powComplex, imaginary part

Percentage Accurate: 41.2% → 64.8%

Time: 21.0s

Alternatives: 24

Speedup: 2.9×

• 36.2% 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: 41.2% 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: 64.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0}\\ t_4 := t\_3 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ \mathbf{if}\;t\_4 \leq -0.9:\\ \;\;\;\;t\_3 \cdot \left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.re \cdot \mathsf{fma}\left(y.im, \frac{\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3 \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right) \cdot \left(y.im \cdot 0.5\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3 \cdot \sin t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (exp (- (* t_2 y.re) t_0)))
        (t_4 (* t_3 (sin (+ (* t_2 y.im) t_1)))))
   (if (<= t_4 -0.9)
     (*
      t_3
      (*
       y.re
       (fma
        -0.16666666666666666
        (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
        (atan2 x.im x.re))))
     (if (<= t_4 -2e-159)
       (*
        (exp (* y.im (- 0.0 (atan2 x.im x.re))))
        (sin
         (*
          y.re
          (fma
           y.im
           (/ (log (sqrt (fma x.im x.im (* x.re x.re)))) y.re)
           (atan2 x.im x.re)))))
       (if (<= t_4 0.0)
         (* t_3 t_1)
         (if (<= t_4 1.0)
           (/
            (sin
             (fma
              y.re
              (atan2 x.im x.re)
              (* (log (fma x.re x.re (* x.im x.im))) (* y.im 0.5))))
            (exp (fma (atan2 x.im x.re) y.im 0.0)))
           (if (<= t_4 INFINITY)
             (* t_3 (sin t_1))
             (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double t_4 = t_3 * sin(((t_2 * y_46_im) + t_1));
	double tmp;
	if (t_4 <= -0.9) {
		tmp = t_3 * (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), atan2(x_46_im, x_46_re)));
	} else if (t_4 <= -2e-159) {
		tmp = exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re)))) * sin((y_46_re * fma(y_46_im, (log(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))) / y_46_re), atan2(x_46_im, x_46_re))));
	} else if (t_4 <= 0.0) {
		tmp = t_3 * t_1;
	} else if (t_4 <= 1.0) {
		tmp = sin(fma(y_46_re, atan2(x_46_im, x_46_re), (log(fma(x_46_re, x_46_re, (x_46_im * x_46_im))) * (y_46_im * 0.5)))) / exp(fma(atan2(x_46_im, x_46_re), y_46_im, 0.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3 * sin(t_1);
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	t_4 = Float64(t_3 * sin(Float64(Float64(t_2 * y_46_im) + t_1)))
	tmp = 0.0
	if (t_4 <= -0.9)
		tmp = Float64(t_3 * Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), atan(x_46_im, x_46_re))));
	elseif (t_4 <= -2e-159)
		tmp = Float64(exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re)))) * sin(Float64(y_46_re * fma(y_46_im, Float64(log(sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))) / y_46_re), atan(x_46_im, x_46_re)))));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_3 * t_1);
	elseif (t_4 <= 1.0)
		tmp = Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))) * Float64(y_46_im * 0.5)))) / exp(fma(atan(x_46_im, x_46_re), y_46_im, 0.0)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_3 * sin(t_1));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -0.9], N[(t$95$3 * N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -2e-159], N[(N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[(y$46$im * N[(N[Log[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / y$46$re), $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$3 * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(y$46$im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$3 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.900000000000000022

   1. Initial program 50.1%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 50.1

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

   5. Simplified 50.1%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. atan2-lowering-atan2.f64 63.8

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

   8. Simplified 63.8%

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

   if -0.900000000000000022 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -1.99999999999999998e-159

   1. Initial program 94.1%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. atan2-lowering-atan2.f64 94.1

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

   5. Simplified 94.1%

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

   6. Taylor expanded in y.re around inf

      \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \color{blue}{\left(y.re \cdot \left(\frac{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.re} + \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]
   7. Step-by-step derivation

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

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \color{blue}{\left(y.re \cdot \left(\frac{y.im \cdot \log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.re} + \tan^{-1}_* \frac{x.im}{x.re}\right)\right)} \]

      2. associate-/l* N/A

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \left(\color{blue}{y.im \cdot \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.re}} + \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

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

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \color{blue}{\mathsf{fma}\left(y.im, \frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

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

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \mathsf{fma}\left(y.im, \color{blue}{\frac{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}{y.re}}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

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

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \mathsf{fma}\left(y.im, \frac{\color{blue}{\log \left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

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

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \mathsf{fma}\left(y.im, \frac{\log \color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      7. unpow2 N/A

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \mathsf{fma}\left(y.im, \frac{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

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

         \[\leadsto e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \mathsf{fma}\left(y.im, \frac{\log \left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      9. unpow2 N/A

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

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

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

      11. atan2-lowering-atan2.f64 95.1

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

   8. Simplified 95.1%

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

   if -1.99999999999999998e-159 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. atan2-lowering-atan2.f64 85.9

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

   5. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. 1-exp N/A

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

      4. *-commutative N/A

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

      5. un-div-inv N/A

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

      6. *-commutative N/A

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

      7. pow1/2 N/A

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

      8. log-pow N/A

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

      9. associate-*l* N/A

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

   7. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 73.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 73.9

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

   5. Simplified 73.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y.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 39.7

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

   5. Simplified 39.7%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 41.9

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

   8. Simplified 41.9%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 43.7

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

   11. Simplified 43.7%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 58.9

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

   13. Applied egg-rr 58.9%

       \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 6 regimes into one program.

