powComplex, imaginary part

Percentage Accurate: 40.3% → 67.0%

Time: 20.0s

Alternatives: 16

Speedup: 2.8×

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

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    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

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    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: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{-1}{x.re}\right)\\ \mathbf{if}\;x.re \leq -2.25 \cdot 10^{-238}:\\ \;\;\;\;e^{-\mathsf{fma}\left(y.re, t\_2, t\_0\right)} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t\_2 \cdot \left(-y.im\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 4.5 \cdot 10^{-37}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im}\right) - t\_0} \cdot \sin t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ -1.0 x.re))))
   (if (<= x.re -2.25e-238)
     (*
      (exp (- (fma y.re t_2 t_0)))
      (sin (fma y.re (atan2 x.im x.re) (* t_2 (- y.im)))))
     (if (<= x.re 4.5e-37)
       (* (exp (- (* y.re (log (sqrt (* x.im x.im)))) t_0)) (sin t_1))
       (*
        (exp (- (* y.re (log x.re)) t_0))
        (sin (fma y.im (log x.re) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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 tmp;
	if (x_46_re <= -2.25e-238) {
		tmp = exp(-fma(y_46_re, t_2, t_0)) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (t_2 * -y_46_im)));
	} else if (x_46_re <= 4.5e-37) {
		tmp = exp(((y_46_re * log(sqrt((x_46_im * x_46_im)))) - t_0)) * sin(t_1);
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 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))
	tmp = 0.0
	if (x_46_re <= -2.25e-238)
		tmp = Float64(exp(Float64(-fma(y_46_re, t_2, t_0))) * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(t_2 * Float64(-y_46_im)))));
	elseif (x_46_re <= 4.5e-37)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(x_46_im * x_46_im)))) - t_0)) * sin(t_1));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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]}, If[LessEqual[x$46$re, -2.25e-238], N[(N[Exp[(-N[(y$46$re * t$95$2 + t$95$0), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(t$95$2 * (-y$46$im)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 4.5e-37], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.24999999999999998e-238

   1. Initial program 47.3%

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

   3. Taylor expanded in x.re around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. atan2-lowering-atan2.f64 N/A

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

      12. sin-lowering-sin.f64 N/A

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

   5. Simplified 74.1%

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

   if -2.24999999999999998e-238 < x.re < 4.5000000000000004e-37

   1. Initial program 55.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 69.6

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

   5. Simplified 69.6%

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

   6. Taylor expanded in x.re around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 69.6

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

   8. Simplified 69.6%

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

   if 4.5000000000000004e-37 < x.re

   1. Initial program 36.7%

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

   3. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. atan2-lowering-atan2.f64 72.7

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

   5. Simplified 72.7%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.25 \cdot 10^{-238}:\\ \;\;\;\;e^{-\mathsf{fma}\left(y.re, \log \left(\frac{-1}{x.re}\right), y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\frac{-1}{x.re}\right) \cdot \left(-y.im\right)\right)\right)\\ \mathbf{elif}\;x.re \leq 4.5 \cdot 10^{-37}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 62.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\frac{-1}{x.im}\right)\\ \mathbf{if}\;x.im \leq -9.1 \cdot 10^{-243}:\\ \;\;\;\;e^{-\mathsf{fma}\left(y.re, t\_2, t\_0\right)} \cdot \sin \left(\mathsf{fma}\left(t\_2, -y.im, t\_1\right)\right)\\ \mathbf{elif}\;x.im \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t\_0} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (/ -1.0 x.im))))
   (if (<= x.im -9.1e-243)
     (* (exp (- (fma y.re t_2 t_0))) (sin (fma t_2 (- y.im) t_1)))
     (if (<= x.im 2.1e-303)
       (* (exp (- (* y.re (log x.re)) t_0)) (sin (fma y.im (log x.re) t_1)))
       (*
        (exp (- (* y.re (log x.im)) t_0))
        (sin (fma y.im (log x.im) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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_im));
	double tmp;
	if (x_46_im <= -9.1e-243) {
		tmp = exp(-fma(y_46_re, t_2, t_0)) * sin(fma(t_2, -y_46_im, t_1));
	} else if (x_46_im <= 2.1e-303) {
		tmp = exp(((y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), t_1));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0)) * sin(fma(y_46_im, log(x_46_im), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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_im))
	tmp = 0.0
	if (x_46_im <= -9.1e-243)
		tmp = Float64(exp(Float64(-fma(y_46_re, t_2, t_0))) * sin(fma(t_2, Float64(-y_46_im), t_1)));
	elseif (x_46_im <= 2.1e-303)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)) * sin(fma(y_46_im, log(x_46_re), t_1)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0)) * sin(fma(y_46_im, log(x_46_im), t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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$46$im), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -9.1e-243], N[(N[Exp[(-N[(y$46$re * t$95$2 + t$95$0), $MachinePrecision])], $MachinePrecision] * N[Sin[N[(t$95$2 * (-y$46$im) + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.1e-303], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -9.10000000000000018e-243

   1. Initial program 55.5%

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

   3. Taylor expanded in x.im around -inf

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. distribute-neg-out N/A

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

      6. neg-lowering-neg.f64 N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. /-lowering-/.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. atan2-lowering-atan2.f64 N/A

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

      12. sin-lowering-sin.f64 N/A

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

   5. Simplified 73.1%

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

   if -9.10000000000000018e-243 < x.im < 2.1e-303

   1. Initial program 38.8%

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

   3. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. atan2-lowering-atan2.f64 61.1

