_46_im * Math.log(x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -math.log(x_46_re)
	t_1 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_2 = math.exp(-t_1)
	t_3 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -2.8e+94:
		tmp = t_2 * t_3
	elif x_46_im <= -9.2e-146:
		tmp = t_2 * math.sin((y_46_im * math.log(math.fabs(x_46_re))))
	elif x_46_im <= -9.5e-298:
		tmp = math.sin(((math.atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * math.exp(-(t_0 * y_46_re))
	elif x_46_im <= 1e-161:
		tmp = t_3 * math.pow(math.fabs(x_46_re), y_46_re)
	else:
		tmp = math.exp(((-1.0 * (y_46_re * math.log((1.0 / x_46_im)))) - t_1)) * math.sin((y_46_im * math.log(x_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(-log(x_46_re))
	t_1 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_2 = exp(Float64(-t_1))
	t_3 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -2.8e+94)
		tmp = Float64(t_2 * t_3);
	elseif (x_46_im <= -9.2e-146)
		tmp = Float64(t_2 * sin(Float64(y_46_im * log(abs(x_46_re)))));
	elseif (x_46_im <= -9.5e-298)
		tmp = Float64(sin(Float64(Float64(atan(x_46_im, x_46_re) * y_46_re) - Float64(t_0 * y_46_im))) * exp(Float64(-Float64(t_0 * y_46_re))));
	elseif (x_46_im <= 1e-161)
		tmp = Float64(t_3 * (abs(x_46_re) ^ y_46_re));
	else
		tmp = Float64(exp(Float64(Float64(-1.0 * Float64(y_46_re * log(Float64(1.0 / x_46_im)))) - t_1)) * sin(Float64(y_46_im * log(x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -log(x_46_re);
	t_1 = y_46_im * atan2(x_46_im, x_46_re);
	t_2 = exp(-t_1);
	t_3 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -2.8e+94)
		tmp = t_2 * t_3;
	elseif (x_46_im <= -9.2e-146)
		tmp = t_2 * sin((y_46_im * log(abs(x_46_re))));
	elseif (x_46_im <= -9.5e-298)
		tmp = sin(((atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * exp(-(t_0 * y_46_re));
	elseif (x_46_im <= 1e-161)
		tmp = t_3 * (abs(x_46_re) ^ y_46_re);
	else
		tmp = exp(((-1.0 * (y_46_re * log((1.0 / x_46_im)))) - t_1)) * sin((y_46_im * log(x_46_im)));
	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[x$46$re], $MachinePrecision])}, Block[{t$95$1 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$1)], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -2.8e+94], N[(t$95$2 * t$95$3), $MachinePrecision], If[LessEqual[x$46$im, -9.2e-146], N[(t$95$2 * N[Sin[N[(y$46$im * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -9.5e-298], N[(N[Sin[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(t$95$0 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[(-N[(t$95$0 * y$46$re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1e-161], N[(t$95$3 * N[Power[N[Abs[x$46$re], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(-1.0 * N[(y$46$re * N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.im < -2.8e94

   1. Initial program 40.3%

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

   2. 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 52.6%

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 39.7%

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

   7. Applied rewrites 39.7%

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

   if -2.8e94 < x.im < -9.2000000000000003e-146

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-- N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-fabs.f64 32.7%

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

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. remove-sound-/ N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-fabs.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-+.f64 N/A

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

       17. lower-log.f64 N/A

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

       18. lower-fabs.f64 65.6%

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in y.re around 0

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

       1. lower-*.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-atan2.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. lower-fabs.f64 38.4%

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

   14. Applied rewrites 38.4%

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

   if -9.2000000000000003e-146 < x.im < -9.5000000000000001e-298

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in y.im around 0

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

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

   10. Applied rewrites 24.7%

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

   12. Applied rewrites 24.7%

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

   if -9.5000000000000001e-298 < x.im < 1e-161

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-- N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-fabs.f64 32.7%

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

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. remove-sound-/ N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-fabs.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-+.f64 N/A

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

       17. lower-log.f64 N/A

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

       18. lower-fabs.f64 65.6%

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in y.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-fabs.f64 37.1%

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

   14. Applied rewrites 37.1%

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

   if 1e-161 < x.im

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-fma.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. flip-+ N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