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.9:\\ \;\;\;\;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 \left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -2 \cdot 10^{-159}:\\ \;\;\;\;e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.re \cdot \mathsf{fma}\left(y.im, \frac{\log \left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}{y.re}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 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 \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right) \cdot \left(y.im \cdot 0.5\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq \infty:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 64.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0}\\ t_4 := t\_3 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ t_5 := \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, t\_5, t\_1\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3 \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t\_5 \cdot \left(y.im \cdot 0.5\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_3 \cdot \sin t\_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (exp (- (* t_2 y.re) t_0)))
        (t_4 (* t_3 (sin (+ (* t_2 y.im) t_1))))
        (t_5 (log (fma x.re x.re (* x.im x.im)))))
   (if (<= t_4 -2e-159)
     (*
      (sin (fma (* y.im 0.5) t_5 t_1))
      (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
     (if (<= t_4 0.0)
       (* t_3 t_1)
       (if (<= t_4 1.0)
         (/
          (sin (fma y.re (atan2 x.im x.re) (* t_5 (* y.im 0.5))))
          (exp (fma (atan2 x.im x.re) y.im 0.0)))
         (if (<= t_4 INFINITY)
           (* t_3 (sin t_1))
           (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double t_4 = t_3 * sin(((t_2 * y_46_im) + t_1));
	double t_5 = log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (t_4 <= -2e-159) {
		tmp = sin(fma((y_46_im * 0.5), t_5, t_1)) * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	} else if (t_4 <= 0.0) {
		tmp = t_3 * t_1;
	} else if (t_4 <= 1.0) {
		tmp = sin(fma(y_46_re, atan2(x_46_im, x_46_re), (t_5 * (y_46_im * 0.5)))) / exp(fma(atan2(x_46_im, x_46_re), y_46_im, 0.0));
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3 * sin(t_1);
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	t_4 = Float64(t_3 * sin(Float64(Float64(t_2 * y_46_im) + t_1)))
	t_5 = log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_4 <= -2e-159)
		tmp = Float64(sin(fma(Float64(y_46_im * 0.5), t_5, t_1)) * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_3 * t_1);
	elseif (t_4 <= 1.0)
		tmp = Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(t_5 * Float64(y_46_im * 0.5)))) / exp(fma(atan(x_46_im, x_46_re), y_46_im, 0.0)));
	elseif (t_4 <= Inf)
		tmp = Float64(t_3 * sin(t_1));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -2e-159], N[(N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * t$95$5 + t$95$1), $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[t$95$4, 0.0], N[(t$95$3 * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(t$95$5 * N[(y$46$im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(t$95$3 * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -1.99999999999999998e-159

   1. Initial program 69.9%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. log-pow N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 69.9

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 63.6

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

   7. Simplified 63.6%

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

   if -1.99999999999999998e-159 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. atan2-lowering-atan2.f64 85.9

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

   5. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. 1-exp N/A

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

      4. *-commutative N/A

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

      5. un-div-inv N/A

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

      6. *-commutative N/A

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

      7. pow1/2 N/A

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

      8. log-pow N/A

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

      9. associate-*l* N/A

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

   7. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 73.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 73.9

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

   5. Simplified 73.9%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in y.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 39.7