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

   5. Simplified 61.1%

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

   if 2.1e-303 < x.im

   1. Initial program 40.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. log-lowering-log.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. atan2-lowering-atan2.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. atan2-lowering-atan2.f64 67.2

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

   5. Simplified 67.2%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -9.1 \cdot 10^{-243}:\\ \;\;\;\;e^{-\mathsf{fma}\left(y.re, \log \left(\frac{-1}{x.im}\right), y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\frac{-1}{x.im}\right), -y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 2.1 \cdot 10^{-303}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq -1.4 \cdot 10^{+102}:\\ \;\;\;\;\sin t\_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.im \leq -2.2 \cdot 10^{-131}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - t\_2} \cdot \sin \left(y.im \cdot \log t\_0\right)\\ \mathbf{elif}\;x.im \leq -2.95 \cdot 10^{-226}:\\ \;\;\;\;t\_1 \cdot {t\_0}^{y.re}\\ \mathbf{elif}\;x.im \leq 6.8 \cdot 10^{-303}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_2} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t\_2} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, t\_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* y.im (atan2 x.im x.re))))
   (if (<= x.im -1.4e+102)
     (* (sin t_1) (exp (* (atan2 x.im x.re) (- y.im))))
     (if (<= x.im -2.2e-131)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re))))) t_2))
        (sin (* y.im (log t_0))))
       (if (<= x.im -2.95e-226)
         (* t_1 (pow t_0 y.re))
         (if (<= x.im 6.8e-303)
           (*
            (exp (- (* y.re (log x.re)) t_2))
            (sin (fma y.im (log x.re) t_1)))
           (*
            (exp (- (* y.re (log x.im)) t_2))
            (sin (fma y.im (log x.im) t_1)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= -1.4e+102) {
		tmp = sin(t_1) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (x_46_im <= -2.2e-131) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_2)) * sin((y_46_im * log(t_0)));
	} else if (x_46_im <= -2.95e-226) {
		tmp = t_1 * pow(t_0, y_46_re);
	} else if (x_46_im <= 6.8e-303) {
		tmp = exp(((y_46_re * log(x_46_re)) - t_2)) * sin(fma(y_46_im, log(x_46_re), t_1));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_im, log(x_46_im), t_1));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= -1.4e+102)
		tmp = Float64(sin(t_1) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	elseif (x_46_im <= -2.2e-131)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - t_2)) * sin(Float64(y_46_im * log(t_0))));
	elseif (x_46_im <= -2.95e-226)
		tmp = Float64(t_1 * (t_0 ^ y_46_re));
	elseif (x_46_im <= 6.8e-303)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_2)) * sin(fma(y_46_im, log(x_46_re), t_1)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_im, log(x_46_im), t_1)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $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[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -1.4e+102], N[(N[Sin[t$95$1], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -2.2e-131], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, -2.95e-226], N[(t$95$1 * N[Power[t$95$0, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 6.8e-303], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$im], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.im < -1.40000000000000009e102

   1. Initial program 28.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 50.4

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

   5. Simplified 50.4%

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

   6. Taylor expanded in y.re around 0

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

      1. exp-lowering-exp.f64 N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 61.2

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

   8. Simplified 61.2%

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

   if -1.40000000000000009e102 < x.im < -2.2e-131

   1. Initial program 78.1%

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

   3. Taylor expanded in y.im around inf

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

      1. *-lowering-*.f64 N/A

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

      2. log-lowering-log.f64 N/A

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

      3. sqrt-lowering-sqrt.f64 N/A

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

      4. unpow2 N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. *-lowering-*.f64 74.5

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

   5. Simplified 74.5%

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

   if -2.2e-131 < x.im < -2.95e-226

   1. Initial program 43.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 43.6

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

   5. Simplified 43.6%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 57.9

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

   8. Simplified 57.9%

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

   if -2.95e-226 < x.im < 6.8e-303

   1. Initial program 38.8%

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

   3. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. atan2-lowering-atan2.f64 61.1

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

   5. Simplified 61.1%

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

   if 6.8e-303 < x.im

   1. Initial program 40.8%

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

   3. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 N/A

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

      6. log-lowering-log.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. atan2-lowering-atan2.f64 N/A

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

      9. sin-lowering-sin.f64 N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

      11. log-lowering-log.f64 N/A

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

      12. *-lowering-*.f64 N/A

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

      13. atan2-lowering-atan2.f64 67.2

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

   5. Simplified 67.2%

      \[\leadsto \color{blue}{e^{\log x.im \cdot y.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.4 \cdot 10^{+102}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.im \leq -2.2 \cdot 10^{-131}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)\\ \mathbf{elif}\;x.im \leq -2.95 \cdot 10^{-226}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 6.8 \cdot 10^{-303}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ t_2 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq -2.2 \cdot 10^{+154}:\\ \;\;\;\;t\_1 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.re \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;t\_1 \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - t\_2}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_2} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (* y.im (atan2 x.im x.re))))
   (if (<= x.re -2.2e+154)
     (* t_1 (exp (* (atan2 x.im x.re) (- y.im))))
     (if (<= x.re 3.4e-37)
       (*
        t_1
        (exp (- (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re))))) t_2)))
       (*
        (exp (- (* y.re (log x.re)) t_2))
        (sin (fma y.im (log x.re) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = y_46_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -2.2e+154) {
		tmp = t_1 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else if (x_46_re <= 3.4e-37) {
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - t_2));
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - t_2)) * sin(fma(y_46_im, log(x_46_re), t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= -2.2e+154)
		tmp = Float64(t_1 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	elseif (x_46_re <= 3.4e-37)
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - t_2)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_2)) * sin(fma(y_46_im, log(x_46_re), t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -2.2e+154], N[(t$95$1 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 3.4e-37], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -2.2000000000000001e154