   6. Applied rewrites 16.7%

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

   7. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 27.1%

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

   9. Applied rewrites 27.1%

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

Alternative 12: 50.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := -\log x.re\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.re \leq -14:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_1\\ \mathbf{elif}\;x.re \leq 29500000000:\\ \;\;\;\;t\_1 \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - t\_0 \cdot y.im\right) \cdot e^{-t\_0 \cdot y.re}\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (- (log x.re))) (t_1 (sin (* y.re (atan2 x.im x.re)))))
  (if (<= x.re -14.0)
    (* (exp (- (* y.im (atan2 x.im x.re)))) t_1)
    (if (<= x.re 29500000000.0)
      (* t_1 (pow (fabs x.im) y.re))
      (*
       (sin (- (* (atan2 x.im x.re) y.re) (* t_0 y.im)))
       (exp (- (* t_0 y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -log(x_46_re);
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -14.0) {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	} else if (x_46_re <= 29500000000.0) {
		tmp = t_1 * pow(fabs(x_46_im), y_46_re);
	} else {
		tmp = sin(((atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * exp(-(t_0 * y_46_re));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -log(x_46re)
    t_1 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46re <= (-14.0d0)) then
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * t_1
    else if (x_46re <= 29500000000.0d0) then
        tmp = t_1 * (abs(x_46im) ** y_46re)
    else
        tmp = sin(((atan2(x_46im, x_46re) * y_46re) - (t_0 * y_46im))) * exp(-(t_0 * y_46re))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = -Math.log(x_46_re);
	double t_1 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -14.0) {
		tmp = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * t_1;
	} else if (x_46_re <= 29500000000.0) {
		tmp = t_1 * Math.pow(Math.abs(x_46_im), y_46_re);
	} else {
		tmp = Math.sin(((Math.atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * Math.exp(-(t_0 * y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = -math.log(x_46_re)
	t_1 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_re <= -14.0:
		tmp = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))) * t_1
	elif x_46_re <= 29500000000.0:
		tmp = t_1 * math.pow(math.fabs(x_46_im), y_46_re)
	else:
		tmp = math.sin(((math.atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * math.exp(-(t_0 * y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(-log(x_46_re))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_re <= -14.0)
		tmp = Float64(exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_1);
	elseif (x_46_re <= 29500000000.0)
		tmp = Float64(t_1 * (abs(x_46_im) ^ y_46_re));
	else
		tmp = Float64(sin(Float64(Float64(atan(x_46_im, x_46_re) * y_46_re) - Float64(t_0 * y_46_im))) * exp(Float64(-Float64(t_0 * y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = -log(x_46_re);
	t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_re <= -14.0)
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * t_1;
	elseif (x_46_re <= 29500000000.0)
		tmp = t_1 * (abs(x_46_im) ^ y_46_re);
	else
		tmp = sin(((atan2(x_46_im, x_46_re) * y_46_re) - (t_0 * y_46_im))) * exp(-(t_0 * y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = (-N[Log[x$46$re], $MachinePrecision])}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -14.0], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x$46$re, 29500000000.0], N[(t$95$1 * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(t$95$0 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[(-N[(t$95$0 * y$46$re), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -14

   1. Initial program 40.3%

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

   2. 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 52.6%

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

   4. Applied rewrites 52.6%

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

   5. Taylor expanded in y.re around 0

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

      1. lower-exp.f64 N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-atan2.f64 39.7%

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

   7. Applied rewrites 39.7%

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

   if -14 < x.re < 2.95e10

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in y.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-fabs.f64 37.1%