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

   5. Simplified 39.7%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 41.9

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

   8. Simplified 41.9%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 43.7

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

   11. Simplified 43.7%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 58.9

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

   13. Applied egg-rr 58.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 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 \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right) \cdot \left(y.im \cdot 0.5\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq \infty:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t\_1 \cdot y.re - t\_0}\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := \sin \left(t\_1 \cdot y.im + t\_3\right)\\ t_5 := t\_2 \cdot t\_4\\ t_6 := t\_4 \cdot \mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, 0 - y.im, 1\right)\\ \mathbf{if}\;t\_5 \leq -0.5:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;t\_6\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_2 \cdot t\_3\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;t\_6\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (exp (- (* t_1 y.re) t_0)))
        (t_3 (* y.re (atan2 x.im x.re)))
        (t_4 (sin (+ (* t_1 y.im) t_3)))
        (t_5 (* t_2 t_4))
        (t_6 (* t_4 (fma (atan2 x.im x.re) (- 0.0 y.im) 1.0))))
   (if (<= t_5 -0.5)
     (*
      (*
       y.re
       (fma
        -0.16666666666666666
        (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
        (atan2 x.im x.re)))
      (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
     (if (<= t_5 -2e-159)
       t_6
       (if (<= t_5 0.0)
         (* t_2 t_3)
         (if (<= t_5 1.0)
           t_6
           (* (exp (- (* y.re (log (fabs x.re))) 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 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((t_1 * y_46_re) - t_0));
	double t_3 = y_46_re * atan2(x_46_im, x_46_re);
	double t_4 = sin(((t_1 * y_46_im) + t_3));
	double t_5 = t_2 * t_4;
	double t_6 = t_4 * fma(atan2(x_46_im, x_46_re), (0.0 - y_46_im), 1.0);
	double tmp;
	if (t_5 <= -0.5) {
		tmp = (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), 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);
	} else if (t_5 <= -2e-159) {
		tmp = t_6;
	} else if (t_5 <= 0.0) {
		tmp = t_2 * t_3;
	} else if (t_5 <= 1.0) {
		tmp = t_6;
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_3;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = exp(Float64(Float64(t_1 * y_46_re) - t_0))
	t_3 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_4 = sin(Float64(Float64(t_1 * y_46_im) + t_3))
	t_5 = Float64(t_2 * t_4)
	t_6 = Float64(t_4 * fma(atan(x_46_im, x_46_re), Float64(0.0 - y_46_im), 1.0))
	tmp = 0.0
	if (t_5 <= -0.5)
		tmp = Float64(Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), 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));
	elseif (t_5 <= -2e-159)
		tmp = t_6;
	elseif (t_5 <= 0.0)
		tmp = Float64(t_2 * t_3);
	elseif (t_5 <= 1.0)
		tmp = t_6;
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$2 * t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(0.0 - y$46$im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, -0.5], N[(N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + 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[t$95$5, -2e-159], t$95$6, If[LessEqual[t$95$5, 0.0], N[(t$95$2 * t$95$3), $MachinePrecision], If[LessEqual[t$95$5, 1.0], t$95$6, N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 48.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. atan2-lowering-atan2.f64 61.1

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

   8. Simplified 61.1%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \mathsf{fma}\left(\frac{-1}{6}, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

          \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \mathsf{fma}\left(\frac{-1}{6}, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. *-lowering-*.f64 48.5

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

   11. Simplified 48.5%

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

   if -0.5 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -1.99999999999999998e-159 or -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 93.0%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. atan2-lowering-atan2.f64 91.9

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

   5. Simplified 91.9%

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

   6. Taylor expanded in y.im around 0

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

      1. neg-mul-1 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. distribute-rgt-neg-in N/A

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

      5. neg-mul-1 N/A

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

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

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

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

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

      8. neg-mul-1 N/A

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

      9. neg-sub0 N/A

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

      10. --lowering--.f64 89.8

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

   8. Simplified 89.8%

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

   if -1.99999999999999998e-159 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 10.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 44.0

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

   8. Simplified 44.0%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 43.2

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

   11. Simplified 43.2%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 58.0

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

   13. Applied egg-rr 58.0%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.5:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, 0 - y.im, 1\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 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 \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, 0 - y.im, 1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0}\\ t_4 := t\_3 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ t_5 := \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;t\_4 \leq -0.5:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, t\_5, y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3 \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(t\_5, -0.5, \frac{t\_1}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (exp (- (* t_2 y.re) t_0)))
        (t_4 (* t_3 (sin (+ (* t_2 y.im) t_1))))
        (t_5 (log (fma x.re x.re (* x.im x.im)))))
   (if (<= t_4 -0.5)
     (*
      (*
       y.re
       (fma
        -0.16666666666666666
        (* (* y.re y.re) (pow (atan2 x.im x.re) 3.0))
        (atan2 x.im x.re)))
      (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
     (if (<= t_4 -5e-179)
       (sin (* y.im (fma 0.5 t_5 (* y.re (/ (atan2 x.im x.re) y.im)))))
       (if (<= t_4 0.0)
         (* t_3 t_1)
         (if (<= t_4 1.0)
           (sin (* (- 0.0 y.im) (fma t_5 -0.5 (/ t_1 (- 0.0 y.im)))))
           (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double t_4 = t_3 * sin(((t_2 * y_46_im) + t_1));
	double t_5 = log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (t_4 <= -0.5) {
		tmp = (y_46_re * fma(-0.16666666666666666, ((y_46_re * y_46_re) * pow(atan2(x_46_im, x_46_re), 3.0)), 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);
	} else if (t_4 <= -5e-179) {
		tmp = sin((y_46_im * fma(0.5, t_5, (y_46_re * (atan2(x_46_im, x_46_re) / y_46_im)))));
	} else if (t_4 <= 0.0) {
		tmp = t_3 * t_1;
	} else if (t_4 <= 1.0) {
		tmp = sin(((0.0 - y_46_im) * fma(t_5, -0.5, (t_1 / (0.0 - y_46_im)))));
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	t_4 = Float64(t_3 * sin(Float64(Float64(t_2 * y_46_im) + t_1)))
	t_5 = log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_4 <= -0.5)
		tmp = Float64(Float64(y_46_re * fma(-0.16666666666666666, Float64(Float64(y_46_re * y_46_re) * (atan(x_46_im, x_46_re) ^ 3.0)), 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));
	elseif (t_4 <= -5e-179)
		tmp = sin(Float64(y_46_im * fma(0.5, t_5, Float64(y_46_re * Float64(atan(x_46_im, x_46_re) / y_46_im)))));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_3 * t_1);
	elseif (t_4 <= 1.0)
		tmp = sin(Float64(Float64(0.0 - y_46_im) * fma(t_5, -0.5, Float64(t_1 / Float64(0.0 - y_46_im)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -0.5], N[(N[(y$46$re * N[(-0.16666666666666666 * N[(N[(y$46$re * y$46$re), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] + 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[t$95$4, -5e-179], N[Sin[N[(y$46$im * N[(0.5 * t$95$5 + N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$3 * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[Sin[N[(N[(0.0 - y$46$im), $MachinePrecision] * N[(t$95$5 * -0.5 + N[(t$95$1 / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 48.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in y.re around 0