   1. Initial program 0.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 23.6

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

   5. Simplified 23.6%

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

   6. Taylor expanded in y.re around 0

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

      1. exp-lowering-exp.f64 N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 49.2

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

   8. Simplified 49.2%

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

   if -2.2000000000000001e154 < x.re < 3.40000000000000018e-37

   1. Initial program 59.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 61.1

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

   5. Simplified 61.1%

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

   if 3.40000000000000018e-37 < x.re

   1. Initial program 36.7%

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

   3. Taylor expanded in x.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. log-lowering-log.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. sin-lowering-sin.f64 N/A

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

      9. accelerator-lowering-fma.f64 N/A

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

      10. log-lowering-log.f64 N/A

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

      11. *-lowering-*.f64 N/A

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

      12. atan2-lowering-atan2.f64 72.7

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

   5. Simplified 72.7%

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

4. Final simplification 62.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2.2 \cdot 10^{+154}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;x.re \leq 3.4 \cdot 10^{-37}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ t_2 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+57}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-99}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot t\_2\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-229}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \left(y.im \cdot t\_2\right)\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;t\_0 \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1
         (*
          t_0
          (exp
           (-
            (* y.re (log (sqrt (+ (* x.im x.im) (* x.re x.re)))))
            (* y.im (atan2 x.im x.re))))))
        (t_2 (exp (* (atan2 x.im x.re) (- y.im)))))
   (if (<= y.re -5.5e+57)
     t_1
     (if (<= y.re -1.4e-99)
       (* y.re (* (atan2 x.im x.re) t_2))
       (if (<= y.re 5.8e-229)
         (* (log (sqrt (fma x.im x.im (* x.re x.re)))) (* y.im t_2))
         (if (<= y.re 6.8e+43) (* t_0 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 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * exp(((y_46_re * log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re))))) - (y_46_im * atan2(x_46_im, x_46_re))));
	double t_2 = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double tmp;
	if (y_46_re <= -5.5e+57) {
		tmp = t_1;
	} else if (y_46_re <= -1.4e-99) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * t_2);
	} else if (y_46_re <= 5.8e-229) {
		tmp = log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) * (y_46_im * t_2);
	} else if (y_46_re <= 6.8e+43) {
		tmp = t_0 * t_2;
	} 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 * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re))))) - Float64(y_46_im * atan(x_46_im, x_46_re)))))
	t_2 = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))
	tmp = 0.0
	if (y_46_re <= -5.5e+57)
		tmp = t_1;
	elseif (y_46_re <= -1.4e-99)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * t_2));
	elseif (y_46_re <= 5.8e-229)
		tmp = Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) * Float64(y_46_im * t_2));
	elseif (y_46_re <= 6.8e+43)
		tmp = Float64(t_0 * t_2);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[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[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -5.5e+57], t$95$1, If[LessEqual[y$46$re, -1.4e-99], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e-229], N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(y$46$im * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.8e+43], N[(t$95$0 * t$95$2), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.5000000000000002e57 or 6.80000000000000024e43 < y.re

   1. Initial program 45.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 72.1

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

   5. Simplified 72.1%

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

   if -5.5000000000000002e57 < y.re < -1.4e-99

   1. Initial program 43.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 53.3

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. atan2-lowering-atan2.f64 N/A

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

      7. atan2-lowering-atan2.f64 68.1

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

   8. Simplified 68.1%

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

   if -1.4e-99 < y.re < 5.7999999999999999e-229

   1. Initial program 60.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. sin-lowering-sin.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. atan2-lowering-atan2.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in y.re around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. exp-lowering-exp.f64 N/A

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

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

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. atan2-lowering-atan2.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 59.6

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

   8. Simplified 59.6%

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

   if 5.7999999999999999e-229 < y.re < 6.80000000000000024e43

   1. Initial program 38.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 40.7

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

   5. Simplified 40.7%

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

   6. Taylor expanded in y.re around 0

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

      1. exp-lowering-exp.f64 N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 57.5