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

   14. Applied rewrites 37.1%

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

   if 2.95e10 < x.re

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

   8. Taylor expanded in y.im around 0

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

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

   10. Applied rewrites 24.7%

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

   12. Applied rewrites 24.7%

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

Alternative 13: 49.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+30}:\\ \;\;\;\;t\_0 \cdot \sin \left(y.im \cdot \log \left(\left|x.im\right|\right)\right)\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-13}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sin \left(y.im \cdot \log \left(\left|x.re\right|\right)\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (exp (- (* y.im (atan2 x.im x.re))))))
  (if (<= y.im -3.6e+30)
    (* t_0 (sin (* y.im (log (fabs x.im)))))
    (if (<= y.im 1.25e-13)
      (* (sin (* y.re (atan2 x.im x.re))) (pow (fabs x.im) y.re))
      (* t_0 (sin (* y.im (log (fabs x.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(-(y_46_im * atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0 * sin((y_46_im * log(fabs(x_46_im))));
	} else if (y_46_im <= 1.25e-13) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(fabs(x_46_im), y_46_re);
	} else {
		tmp = t_0 * sin((y_46_im * log(fabs(x_46_re))));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-(y_46im * atan2(x_46im, x_46re)))
    if (y_46im <= (-3.6d+30)) then
        tmp = t_0 * sin((y_46im * log(abs(x_46im))))
    else if (y_46im <= 1.25d-13) then
        tmp = sin((y_46re * atan2(x_46im, x_46re))) * (abs(x_46im) ** y_46re)
    else
        tmp = t_0 * sin((y_46im * log(abs(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 t_0 = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0 * Math.sin((y_46_im * Math.log(Math.abs(x_46_im))));
	} else if (y_46_im <= 1.25e-13) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.abs(x_46_im), y_46_re);
	} else {
		tmp = t_0 * Math.sin((y_46_im * Math.log(Math.abs(x_46_re))));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -3.6e+30:
		tmp = t_0 * math.sin((y_46_im * math.log(math.fabs(x_46_im))))
	elif y_46_im <= 1.25e-13:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.fabs(x_46_im), y_46_re)
	else:
		tmp = t_0 * math.sin((y_46_im * math.log(math.fabs(x_46_re))))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_im <= -3.6e+30)
		tmp = Float64(t_0 * sin(Float64(y_46_im * log(abs(x_46_im)))));
	elseif (y_46_im <= 1.25e-13)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (abs(x_46_im) ^ y_46_re));
	else
		tmp = Float64(t_0 * sin(Float64(y_46_im * log(abs(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 = exp(-(y_46_im * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -3.6e+30)
		tmp = t_0 * sin((y_46_im * log(abs(x_46_im))));
	elseif (y_46_im <= 1.25e-13)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (abs(x_46_im) ^ y_46_re);
	else
		tmp = t_0 * sin((y_46_im * log(abs(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[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[y$46$im, -3.6e+30], N[(t$95$0 * N[Sin[N[(y$46$im * N[Log[N[Abs[x$46$im], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.25e-13], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Sin[N[(y$46$im * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.6000000000000002e30

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in y.re around 0

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

       1. lower-*.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-atan2.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. lower-fabs.f64 36.8%

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

   14. Applied rewrites 36.8%

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

   if -3.6000000000000002e30 < y.im < 1.25e-13

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in y.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-fabs.f64 37.1%

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

   14. Applied rewrites 37.1%

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

   if 1.25e-13 < y.im

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-- N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-fabs.f64 32.7%

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

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. remove-sound-/ N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-fabs.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-+.f64 N/A

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

       17. lower-log.f64 N/A

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

       18. lower-fabs.f64 65.6%

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in y.re around 0

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

       1. lower-*.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-atan2.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. lower-fabs.f64 38.4%

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

   14. Applied rewrites 38.4%

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

Alternative 14: 47.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\left|x.re\right|\right)\right)\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+30}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-13}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0
        (*
         (exp (- (* y.im (atan2 x.im x.re))))
         (sin (* y.im (log (fabs x.re)))))))
  (if (<= y.im -3.6e+30)
    t_0
    (if (<= y.im 1.25e-13)
      (* (sin (* y.re (atan2 x.im x.re))) (pow (fabs x.im) y.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(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(fabs(x_46_re))));
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e-13) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(fabs(x_46_im), y_46_re);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-(y_46im * atan2(x_46im, x_46re))) * sin((y_46im * log(abs(x_46re))))
    if (y_46im <= (-3.6d+30)) then
        tmp = t_0
    else if (y_46im <= 1.25d-13) then
        tmp = sin((y_46re * atan2(x_46im, x_46re))) * (abs(x_46im) ** y_46re)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((y_46_im * Math.log(Math.abs(x_46_re))));
	double tmp;
	if (y_46_im <= -3.6e+30) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e-13) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.abs(x_46_im), y_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(-(y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((y_46_im * math.log(math.fabs(x_46_re))))
	tmp = 0
	if y_46_im <= -3.6e+30:
		tmp = t_0
	elif y_46_im <= 1.25e-13:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.fabs(x_46_im), y_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(y_46_im * atan(x_46_im, x_46_re)))) * sin(Float64(y_46_im * log(abs(x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 1.25e-13)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (abs(x_46_im) ^ y_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(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(abs(x_46_re))));
	tmp = 0.0;
	if (y_46_im <= -3.6e+30)
		tmp = t_0;
	elseif (y_46_im <= 1.25e-13)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (abs(x_46_im) ^ y_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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.6e+30], t$95$0, If[LessEqual[y$46$im, 1.25e-13], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.6000000000000002e30 or 1.25e-13 < y.im