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

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

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

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

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

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

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

      4. unpow2 N/A

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

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

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

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

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

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

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

      8. atan2-lowering-atan2.f64 61.1

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

   8. Simplified 61.1%

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

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \mathsf{fma}\left(\frac{-1}{6}, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   10. Step-by-step derivation

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

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

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

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

       3. unpow2 N/A

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

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

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

       5. unpow2 N/A

          \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \mathsf{fma}\left(\frac{-1}{6}, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

       6. *-lowering-*.f64 48.5

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

   11. Simplified 48.5%

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

   if -0.5 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -4.9999999999999998e-179

   1. Initial program 99.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.0

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 96.0%

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

   11. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

       9. associate-/l* N/A

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

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

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

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

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

       12. atan2-lowering-atan2.f64 96.3

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

   13. Simplified 96.3%

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

   if -4.9999999999999998e-179 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.2

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

   7. Simplified 85.2%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 83.1%

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

   11. Taylor expanded in y.im around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

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

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

   13. Simplified 83.3%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im}\right) \cdot \left(0 - y.im\right)\right)} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 10.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 44.0

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

   8. Simplified 44.0%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 43.2

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

   11. Simplified 43.2%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 58.0

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

   13. Applied egg-rr 58.0%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.5:\\ \;\;\;\;\left(y.re \cdot \mathsf{fma}\left(-0.16666666666666666, \left(y.re \cdot y.re\right) \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 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 \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := \mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\\ t_4 := e^{t\_2 \cdot y.re - t\_0}\\ t_5 := t\_4 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ t_6 := \log t\_3\\ \mathbf{if}\;t\_5 \leq -0.5:\\ \;\;\;\;t\_1 \cdot \frac{{t\_3}^{\left(y.re \cdot 0.5\right)}}{e^{t\_0}}\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, t\_6, y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_4 \cdot t\_1\\ \mathbf{elif}\;t\_5 \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(t\_6, -0.5, \frac{t\_1}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (fma x.re x.re (* x.im x.im)))
        (t_4 (exp (- (* t_2 y.re) t_0)))
        (t_5 (* t_4 (sin (+ (* t_2 y.im) t_1))))
        (t_6 (log t_3)))
   (if (<= t_5 -0.5)
     (* t_1 (/ (pow t_3 (* y.re 0.5)) (exp t_0)))
     (if (<= t_5 -5e-179)
       (sin (* y.im (fma 0.5 t_6 (* y.re (/ (atan2 x.im x.re) y.im)))))
       (if (<= t_5 0.0)
         (* t_4 t_1)
         (if (<= t_5 1.0)
           (sin (* (- 0.0 y.im) (fma t_6 -0.5 (/ t_1 (- 0.0 y.im)))))
           (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = fma(x_46_re, x_46_re, (x_46_im * x_46_im));
	double t_4 = exp(((t_2 * y_46_re) - t_0));
	double t_5 = t_4 * sin(((t_2 * y_46_im) + t_1));
	double t_6 = log(t_3);
	double tmp;
	if (t_5 <= -0.5) {
		tmp = t_1 * (pow(t_3, (y_46_re * 0.5)) / exp(t_0));
	} else if (t_5 <= -5e-179) {
		tmp = sin((y_46_im * fma(0.5, t_6, (y_46_re * (atan2(x_46_im, x_46_re) / y_46_im)))));
	} else if (t_5 <= 0.0) {
		tmp = t_4 * t_1;
	} else if (t_5 <= 1.0) {
		tmp = sin(((0.0 - y_46_im) * fma(t_6, -0.5, (t_1 / (0.0 - y_46_im)))));
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))
	t_4 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	t_5 = Float64(t_4 * sin(Float64(Float64(t_2 * y_46_im) + t_1)))
	t_6 = log(t_3)
	tmp = 0.0
	if (t_5 <= -0.5)
		tmp = Float64(t_1 * Float64((t_3 ^ Float64(y_46_re * 0.5)) / exp(t_0)));
	elseif (t_5 <= -5e-179)
		tmp = sin(Float64(y_46_im * fma(0.5, t_6, Float64(y_46_re * Float64(atan(x_46_im, x_46_re) / y_46_im)))));
	elseif (t_5 <= 0.0)
		tmp = Float64(t_4 * t_1);
	elseif (t_5 <= 1.0)
		tmp = sin(Float64(Float64(0.0 - y_46_im) * fma(t_6, -0.5, Float64(t_1 / Float64(0.0 - y_46_im)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Log[t$95$3], $MachinePrecision]}, If[LessEqual[t$95$5, -0.5], N[(t$95$1 * N[(N[Power[t$95$3, N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, -5e-179], N[Sin[N[(y$46$im * N[(0.5 * t$95$6 + N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(t$95$4 * t$95$1), $MachinePrecision], If[LessEqual[t$95$5, 1.0], N[Sin[N[(N[(0.0 - y$46$im), $MachinePrecision] * N[(t$95$6 * -0.5 + N[(t$95$1 / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 48.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 48.1