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

   8. Simplified 57.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+57}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.4 \cdot 10^{-99}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-229}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \left(y.im \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+43}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 55.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ \mathbf{if}\;y.re \leq -1 \cdot 10^{+58}:\\ \;\;\;\;t\_1 \cdot {t\_2}^{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-102}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot t\_0\right)\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-227}:\\ \;\;\;\;\log t\_2 \cdot \left(y.im \cdot t\_0\right)\\ \mathbf{elif}\;y.re \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (atan2 x.im x.re) (- y.im))))
        (t_1 (sin (* y.re (atan2 x.im x.re))))
        (t_2 (sqrt (fma x.im x.im (* x.re x.re)))))
   (if (<= y.re -1e+58)
     (* t_1 (pow t_2 y.re))
     (if (<= y.re -9e-102)
       (* y.re (* (atan2 x.im x.re) t_0))
       (if (<= y.re 1.9e-227)
         (* (log t_2) (* y.im t_0))
         (if (<= y.re 2.65e-30)
           (* t_1 t_0)
           (*
            (exp
             (-
              (* y.re (log (sqrt (* x.im x.im))))
              (* y.im (atan2 x.im x.re))))
            t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double tmp;
	if (y_46_re <= -1e+58) {
		tmp = t_1 * pow(t_2, y_46_re);
	} else if (y_46_re <= -9e-102) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * t_0);
	} else if (y_46_re <= 1.9e-227) {
		tmp = log(t_2) * (y_46_im * t_0);
	} else if (y_46_re <= 2.65e-30) {
		tmp = t_1 * t_0;
	} else {
		tmp = exp(((y_46_re * log(sqrt((x_46_im * x_46_im)))) - (y_46_im * atan2(x_46_im, x_46_re)))) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_2 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	tmp = 0.0
	if (y_46_re <= -1e+58)
		tmp = Float64(t_1 * (t_2 ^ y_46_re));
	elseif (y_46_re <= -9e-102)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * t_0));
	elseif (y_46_re <= 1.9e-227)
		tmp = Float64(log(t_2) * Float64(y_46_im * t_0));
	elseif (y_46_re <= 2.65e-30)
		tmp = Float64(t_1 * t_0);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(x_46_im * x_46_im)))) - Float64(y_46_im * atan(x_46_im, x_46_re)))) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -1e+58], N[(t$95$1 * N[Power[t$95$2, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9e-102], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.9e-227], N[(N[Log[t$95$2], $MachinePrecision] * N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.65e-30], N[(t$95$1 * t$95$0), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(x$46$im * x$46$im), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -9.99999999999999944e57

   1. Initial program 53.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 77.9

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

   5. Simplified 77.9%

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

   if -9.99999999999999944e57 < y.re < -8.99999999999999999e-102

   1. Initial program 43.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 53.3

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. atan2-lowering-atan2.f64 N/A

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

      7. atan2-lowering-atan2.f64 68.1

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

   8. Simplified 68.1%

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

   if -8.99999999999999999e-102 < y.re < 1.90000000000000005e-227

   1. Initial program 60.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. sin-lowering-sin.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. atan2-lowering-atan2.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in y.re around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. exp-lowering-exp.f64 N/A

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

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

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. atan2-lowering-atan2.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 59.6

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

   8. Simplified 59.6%

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

   if 1.90000000000000005e-227 < y.re < 2.64999999999999987e-30

   1. Initial program 41.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 40.4

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

   5. Simplified 40.4%

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

   6. Taylor expanded in y.re around 0

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

      1. exp-lowering-exp.f64 N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 60.3

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

   8. Simplified 60.3%

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

   if 2.64999999999999987e-30 < y.re

   1. Initial program 35.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 53.7

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

   5. Simplified 53.7%

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

   6. Taylor expanded in x.re around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 55.3

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

   8. Simplified 55.3%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.im \cdot x.im}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 63.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-102}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-227}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \left(y.im \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 2.65 \cdot 10^{-30}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_2 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_3 := t\_1 \cdot {t\_2}^{y.re}\\ \mathbf{if}\;y.re \leq -7 \cdot 10^{+57}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-102}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot t\_0\right)\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-228}:\\ \;\;\;\;\log t\_2 \cdot \left(y.im \cdot t\_0\right)\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+43}:\\ \;\;\;\;t\_1 \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (atan2 x.im x.re) (- y.im))))
        (t_1 (sin (* y.re (atan2 x.im x.re))))
        (t_2 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_3 (* t_1 (pow t_2 y.re))))
   (if (<= y.re -7e+57)
     t_3
     (if (<= y.re -9e-102)
       (* y.re (* (atan2 x.im x.re) t_0))
       (if (<= y.re 2e-228)
         (* (log t_2) (* y.im t_0))
         (if (<= y.re 7.2e+43) (* t_1 t_0) t_3))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_2 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_3 = t_1 * pow(t_2, y_46_re);
	double tmp;
	if (y_46_re <= -7e+57) {
		tmp = t_3;
	} else if (y_46_re <= -9e-102) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * t_0);
	} else if (y_46_re <= 2e-228) {
		tmp = log(t_2) * (y_46_im * t_0);
	} else if (y_46_re <= 7.2e+43) {
		tmp = t_1 * t_0;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_2 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_3 = Float64(t_1 * (t_2 ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -7e+57)
		tmp = t_3;
	elseif (y_46_re <= -9e-102)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * t_0));
	elseif (y_46_re <= 2e-228)
		tmp = Float64(log(t_2) * Float64(y_46_im * t_0));
	elseif (y_46_re <= 7.2e+43)
		tmp = Float64(t_1 * t_0);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * N[Power[t$95$2, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -7e+57], t$95$3, If[LessEqual[y$46$re, -9e-102], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2e-228], N[(N[Log[t$95$2], $MachinePrecision] * N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7.2e+43], N[(t$95$1 * t$95$0), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.9999999999999995e57 or 7.2000000000000002e43 < y.re

   1. Initial program 45.8%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 67.5

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

   5. Simplified 67.5%

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

   if -6.9999999999999995e57 < y.re < -8.99999999999999999e-102

   1. Initial program 43.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 53.3

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

   5. Simplified 53.3%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. atan2-lowering-atan2.f64 N/A

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

      7. atan2-lowering-atan2.f64 68.1

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

   8. Simplified 68.1%

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

   if -8.99999999999999999e-102 < y.re < 2.00000000000000007e-228

   1. Initial program 60.9%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. sin-lowering-sin.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. atan2-lowering-atan2.f64 59.2