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-- N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-fabs.f64 32.7%

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

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. remove-sound-/ N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-fabs.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-+.f64 N/A

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

       17. lower-log.f64 N/A

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

       18. lower-fabs.f64 65.6%

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in y.re around 0

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

       1. lower-*.f64 N/A

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

       2. lower-exp.f64 N/A

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

       3. lower-neg.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-atan2.f64 N/A

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

       6. lower-sin.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-log.f64 N/A

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

       9. lower-fabs.f64 38.4%

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

   14. Applied rewrites 38.4%

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

   if -3.6000000000000002e30 < y.im < 1.25e-13

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in y.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-fabs.f64 37.1%

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

   14. Applied rewrites 37.1%

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

Alternative 15: 44.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -7400000:\\ \;\;\;\;t\_0 \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 3 \cdot 10^{-11}:\\ \;\;\;\;t\_0 \cdot {\left(\left|x.re\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re)))))
  (if (<= x.im -7400000.0)
    (* t_0 (pow (fabs x.im) y.re))
    (if (<= x.im 3e-11)
      (* t_0 (pow (fabs x.re) y.re))
      (*
       (exp (- (* y.im (atan2 x.im x.re))))
       (sin (* y.im (log x.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 (x_46_im <= -7400000.0) {
		tmp = t_0 * pow(fabs(x_46_im), y_46_re);
	} else if (x_46_im <= 3e-11) {
		tmp = t_0 * pow(fabs(x_46_re), y_46_re);
	} else {
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(x_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    if (x_46im <= (-7400000.0d0)) then
        tmp = t_0 * (abs(x_46im) ** y_46re)
    else if (x_46im <= 3d-11) then
        tmp = t_0 * (abs(x_46re) ** y_46re)
    else
        tmp = exp(-(y_46im * atan2(x_46im, x_46re))) * sin((y_46im * log(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 t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -7400000.0) {
		tmp = t_0 * Math.pow(Math.abs(x_46_im), y_46_re);
	} else if (x_46_im <= 3e-11) {
		tmp = t_0 * Math.pow(Math.abs(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_im * Math.log(x_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -7400000.0:
		tmp = t_0 * math.pow(math.fabs(x_46_im), y_46_re)
	elif x_46_im <= 3e-11:
		tmp = t_0 * math.pow(math.fabs(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_im * math.log(x_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 (x_46_im <= -7400000.0)
		tmp = Float64(t_0 * (abs(x_46_im) ^ y_46_re));
	elseif (x_46_im <= 3e-11)
		tmp = Float64(t_0 * (abs(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_im * log(x_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -7400000.0)
		tmp = t_0 * (abs(x_46_im) ^ y_46_re);
	elseif (x_46_im <= 3e-11)
		tmp = t_0 * (abs(x_46_re) ^ y_46_re);
	else
		tmp = exp(-(y_46_im * atan2(x_46_im, x_46_re))) * sin((y_46_im * log(x_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -7400000.0], N[(t$95$0 * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 3e-11], N[(t$95$0 * N[Power[N[Abs[x$46$re], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -7.4e6

   1. Initial program 40.3%

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

   2. 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 31.2%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in y.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 N/A

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

       5. lower-pow.f64 N/A

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

       6. lower-fabs.f64 37.1%

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

   14. Applied rewrites 37.1%

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

   if -7.4e6 < x.im < 3e-11

   1. Initial program 40.3%

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

   2. 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 31.2%

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

   5. Taylor expanded in x.re around inf

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

      1. lower-*.f64 N/A

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

   7. Applied rewrites 32.7%

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

      1. lift-log.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. log-div N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. flip-- N/A