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

   5. Simplified 48.1%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 35.1

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

   8. Simplified 35.1%

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

      1. exp-diff N/A

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

      2. +-commutative N/A

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

      3. pow-to-exp N/A

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

      4. frac-2neg N/A

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

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

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

   10. Applied egg-rr 39.4%

       \[\leadsto \color{blue}{\frac{0 - {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(y.re \cdot 0.5\right)}}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -0.5 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -4.9999999999999998e-179

   1. Initial program 99.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.0

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 96.0%

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

   11. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

       9. associate-/l* N/A

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

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

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

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

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

       12. atan2-lowering-atan2.f64 96.3

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

   13. Simplified 96.3%

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

   if -4.9999999999999998e-179 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.2

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

   7. Simplified 85.2%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 83.1%

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

   11. Taylor expanded in y.im around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

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

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

   13. Simplified 83.3%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im}\right) \cdot \left(0 - y.im\right)\right)} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 10.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 44.0

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

   8. Simplified 44.0%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 43.2

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

   11. Simplified 43.2%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 58.0

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

   13. Applied egg-rr 58.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.5:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(y.re \cdot 0.5\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 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 \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := e^{t\_1 \cdot y.re - t\_0}\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := t\_2 \cdot \sin \left(t\_1 \cdot y.im + t\_3\right)\\ t_5 := \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;t\_4 \leq -0.5:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(y.re \cdot y.re\right), {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, t\_5, y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_2 \cdot t\_3\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(t\_5, -0.5, \frac{t\_3}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (exp (- (* t_1 y.re) t_0)))
        (t_3 (* y.re (atan2 x.im x.re)))
        (t_4 (* t_2 (sin (+ (* t_1 y.im) t_3))))
        (t_5 (log (fma x.re x.re (* x.im x.im)))))
   (if (<= t_4 -0.5)
     (*
      y.re
      (fma
       (* -0.16666666666666666 (* y.re y.re))
       (pow (atan2 x.im x.re) 3.0)
       (atan2 x.im x.re)))
     (if (<= t_4 -5e-179)
       (sin (* y.im (fma 0.5 t_5 (* y.re (/ (atan2 x.im x.re) y.im)))))
       (if (<= t_4 0.0)
         (* t_2 t_3)
         (if (<= t_4 1.0)
           (sin (* (- 0.0 y.im) (fma t_5 -0.5 (/ t_3 (- 0.0 y.im)))))
           (* (exp (- (* y.re (log (fabs x.re))) 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 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = exp(((t_1 * y_46_re) - t_0));
	double t_3 = y_46_re * atan2(x_46_im, x_46_re);
	double t_4 = t_2 * sin(((t_1 * y_46_im) + t_3));
	double t_5 = log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (t_4 <= -0.5) {
		tmp = y_46_re * fma((-0.16666666666666666 * (y_46_re * y_46_re)), pow(atan2(x_46_im, x_46_re), 3.0), atan2(x_46_im, x_46_re));
	} else if (t_4 <= -5e-179) {
		tmp = sin((y_46_im * fma(0.5, t_5, (y_46_re * (atan2(x_46_im, x_46_re) / y_46_im)))));
	} else if (t_4 <= 0.0) {
		tmp = t_2 * t_3;
	} else if (t_4 <= 1.0) {
		tmp = sin(((0.0 - y_46_im) * fma(t_5, -0.5, (t_3 / (0.0 - y_46_im)))));
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_3;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = exp(Float64(Float64(t_1 * y_46_re) - t_0))
	t_3 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_4 = Float64(t_2 * sin(Float64(Float64(t_1 * y_46_im) + t_3)))
	t_5 = log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_4 <= -0.5)
		tmp = Float64(y_46_re * fma(Float64(-0.16666666666666666 * Float64(y_46_re * y_46_re)), (atan(x_46_im, x_46_re) ^ 3.0), atan(x_46_im, x_46_re)));
	elseif (t_4 <= -5e-179)
		tmp = sin(Float64(y_46_im * fma(0.5, t_5, Float64(y_46_re * Float64(atan(x_46_im, x_46_re) / y_46_im)))));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_2 * t_3);
	elseif (t_4 <= 1.0)
		tmp = sin(Float64(Float64(0.0 - y_46_im) * fma(t_5, -0.5, Float64(t_3 / Float64(0.0 - y_46_im)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * N[Sin[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -0.5], N[(y$46$re * N[(N[(-0.16666666666666666 * N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, -5e-179], N[Sin[N[(y$46$im * N[(0.5 * t$95$5 + N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(t$95$2 * t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[Sin[N[(N[(0.0 - y$46$im), $MachinePrecision] * N[(t$95$5 * -0.5 + N[(t$95$3 / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 48.0%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. atan2-lowering-atan2.f64 14.5