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

   5. Simplified 59.2%

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

   6. Taylor expanded in y.re around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. exp-lowering-exp.f64 N/A

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

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

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. atan2-lowering-atan2.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 59.6

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

   8. Simplified 59.6%

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

   if 2.00000000000000007e-228 < y.re < 7.2000000000000002e43

   1. Initial program 38.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 40.7

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

   5. Simplified 40.7%

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

   6. Taylor expanded in y.re around 0

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

      1. exp-lowering-exp.f64 N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 57.5

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

   8. Simplified 57.5%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{+57}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-102}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-228}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \left(y.im \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+43}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 54.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ t_1 := y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot t\_0\right)\\ t_2 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_3 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {t\_2}^{y.re}\\ t_4 := \log t\_2 \cdot \left(y.im \cdot t\_0\right)\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+57}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-227}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+193}:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (exp (* (atan2 x.im x.re) (- y.im))))
        (t_1 (* y.re (* (atan2 x.im x.re) t_0)))
        (t_2 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_3 (* (sin (* y.re (atan2 x.im x.re))) (pow t_2 y.re)))
        (t_4 (* (log t_2) (* y.im t_0))))
   (if (<= y.re -9e+57)
     t_3
     (if (<= y.re -9e-102)
       t_1
       (if (<= y.re 2.2e-227)
         t_4
         (if (<= y.re 2.8e+17) t_1 (if (<= y.re 3.6e+193) t_4 t_3)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double t_1 = y_46_re * (atan2(x_46_im, x_46_re) * t_0);
	double t_2 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_3 = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(t_2, y_46_re);
	double t_4 = log(t_2) * (y_46_im * t_0);
	double tmp;
	if (y_46_re <= -9e+57) {
		tmp = t_3;
	} else if (y_46_re <= -9e-102) {
		tmp = t_1;
	} else if (y_46_re <= 2.2e-227) {
		tmp = t_4;
	} else if (y_46_re <= 2.8e+17) {
		tmp = t_1;
	} else if (y_46_re <= 3.6e+193) {
		tmp = t_4;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))
	t_1 = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * t_0))
	t_2 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_3 = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (t_2 ^ y_46_re))
	t_4 = Float64(log(t_2) * Float64(y_46_im * t_0))
	tmp = 0.0
	if (y_46_re <= -9e+57)
		tmp = t_3;
	elseif (y_46_re <= -9e-102)
		tmp = t_1;
	elseif (y_46_re <= 2.2e-227)
		tmp = t_4;
	elseif (y_46_re <= 2.8e+17)
		tmp = t_1;
	elseif (y_46_re <= 3.6e+193)
		tmp = t_4;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[t$95$2, y$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Log[t$95$2], $MachinePrecision] * N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9e+57], t$95$3, If[LessEqual[y$46$re, -9e-102], t$95$1, If[LessEqual[y$46$re, 2.2e-227], t$95$4, If[LessEqual[y$46$re, 2.8e+17], t$95$1, If[LessEqual[y$46$re, 3.6e+193], t$95$4, t$95$3]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.99999999999999991e57 or 3.6e193 < y.re

   1. Initial program 50.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 75.8

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

   5. Simplified 75.8%

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

   if -8.99999999999999991e57 < y.re < -8.99999999999999999e-102 or 2.19999999999999981e-227 < y.re < 2.8e17

   1. Initial program 42.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 45.0

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

   5. Simplified 45.0%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. atan2-lowering-atan2.f64 N/A

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

      7. atan2-lowering-atan2.f64 61.6

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

   8. Simplified 61.6%

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

   if -8.99999999999999999e-102 < y.re < 2.19999999999999981e-227 or 2.8e17 < y.re < 3.6e193

   1. Initial program 48.7%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

      2. associate-*r* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. cos-lowering-cos.f64 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. atan2-lowering-atan2.f64 N/A

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

      8. log-lowering-log.f64 N/A

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

      9. sqrt-lowering-sqrt.f64 N/A

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

      10. unpow2 N/A

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

      11. accelerator-lowering-fma.f64 N/A

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

      12. unpow2 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. sin-lowering-sin.f64 N/A

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

      15. *-lowering-*.f64 N/A

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

      16. atan2-lowering-atan2.f64 56.9

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

   5. Simplified 56.9%

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

   6. Taylor expanded in y.re around 0

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

      1. associate-*r* N/A

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

      2. *-lowering-*.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. exp-lowering-exp.f64 N/A

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

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

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

      6. *-lowering-*.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. atan2-lowering-atan2.f64 N/A

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

      9. log-lowering-log.f64 N/A

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

      10. sqrt-lowering-sqrt.f64 N/A

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

      11. unpow2 N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. unpow2 N/A

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

      14. *-lowering-*.f64 57.4

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

   8. Simplified 57.4%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+57}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-102}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-227}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \left(y.im \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+17}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+193}:\\ \;\;\;\;\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right) \cdot \left(y.im \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 110000000000:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (sin (* y.re (atan2 x.im x.re)))
          (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -1.4e+58)
     t_0
     (if (<= y.re 110000000000.0)
       (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -1.4e+58) {
		tmp = t_0;
	} else if (y_46_re <= 110000000000.0) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.4e+58)
		tmp = t_0;
	elseif (y_46_re <= 110000000000.0)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	else
		tmp = 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[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.4e+58], t$95$0, If[LessEqual[y$46$re, 110000000000.0], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.3999999999999999e58 or 1.1e11 < y.re