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

      7. remove-sound-/ N/A

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

      8. lower-/.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-log.f64 N/A

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

      13. lower-fabs.f64 N/A

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

      14. lower-log.f64 N/A

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

      15. lower-fabs.f64 N/A

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

      16. lower-+.f64 N/A

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

      17. lower-log.f64 N/A

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

      18. lower-fabs.f64 32.7%

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

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. log-div N/A

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

       4. metadata-eval N/A

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

       5. metadata-eval N/A

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

       6. flip-- N/A

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

       7. remove-sound-/ N/A

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

       8. lower-/.f64 N/A

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

       9. metadata-eval N/A

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

       10. lower--.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-log.f64 N/A

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

       13. lower-fabs.f64 N/A

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

       14. lower-log.f64 N/A

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

       15. lower-fabs.f64 N/A

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

       16. lower-+.f64 N/A

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

       17. lower-log.f64 N/A

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

       18. lower-fabs.f64 65.6%

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

   11. Applied rewrites 65.6%

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

   12. Taylor expanded in y.im around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sin.f64 N/A

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

       3. lower-*.f64 N/A

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

       4. lower-atan2.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       5. lower-pow.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       6. lower-fabs.f64 37.1%

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

   14. Applied rewrites 37.1%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\left|x.re\right|\right)}^{y.re}} \]

   if 3e-11 < x.im

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. *-commutative N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. lift-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right) + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. +-commutative N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re + -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \]

      6. flip-+ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\frac{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \cdot \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right)}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)}\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\frac{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \cdot \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right)}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\frac{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right) \cdot \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)\right)}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - -1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.im}\right)\right)}\right) \]

   6. Applied rewrites 16.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\frac{{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{2} - \left(-\log x.im \cdot y.im\right) \cdot \left(-\log x.im \cdot y.im\right)}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re - \left(-y.im\right) \cdot \left(-\log x.im\right)}\right) \]

   7. Taylor expanded in y.re around 0

      \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \color{blue}{\sin \left(y.im \cdot \log x.im\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log x.im\right) \]

      2. lower-exp.f64 N/A

         \[\leadsto e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log x.im\right) \]

      3. lower-neg.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right) \]

      4. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right) \]

      5. lower-atan2.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right) \]

      7. lower-*.f64 N/A

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right) \]

      8. lower-log.f64 18.3%

         \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log x.im\right) \]

   9. Applied rewrites 18.3%

      \[\leadsto e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(y.im \cdot \log x.im\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 43.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot {\left(\left|x.im\right|\right)}^{y.re}\\ \mathbf{if}\;x.im \leq -7400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.im \leq 5.2 \cdot 10^{-27}:\\ \;\;\;\;t\_0 \cdot {\left(\left|x.re\right|\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
       (t_1 (* t_0 (pow (fabs x.im) y.re))))
  (if (<= x.im -7400000.0)
    t_1
    (if (<= x.im 5.2e-27) (* t_0 (pow (fabs x.re) y.re)) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * pow(fabs(x_46_im), y_46_re);
	double tmp;
	if (x_46_im <= -7400000.0) {
		tmp = t_1;
	} else if (x_46_im <= 5.2e-27) {
		tmp = t_0 * pow(fabs(x_46_re), y_46_re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = t_0 * (abs(x_46im) ** y_46re)
    if (x_46im <= (-7400000.0d0)) then
        tmp = t_1
    else if (x_46im <= 5.2d-27) then
        tmp = t_0 * (abs(x_46re) ** y_46re)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * Math.pow(Math.abs(x_46_im), y_46_re);
	double tmp;
	if (x_46_im <= -7400000.0) {
		tmp = t_1;
	} else if (x_46_im <= 5.2e-27) {
		tmp = t_0 * Math.pow(Math.abs(x_46_re), y_46_re);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = t_0 * math.pow(math.fabs(x_46_im), y_46_re)
	tmp = 0
	if x_46_im <= -7400000.0:
		tmp = t_1
	elif x_46_im <= 5.2e-27:
		tmp = t_0 * math.pow(math.fabs(x_46_re), y_46_re)
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_0 * (abs(x_46_im) ^ y_46_re))
	tmp = 0.0
	if (x_46_im <= -7400000.0)
		tmp = t_1;
	elseif (x_46_im <= 5.2e-27)
		tmp = Float64(t_0 * (abs(x_46_re) ^ y_46_re));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = t_0 * (abs(x_46_im) ^ y_46_re);
	tmp = 0.0;
	if (x_46_im <= -7400000.0)
		tmp = t_1;
	elseif (x_46_im <= 5.2e-27)
		tmp = t_0 * (abs(x_46_re) ^ y_46_re);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Abs[x$46$im], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -7400000.0], t$95$1, If[LessEqual[x$46$im, 5.2e-27], N[(t$95$0 * N[Power[N[Abs[x$46$re], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -7.4e6 or 5.2000000000000003e-27 < x.im