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

   5. Simplified 14.5%

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

   6. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 2.0

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

   8. Simplified 2.0%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(\frac{-1}{6} \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   10. Step-by-step derivation

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

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

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

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

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

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

       9. atan2-lowering-atan2.f64 36.1

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

   11. Simplified 36.1%

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

   if -0.5 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -4.9999999999999998e-179

   1. Initial program 99.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.0

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 96.0%

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

   11. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

       9. associate-/l* N/A

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

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

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

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

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

       12. atan2-lowering-atan2.f64 96.3

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

   13. Simplified 96.3%

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

   if -4.9999999999999998e-179 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.2

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

   7. Simplified 85.2%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 83.1%

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

   11. Taylor expanded in y.im around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

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

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

   13. Simplified 83.3%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im}\right) \cdot \left(0 - y.im\right)\right)} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 10.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 44.0

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

   8. Simplified 44.0%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 43.2

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

   11. Simplified 43.2%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 58.0

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

   13. Applied egg-rr 58.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.5:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(y.re \cdot y.re\right), {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 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 \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 59.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := e^{t\_1 \cdot y.re - t\_0} \cdot \sin \left(t\_1 \cdot y.im + t\_2\right)\\ t_4 := e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_2\\ t_5 := \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;t\_3 \leq -0.5:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(y.re \cdot y.re\right), {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, t\_5, y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(t\_5, -0.5, \frac{t\_2}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_4\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (* (exp (- (* t_1 y.re) t_0)) (sin (+ (* t_1 y.im) t_2))))
        (t_4 (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_2))
        (t_5 (log (fma x.re x.re (* x.im x.im)))))
   (if (<= t_3 -0.5)
     (*
      y.re
      (fma
       (* -0.16666666666666666 (* y.re y.re))
       (pow (atan2 x.im x.re) 3.0)
       (atan2 x.im x.re)))
     (if (<= t_3 -5e-179)
       (sin (* y.im (fma 0.5 t_5 (* y.re (/ (atan2 x.im x.re) y.im)))))
       (if (<= t_3 0.0)
         t_4
         (if (<= t_3 1.0)
           (sin (* (- 0.0 y.im) (fma t_5 -0.5 (/ t_2 (- 0.0 y.im)))))
           t_4))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = exp(((t_1 * y_46_re) - t_0)) * sin(((t_1 * y_46_im) + t_2));
	double t_4 = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_2;
	double t_5 = log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (t_3 <= -0.5) {
		tmp = y_46_re * fma((-0.16666666666666666 * (y_46_re * y_46_re)), pow(atan2(x_46_im, x_46_re), 3.0), atan2(x_46_im, x_46_re));
	} else if (t_3 <= -5e-179) {
		tmp = sin((y_46_im * fma(0.5, t_5, (y_46_re * (atan2(x_46_im, x_46_re) / y_46_im)))));
	} else if (t_3 <= 0.0) {
		tmp = t_4;
	} else if (t_3 <= 1.0) {
		tmp = sin(((0.0 - y_46_im) * fma(t_5, -0.5, (t_2 / (0.0 - y_46_im)))));
	} else {
		tmp = t_4;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_0)) * sin(Float64(Float64(t_1 * y_46_im) + t_2)))
	t_4 = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_2)
	t_5 = log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_3 <= -0.5)
		tmp = Float64(y_46_re * fma(Float64(-0.16666666666666666 * Float64(y_46_re * y_46_re)), (atan(x_46_im, x_46_re) ^ 3.0), atan(x_46_im, x_46_re)));
	elseif (t_3 <= -5e-179)
		tmp = sin(Float64(y_46_im * fma(0.5, t_5, Float64(y_46_re * Float64(atan(x_46_im, x_46_re) / y_46_im)))));
	elseif (t_3 <= 0.0)
		tmp = t_4;
	elseif (t_3 <= 1.0)
		tmp = sin(Float64(Float64(0.0 - y_46_im) * fma(t_5, -0.5, Float64(t_2 / Float64(0.0 - y_46_im)))));
	else
		tmp = t_4;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(t$95$1 * y$46$im), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$5 = N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$3, -0.5], N[(y$46$re * N[(N[(-0.16666666666666666 * N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] * N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision] + N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, -5e-179], N[Sin[N[(y$46$im * N[(0.5 * t$95$5 + N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$4, If[LessEqual[t$95$3, 1.0], N[Sin[N[(N[(0.0 - y$46$im), $MachinePrecision] * N[(t$95$5 * -0.5 + N[(t$95$2 / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$4]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 48.0%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. atan2-lowering-atan2.f64 14.5