   1. Initial program 44.0%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 65.8

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

   5. Simplified 65.8%

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

   if -1.3999999999999999e58 < y.re < 1.1e11

   1. Initial program 49.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 39.8

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

   5. Simplified 39.8%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. atan2-lowering-atan2.f64 N/A

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

      7. atan2-lowering-atan2.f64 53.0

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

   8. Simplified 53.0%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 110000000000:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 55.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+40}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -3.3e+40)
     (* t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
     (if (<= y.re 3.6e+16)
       (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))
       (* (sin t_0) (pow x.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -3.3e+40) {
		tmp = t_0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	} else if (y_46_re <= 3.6e+16) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	} else {
		tmp = sin(t_0) * pow(x_46_im, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -3.3e+40)
		tmp = Float64(t_0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	elseif (y_46_re <= 3.6e+16)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	else
		tmp = Float64(sin(t_0) * (x_46_im ^ 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]}, If[LessEqual[y$46$re, -3.3e+40], N[(t$95$0 * N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.6e+16], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.2999999999999998e40

   1. Initial program 54.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 75.7

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

   5. Simplified 75.7%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 72.4

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

   8. Simplified 72.4%

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

   if -3.2999999999999998e40 < y.re < 3.6e16

   1. Initial program 48.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 40.0

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

   5. Simplified 40.0%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. exp-lowering-exp.f64 N/A

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

      4. neg-lowering-neg.f64 N/A

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

      5. *-lowering-*.f64 N/A

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

      6. atan2-lowering-atan2.f64 N/A

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

      7. atan2-lowering-atan2.f64 53.4

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

   8. Simplified 53.4%

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

   if 3.6e16 < y.re

   1. Initial program 34.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 53.7

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

   5. Simplified 53.7%

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

   6. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 52.2

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

   8. Simplified 52.2%

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

4. Final simplification 57.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+40}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+16}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ \mathbf{if}\;y.re \leq -4.1 \cdot 10^{-103}:\\ \;\;\;\;t\_0 \cdot {t\_1}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{-227}:\\ \;\;\;\;\sin \left(y.im \cdot \log t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {x.im}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sqrt (fma x.im x.im (* x.re x.re)))))
   (if (<= y.re -4.1e-103)
     (* t_0 (pow t_1 y.re))
     (if (<= y.re 4.4e-227)
       (sin (* y.im (log t_1)))
       (* (sin t_0) (pow x.im 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 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double tmp;
	if (y_46_re <= -4.1e-103) {
		tmp = t_0 * pow(t_1, y_46_re);
	} else if (y_46_re <= 4.4e-227) {
		tmp = sin((y_46_im * log(t_1)));
	} else {
		tmp = sin(t_0) * pow(x_46_im, 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 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	tmp = 0.0
	if (y_46_re <= -4.1e-103)
		tmp = Float64(t_0 * (t_1 ^ y_46_re));
	elseif (y_46_re <= 4.4e-227)
		tmp = sin(Float64(y_46_im * log(t_1)));
	else
		tmp = Float64(sin(t_0) * (x_46_im ^ 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[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -4.1e-103], N[(t$95$0 * N[Power[t$95$1, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.4e-227], N[Sin[N[(y$46$im * N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.09999999999999996e-103

   1. Initial program 50.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 63.1

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

   5. Simplified 63.1%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 60.9

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

   8. Simplified 60.9%

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

   if -4.09999999999999996e-103 < y.re < 4.39999999999999962e-227

   1. Initial program 60.2%

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

   3. Taylor expanded in y.im around 0

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

      1. pow-lowering-pow.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 35.9

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

   5. Simplified 35.9%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)\right) \]

      8. *-lowering-*.f64 34.1

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)\right) \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)} \]

   if 4.39999999999999962e-227 < y.re

   1. Initial program 37.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 39.4

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

   5. Simplified 39.4%

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

   6. Taylor expanded in x.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 38.4

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

   8. Simplified 38.4%

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

Alternative 12: 44.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\\ t_1 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {t\_0}^{y.re}\\ \mathbf{if}\;y.re \leq -1.02 \cdot 10^{-103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.35 \cdot 10^{-227}:\\ \;\;\;\;\sin \left(y.im \cdot \log t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sqrt (fma x.im x.im (* x.re x.re))))
        (t_1 (* (* y.re (atan2 x.im x.re)) (pow t_0 y.re))))
   (if (<= y.re -1.02e-103)
     t_1
     (if (<= y.re 1.35e-227) (sin (* y.im (log 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 = sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)));
	double t_1 = (y_46_re * atan2(x_46_im, x_46_re)) * pow(t_0, y_46_re);
	double tmp;
	if (y_46_re <= -1.02e-103) {
		tmp = t_1;
	} else if (y_46_re <= 1.35e-227) {
		tmp = sin((y_46_im * log(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 = sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))
	t_1 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * (t_0 ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -1.02e-103)
		tmp = t_1;
	elseif (y_46_re <= 1.35e-227)
		tmp = sin(Float64(y_46_im * log(t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.02e-103], t$95$1, If[LessEqual[y$46$re, 1.35e-227], N[Sin[N[(y$46$im * N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.01999999999999998e-103 or 1.35e-227 < y.re