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   5. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(0 - \log \left(\left|x.im\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   6. Applied rewrites 31.2%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.im\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip3-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{{0}^{3} - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-pow.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 \cdot 0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \left(\log \left(\left|x.im\right|\right) \cdot \log \left(\left|x.im\right|\right) + 0 \cdot \log \left(\left|x.im\right|\right)\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   8. Applied rewrites 62.6%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Taylor expanded in y.re around 0

      \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lower-exp.f64 N/A

          \[\leadsto \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, \color{blue}{y.im} \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. lower-atan2.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \color{blue}{\frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-exp.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{\color{blue}{0} + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. lower-neg.f64 N/A

          \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-atan2.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \color{blue}{\mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 41.5%

       \[\leadsto \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} + y.re \cdot \left(e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \log \left(\left|x.im\right|\right)\right)\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - {\log \left(\left|x.im\right|\right)}^{3}}{0 + \mathsf{fma}\left(\log \left(\left|x.im\right|\right), \log \left(\left|x.im\right|\right), 0\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   12. Taylor expanded in y.im around 0

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\left|x.im\right|\right)}^{y.re}} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{\color{blue}{y.re}} \]

       2. lower-sin.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

       3. lower-*.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

       4. lower-atan2.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

       5. lower-pow.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

       6. lower-fabs.f64 37.1%

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.im\right|\right)}^{y.re} \]

   14. Applied rewrites 37.1%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\left|x.im\right|\right)}^{y.re}} \]

   if -7.4e6 < x.im < 5.2000000000000003e-27

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   5. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.re\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(\log 1 - \log \left(\left|x.re\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(0 - \log \left(\left|x.re\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      16. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      17. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      18. lower-fabs.f64 32.7%

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lift-/.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. log-div N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. flip-- N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. remove-sound-/ N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower--.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       13. lower-fabs.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       14. lower-log.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       15. lower-fabs.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       16. lower-+.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       17. lower-log.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       18. lower-fabs.f64 65.6%

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   12. Taylor expanded in y.im around 0

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\left|x.re\right|\right)}^{y.re}} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{\color{blue}{y.re}} \]

       2. lower-sin.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       3. lower-*.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       4. lower-atan2.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       5. lower-pow.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       6. lower-fabs.f64 37.1%

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

   14. Applied rewrites 37.1%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\left|x.re\right|\right)}^{y.re}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 41.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot {\left(\left|x.re\right|\right)}^{y.re}\\ \mathbf{if}\;x.re \leq -600000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.re \leq 8.5 \cdot 10^{-16}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
       (t_1 (* t_0 (pow (fabs x.re) y.re))))
  (if (<= x.re -600000.0)
    t_1
    (if (<= x.re 8.5e-16) (* (pow x.im y.re) 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * pow(fabs(x_46_re), y_46_re);
	double tmp;
	if (x_46_re <= -600000.0) {
		tmp = t_1;
	} else if (x_46_re <= 8.5e-16) {
		tmp = pow(x_46_im, y_46_re) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = t_0 * (abs(x_46re) ** y_46re)
    if (x_46re <= (-600000.0d0)) then
        tmp = t_1
    else if (x_46re <= 8.5d-16) then
        tmp = (x_46im ** y_46re) * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * Math.pow(Math.abs(x_46_re), y_46_re);
	double tmp;
	if (x_46_re <= -600000.0) {
		tmp = t_1;
	} else if (x_46_re <= 8.5e-16) {
		tmp = Math.pow(x_46_im, y_46_re) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = t_0 * math.pow(math.fabs(x_46_re), y_46_re)
	tmp = 0
	if x_46_re <= -600000.0:
		tmp = t_1
	elif x_46_re <= 8.5e-16:
		tmp = math.pow(x_46_im, y_46_re) * 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 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_0 * (abs(x_46_re) ^ y_46_re))
	tmp = 0.0
	if (x_46_re <= -600000.0)
		tmp = t_1;
	elseif (x_46_re <= 8.5e-16)
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = t_0 * (abs(x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (x_46_re <= -600000.0)
		tmp = t_1;
	elseif (x_46_re <= 8.5e-16)
		tmp = (x_46_im ^ y_46_re) * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[Abs[x$46$re], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -600000.0], t$95$1, If[LessEqual[x$46$re, 8.5e-16], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -6e5 or 8.5000000000000001e-16 < x.re