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

   5. Simplified 14.5%

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

   6. Taylor expanded in y.im around 0

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

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

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

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

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

      3. atan2-lowering-atan2.f64 2.0

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

   8. Simplified 2.0%

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

   9. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(\frac{-1}{6} \cdot \left({y.re}^{2} \cdot {\tan^{-1}_* \frac{x.im}{x.re}}^{3}\right) + \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   10. Step-by-step derivation

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

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

       2. associate-*r* N/A

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

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

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

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

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

       5. unpow2 N/A

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

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

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

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

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

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

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

       9. atan2-lowering-atan2.f64 36.1

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

   11. Simplified 36.1%

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

   if -0.5 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -4.9999999999999998e-179

   1. Initial program 99.5%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 99.4%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 96.0

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

   7. Simplified 96.0%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 96.0%

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

   11. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

       4. +-commutative N/A

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

       5. unpow2 N/A

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

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

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

       7. unpow2 N/A

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

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

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

       9. associate-/l* N/A

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

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

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

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

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

       12. atan2-lowering-atan2.f64 96.3

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

   13. Simplified 96.3%

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

   if -4.9999999999999998e-179 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0 or 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 25.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 54.1

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

   5. Simplified 54.1%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 53.6

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

   8. Simplified 53.6%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 50.8

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

   11. Simplified 50.8%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 64.1

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

   13. Applied egg-rr 64.1%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 85.2

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

   7. Simplified 85.2%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 83.1%

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

   11. Taylor expanded in y.im around -inf

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

       1. mul-1-neg N/A

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

       2. *-commutative N/A

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

       3. distribute-rgt-neg-in N/A

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

       4. neg-mul-1 N/A

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

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

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

   13. Simplified 83.3%

       \[\leadsto 1 \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im}\right) \cdot \left(0 - y.im\right)\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -0.5:\\ \;\;\;\;y.re \cdot \mathsf{fma}\left(-0.16666666666666666 \cdot \left(y.re \cdot y.re\right), {\tan^{-1}_* \frac{x.im}{x.re}}^{3}, \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -5 \cdot 10^{-179}:\\ \;\;\;\;\sin \left(y.im \cdot \mathsf{fma}\left(0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}\right)\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 0:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\sin \left(\left(0 - y.im\right) \cdot \mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), -0.5, \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{0 - y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0}\\ t_4 := t\_3 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ t_5 := \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, t\_5, t\_1\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3 \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, t\_5 \cdot \left(y.im \cdot 0.5\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (exp (- (* t_2 y.re) t_0)))
        (t_4 (* t_3 (sin (+ (* t_2 y.im) t_1))))
        (t_5 (log (fma x.re x.re (* x.im x.im)))))
   (if (<= t_4 -2e-159)
     (*
      (sin (fma (* y.im 0.5) t_5 t_1))
      (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
     (if (<= t_4 0.0)
       (* t_3 t_1)
       (if (<= t_4 1.0)
         (/
          (sin (fma y.re (atan2 x.im x.re) (* t_5 (* y.im 0.5))))
          (exp (fma (atan2 x.im x.re) y.im 0.0)))
         (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double t_4 = t_3 * sin(((t_2 * y_46_im) + t_1));
	double t_5 = log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (t_4 <= -2e-159) {
		tmp = sin(fma((y_46_im * 0.5), t_5, t_1)) * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	} else if (t_4 <= 0.0) {
		tmp = t_3 * t_1;
	} else if (t_4 <= 1.0) {
		tmp = sin(fma(y_46_re, atan2(x_46_im, x_46_re), (t_5 * (y_46_im * 0.5)))) / exp(fma(atan2(x_46_im, x_46_re), y_46_im, 0.0));
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	t_4 = Float64(t_3 * sin(Float64(Float64(t_2 * y_46_im) + t_1)))
	t_5 = log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_4 <= -2e-159)
		tmp = Float64(sin(fma(Float64(y_46_im * 0.5), t_5, t_1)) * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	elseif (t_4 <= 0.0)
		tmp = Float64(t_3 * t_1);
	elseif (t_4 <= 1.0)
		tmp = Float64(sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(t_5 * Float64(y_46_im * 0.5)))) / exp(fma(atan(x_46_im, x_46_re), y_46_im, 0.0)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$4, -2e-159], N[(N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * t$95$5 + t$95$1), $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[t$95$4, 0.0], N[(t$95$3 * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1.0], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(t$95$5 * N[(y$46$im * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im + 0.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -1.99999999999999998e-159