   1. Initial program 43.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 50.0

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

   5. Simplified 50.0%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. atan2-lowering-atan2.f64 46.0

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

   8. Simplified 46.0%

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

   if -1.01999999999999998e-103 < y.re < 1.35e-227

   1. Initial program 60.2%

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

   3. Taylor expanded in y.im around 0

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

      1. pow-lowering-pow.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 35.9

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

   5. Simplified 35.9%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)\right) \]

      8. *-lowering-*.f64 34.1

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)\right) \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 16.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -2 \cdot 10^{-104}:\\ \;\;\;\;\sin t\_0\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-227}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.re -2e-104)
     (sin t_0)
     (if (<= y.re 2.2e-227)
       (sin (* y.im (log (sqrt (fma x.im x.im (* x.re x.re))))))
       (if (<= y.re 1.45e-31)
         t_0
         (* y.im (fma 0.5 (/ (* x.im x.im) (* x.re x.re)) (log x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_re <= -2e-104) {
		tmp = sin(t_0);
	} else if (y_46_re <= 2.2e-227) {
		tmp = sin((y_46_im * log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))))));
	} else if (y_46_re <= 1.45e-31) {
		tmp = t_0;
	} else {
		tmp = y_46_im * fma(0.5, ((x_46_im * x_46_im) / (x_46_re * x_46_re)), log(x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -2e-104)
		tmp = sin(t_0);
	elseif (y_46_re <= 2.2e-227)
		tmp = sin(Float64(y_46_im * log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))))));
	elseif (y_46_re <= 1.45e-31)
		tmp = t_0;
	else
		tmp = Float64(y_46_im * fma(0.5, Float64(Float64(x_46_im * x_46_im) / Float64(x_46_re * x_46_re)), log(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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2e-104], N[Sin[t$95$0], $MachinePrecision], If[LessEqual[y$46$re, 2.2e-227], N[Sin[N[(y$46$im * N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 1.45e-31], t$95$0, N[(y$46$im * N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.99999999999999985e-104

   1. Initial program 50.5%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 63.1

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

   5. Simplified 63.1%

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

   6. Taylor expanded in x.re around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 49.6

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

   8. Simplified 49.6%

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

   9. Taylor expanded in y.re around 0

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

   11. Simplified 16.9%

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

       1. *-rgt-identity N/A

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

       2. *-commutative N/A

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

       3. sin-lowering-sin.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. atan2-lowering-atan2.f64 16.9

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

   13. Applied egg-rr 16.9%

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

   if -1.99999999999999985e-104 < y.re < 2.19999999999999981e-227

   1. Initial program 60.2%

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

   3. Taylor expanded in y.im around 0

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

      1. pow-lowering-pow.f64 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

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

      3. unpow2 N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 35.9

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

   5. Simplified 35.9%

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

   6. Taylor expanded in y.re around 0

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

      1. sin-lowering-sin.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. log-lowering-log.f64 N/A

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

      4. sqrt-lowering-sqrt.f64 N/A

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

      5. unpow2 N/A

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

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)\right) \]

      8. *-lowering-*.f64 34.1

         \[\leadsto \sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)\right) \]

   8. Simplified 34.1%

      \[\leadsto \color{blue}{\sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)} \]

   if 2.19999999999999981e-227 < y.re < 1.45e-31

   1. Initial program 42.9%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 24.7

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

   5. Simplified 24.7%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      2. atan2-lowering-atan2.f64 30.9

         \[\leadsto y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 1.45e-31 < y.re

   1. Initial program 34.7%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 19.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. atan2-lowering-atan2.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 11.6%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(\log x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto y.im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{x.im}^{2}}{{x.re}^{2}}, \log x.re\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{x.im}^{2}}{{x.re}^{2}}}, \log x.re\right) \]

       5. unpow2 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}, \log x.re\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}, \log x.re\right) \]

       7. unpow2 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}, \log x.re\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}, \log x.re\right) \]

       9. log-lowering-log.f64 10.5

          \[\leadsto y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \color{blue}{\log x.re}\right) \]

   11. Simplified 10.5%

       \[\leadsto \color{blue}{y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{-104}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-227}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)\right)\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-31}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 14.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 1.65 \cdot 10^{-155}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 1.65e-155)
   (sin (* y.re (atan2 x.im x.re)))
   (* y.im (fma 0.5 (/ (* x.im x.im) (* x.re x.re)) (log x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 1.65e-155) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = y_46_im * fma(0.5, ((x_46_im * x_46_im) / (x_46_re * x_46_re)), log(x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 1.65e-155)
		tmp = sin(Float64(y_46_re * atan(x_46_im, x_46_re)));
	else
		tmp = Float64(y_46_im * fma(0.5, Float64(Float64(x_46_im * x_46_im) / Float64(x_46_re * x_46_re)), log(x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 1.65e-155], N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(y$46$im * N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 1.64999999999999993e-155