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   5. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \color{blue}{\sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   7. Applied rewrites 32.7%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   8. Step-by-step derivation

      1. lift-log.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. lift-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. log-div N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.re\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(\log 1 - \log \left(\left|x.re\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \left(0 - \log \left(\left|x.re\right|\right)\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      6. flip-- N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. remove-sound-/ N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      8. lower-/.f64 N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. metadata-eval N/A

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      10. lower--.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      11. lower-*.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      12. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      13. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      14. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      15. lower-fabs.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      16. lower-+.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      17. lower-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      18. lower-fabs.f64 32.7%

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   9. Applied rewrites 32.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

       1. lift-log.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       2. lift-/.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.re}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       3. log-div N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log \left(\left|1\right|\right) - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       4. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(\log 1 - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       5. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \left(0 - \log \left(\left|x.re\right|\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. flip-- N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       7. remove-sound-/ N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       8. lower-/.f64 N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 \cdot 0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       9. metadata-eval N/A

          \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       10. lower--.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       12. lower-log.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       13. lower-fabs.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       14. lower-log.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       15. lower-fabs.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       16. lower-+.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       17. lower-log.f64 N/A

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       18. lower-fabs.f64 65.6%

           \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   11. Applied rewrites 65.6%

       \[\leadsto e^{-1 \cdot \left(y.re \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \frac{0 - \log \left(\left|x.re\right|\right) \cdot \log \left(\left|x.re\right|\right)}{0 + \log \left(\left|x.re\right|\right)}, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

   12. Taylor expanded in y.im around 0

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\left|x.re\right|\right)}^{y.re}} \]
   13. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{\color{blue}{y.re}} \]

       2. lower-sin.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       3. lower-*.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       4. lower-atan2.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       5. lower-pow.f64 N/A

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

       6. lower-fabs.f64 37.1%

          \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\left|x.re\right|\right)}^{y.re} \]

   14. Applied rewrites 37.1%

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \color{blue}{{\left(\left|x.re\right|\right)}^{y.re}} \]

   if -6e5 < x.re < 8.5000000000000001e-16

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   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 18.7%

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   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 14.1%

       \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Taylor expanded in x.im around 0

       \[\leadsto {x.im}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. lower-pow.f64 30.9%

          \[\leadsto {x.im}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied rewrites 30.9%

       \[\leadsto {x.im}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 36.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := {x.im}^{y.re} \cdot t\_0\\ \mathbf{if}\;y.re \leq -15600000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 11.6:\\ \;\;\;\;1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
       (t_1 (* (pow x.im y.re) t_0)))
  (if (<= y.re -15600000000.0)
    t_1
    (if (<= y.re 11.6) (* 1.0 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = pow(x_46_im, y_46_re) * t_0;
	double tmp;
	if (y_46_re <= -15600000000.0) {
		tmp = t_1;
	} else if (y_46_re <= 11.6) {
		tmp = 1.0 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
use fmin_fmax_functions
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = (x_46im ** y_46re) * t_0
    if (y_46re <= (-15600000000.0d0)) then
        tmp = t_1
    else if (y_46re <= 11.6d0) then
        tmp = 1.0d0 * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.pow(x_46_im, y_46_re) * t_0;
	double tmp;
	if (y_46_re <= -15600000000.0) {
		tmp = t_1;
	} else if (y_46_re <= 11.6) {
		tmp = 1.0 * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.pow(x_46_im, y_46_re) * t_0
	tmp = 0
	if y_46_re <= -15600000000.0:
		tmp = t_1
	elif y_46_re <= 11.6:
		tmp = 1.0 * 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 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64((x_46_im ^ y_46_re) * t_0)
	tmp = 0.0
	if (y_46_re <= -15600000000.0)
		tmp = t_1;
	elseif (y_46_re <= 11.6)
		tmp = Float64(1.0 * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = (x_46_im ^ y_46_re) * t_0;
	tmp = 0.0;
	if (y_46_re <= -15600000000.0)
		tmp = t_1;
	elseif (y_46_re <= 11.6)
		tmp = 1.0 * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -15600000000.0], t$95$1, If[LessEqual[y$46$re, 11.6], N[(1.0 * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.56e10 or 11.6 < y.re

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   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 18.7%

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   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 14.1%

       \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Taylor expanded in x.im around 0

       \[\leadsto {x.im}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. lower-pow.f64 30.9%

          \[\leadsto {x.im}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied rewrites 30.9%

       \[\leadsto {x.im}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -1.56e10 < y.re < 11.6

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   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 18.7%

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   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 14.1%

       \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 16.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x.im \leq 9.5 \cdot 10^{-259}:\\ \;\;\;\;1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (if (<= x.im 9.5e-259)
  (* 1.0 (sin (* y.re (atan2 x.im x.re))))
  (*
   y.re
   (fma
    -1.0
    (* y.re (* (log (/ 1.0 x.im)) (atan2 x.im x.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 tmp;
	if (x_46_im <= 9.5e-259) {
		tmp = 1.0 * sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_re * fma(-1.0, (y_46_re * (log((1.0 / x_46_im)) * atan2(x_46_im, x_46_re))), atan2(x_46_im, x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (x_46_im <= 9.5e-259)
		tmp = Float64(1.0 * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(y_46_re * fma(-1.0, Float64(y_46_re * Float64(log(Float64(1.0 / x_46_im)) * atan(x_46_im, x_46_re))), atan(x_46_im, x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[x$46$im, 9.5e-259], N[(1.0 * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(-1.0 * N[(y$46$re * N[(N[Log[N[(1.0 / x$46$im), $MachinePrecision]], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x.im < 9.4999999999999995e-259

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   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 18.7%

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   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 14.1%

       \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if 9.4999999999999995e-259 < x.im

   1. Initial program 40.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. 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 31.2%

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

   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 18.7%

         \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   7. Applied rewrites 18.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   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 14.1%

       \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Taylor expanded in y.re around 0

       \[\leadsto y.re \cdot \left(-1 \cdot \left(y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto y.re \cdot \left(-1 \cdot \left(y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) + \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{\color{blue}{x.re}}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       5. lower-log.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       6. lower-/.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       7. lower-atan2.f64 N/A

          \[\leadsto y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       8. lower-atan2.f64 9.5%

          \[\leadsto y.re \cdot \mathsf{fma}\left(-1, y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right), \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied rewrites 9.5%

       \[\leadsto y.re \cdot \mathsf{fma}\left(-1, \color{blue}{y.re \cdot \left(\log \left(\frac{1}{x.im}\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}, \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 14.1% accurate, 2.3× speedup

Math

\[1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

FPCore

(FPCore (x.re x.im y.re y.im)
  :precision binary64
  (* 1.0 (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) {
	return 1.0 * sin((y_46_re * atan2(x_46_im, x_46_re)));
}

Fortran

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 * sin((y_46re * atan2(x_46im, x_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0 * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0 * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(1.0 * sin(Float64(y_46_re * atan(x_46_im, x_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0 * sin((y_46_re * atan2(x_46_im, x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(1.0 * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 40.3%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

2. 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 31.2%

   \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(-1, y.im \cdot \log \left(\frac{1}{x.im}\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

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 18.7%

      \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

7. Applied rewrites 18.7%

   \[\leadsto e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.im}\right)\right)} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

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 14.1%

    \[\leadsto 1 \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(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)))))