   1. Initial program 45.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 45.9

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

   5. Simplified 45.9%

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

   6. Taylor expanded in x.re around inf

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

      1. *-lowering-*.f64 N/A

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

      2. +-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 N/A

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

      7. unpow2 N/A

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

      8. *-lowering-*.f64 29.0

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

   8. Simplified 29.0%

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

   9. Taylor expanded in y.re around 0

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

   11. Simplified 17.5%

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

       1. *-rgt-identity N/A

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

       2. *-commutative N/A

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

       3. sin-lowering-sin.f64 N/A

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

       4. *-lowering-*.f64 N/A

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

       5. atan2-lowering-atan2.f64 17.5

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

   13. Applied egg-rr 17.5%

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

   if 1.64999999999999993e-155 < y.im

   1. Initial program 49.9%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 23.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. atan2-lowering-atan2.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 17.3%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(\log x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto y.im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{x.im}^{2}}{{x.re}^{2}}, \log x.re\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{x.im}^{2}}{{x.re}^{2}}}, \log x.re\right) \]

       5. unpow2 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}, \log x.re\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}, \log x.re\right) \]

       7. unpow2 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}, \log x.re\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}, \log x.re\right) \]

       9. log-lowering-log.f64 12.8

          \[\leadsto y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \color{blue}{\log x.re}\right) \]

   11. Simplified 12.8%

       \[\leadsto \color{blue}{y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 1.65 \cdot 10^{-155}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 14.4% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq 1.55 \cdot 10^{-155}:\\ \;\;\;\;y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{else}:\\ \;\;\;\;y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im 1.55e-155)
   (* y.re (atan2 x.im x.re))
   (* y.im (fma 0.5 (/ (* x.im x.im) (* x.re x.re)) (log x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= 1.55e-155) {
		tmp = y_46_re * atan2(x_46_im, x_46_re);
	} else {
		tmp = y_46_im * fma(0.5, ((x_46_im * x_46_im) / (x_46_re * x_46_re)), log(x_46_re));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= 1.55e-155)
		tmp = Float64(y_46_re * atan(x_46_im, x_46_re));
	else
		tmp = Float64(y_46_im * fma(0.5, Float64(Float64(x_46_im * x_46_im) / Float64(x_46_re * x_46_re)), log(x_46_re)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, 1.55e-155], N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$im * N[(0.5 * N[(N[(x$46$im * x$46$im), $MachinePrecision] / N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision] + N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < 1.55e-155

   1. Initial program 45.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-lowering-*.f64 N/A

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

      2. sin-lowering-sin.f64 N/A

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

      3. *-lowering-*.f64 N/A

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

      4. atan2-lowering-atan2.f64 N/A

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

      5. pow-lowering-pow.f64 N/A

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

      6. sqrt-lowering-sqrt.f64 N/A

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

      7. unpow2 N/A

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

      8. accelerator-lowering-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. *-lowering-*.f64 45.9

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

   5. Simplified 45.9%

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

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

      2. atan2-lowering-atan2.f64 17.5

         \[\leadsto y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \]

   8. Simplified 17.5%

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   if 1.55e-155 < y.im

   1. Initial program 49.9%

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

   3. Taylor expanded in x.im around 0

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

      1. +-commutative N/A

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

      2. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 23.5%

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

   6. Taylor expanded in y.re around 0

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

      1. *-lowering-*.f64 N/A

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

      2. exp-lowering-exp.f64 N/A

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

      3. neg-lowering-neg.f64 N/A

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

      4. *-lowering-*.f64 N/A

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

      5. atan2-lowering-atan2.f64 N/A

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

      6. +-commutative N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

   8. Simplified 17.3%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{y.im \cdot \left(\log x.re + \frac{1}{2} \cdot \frac{{x.im}^{2}}{{x.re}^{2}}\right)} \]
   10. Step-by-step derivation

       1. *-lowering-*.f64 N/A

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

       2. +-commutative N/A

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

       3. accelerator-lowering-fma.f64 N/A

          \[\leadsto y.im \cdot \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{{x.im}^{2}}{{x.re}^{2}}, \log x.re\right)} \]

       4. /-lowering-/.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{{x.im}^{2}}{{x.re}^{2}}}, \log x.re\right) \]

       5. unpow2 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}, \log x.re\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{x.im \cdot x.im}}{{x.re}^{2}}, \log x.re\right) \]

       7. unpow2 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}, \log x.re\right) \]

       8. *-lowering-*.f64 N/A

          \[\leadsto y.im \cdot \mathsf{fma}\left(\frac{1}{2}, \frac{x.im \cdot x.im}{\color{blue}{x.re \cdot x.re}}, \log x.re\right) \]

       9. log-lowering-log.f64 12.8

          \[\leadsto y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \color{blue}{\log x.re}\right) \]

   11. Simplified 12.8%

       \[\leadsto \color{blue}{y.im \cdot \mathsf{fma}\left(0.5, \frac{x.im \cdot x.im}{x.re \cdot x.re}, \log x.re\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 14.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* 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 y_46_re * atan2(x_46_im, x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    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 = 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 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 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(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 = 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 46.9%

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

3. Taylor expanded in y.im around 0

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

   1. *-lowering-*.f64 N/A

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

   2. sin-lowering-sin.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. atan2-lowering-atan2.f64 N/A

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

   5. pow-lowering-pow.f64 N/A

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

   6. sqrt-lowering-sqrt.f64 N/A

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

   7. unpow2 N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 41.5

       \[\leadsto \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \]

5. Simplified 41.5%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \]

6. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]

   2. atan2-lowering-atan2.f64 13.6

      \[\leadsto y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \]

8. Simplified 13.6%

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024199 
(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)))))