   1. Initial program 69.9%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. log-pow N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 69.9

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 63.6

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

   7. Simplified 63.6%

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

   if -1.99999999999999998e-159 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

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

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

      5. atan2-lowering-atan2.f64 85.9

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

   5. Simplified 85.9%

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

      1. *-commutative N/A

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

      2. exp-diff N/A

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

      3. 1-exp N/A

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

      4. *-commutative N/A

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

      5. un-div-inv N/A

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

      6. *-commutative N/A

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

      7. pow1/2 N/A

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

      8. log-pow N/A

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

      9. associate-*l* N/A

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

   7. Applied egg-rr 86.0%

      \[\leadsto \color{blue}{\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 10.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 44.6

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

   5. Simplified 44.6%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 44.0

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

   8. Simplified 44.0%

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

   9. Taylor expanded in x.re around inf

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

       1. unpow2 N/A

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

       2. *-lowering-*.f64 43.2

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

   11. Simplified 43.2%

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

       1. rem-sqrt-square N/A

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

       2. fabs-lowering-fabs.f64 58.0

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

   13. Applied egg-rr 58.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\frac{\sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right) \cdot \left(y.im \cdot 0.5\right)\right)\right)}{e^{\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.im, 0\right)}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_4 := e^{t\_3 \cdot y.re - t\_1}\\ t_5 := t\_4 \cdot \sin \left(t\_3 \cdot y.im + t\_2\right)\\ t_6 := \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)\\ \mathbf{if}\;t\_5 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, t\_6, t\_2\right)\right) \cdot t\_0\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;t\_4 \cdot t\_2\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot t\_6, y.im, t\_2\right)}}\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_1} \cdot t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
        (t_1 (* (atan2 x.im x.re) y.im))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_4 (exp (- (* t_3 y.re) t_1)))
        (t_5 (* t_4 (sin (+ (* t_3 y.im) t_2))))
        (t_6 (log (fma x.re x.re (* x.im x.im)))))
   (if (<= t_5 -2e-159)
     (* (sin (fma (* y.im 0.5) t_6 t_2)) t_0)
     (if (<= t_5 0.0)
       (* t_4 t_2)
       (if (<= t_5 INFINITY)
         (* (sin (/ 1.0 (/ 1.0 (fma (* 0.5 t_6) y.im t_2)))) t_0)
         (* (exp (- (* y.re (log (fabs x.re))) t_1)) t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_4 = exp(((t_3 * y_46_re) - t_1));
	double t_5 = t_4 * sin(((t_3 * y_46_im) + t_2));
	double t_6 = log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)));
	double tmp;
	if (t_5 <= -2e-159) {
		tmp = sin(fma((y_46_im * 0.5), t_6, t_2)) * t_0;
	} else if (t_5 <= 0.0) {
		tmp = t_4 * t_2;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sin((1.0 / (1.0 / fma((0.5 * t_6), y_46_im, t_2)))) * t_0;
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_1)) * t_2;
	}
	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))) ^ y_46_re
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_4 = exp(Float64(Float64(t_3 * y_46_re) - t_1))
	t_5 = Float64(t_4 * sin(Float64(Float64(t_3 * y_46_im) + t_2)))
	t_6 = log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))
	tmp = 0.0
	if (t_5 <= -2e-159)
		tmp = Float64(sin(fma(Float64(y_46_im * 0.5), t_6, t_2)) * t_0);
	elseif (t_5 <= 0.0)
		tmp = Float64(t_4 * t_2);
	elseif (t_5 <= Inf)
		tmp = Float64(sin(Float64(1.0 / Float64(1.0 / fma(Float64(0.5 * t_6), y_46_im, t_2)))) * t_0);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_1)) * t_2);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Exp[N[(N[(t$95$3 * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sin[N[(N[(t$95$3 * y$46$im), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$5, -2e-159], N[(N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * t$95$6 + t$95$2), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(t$95$4 * t$95$2), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[Sin[N[(1.0 / N[(1.0 / N[(N[(0.5 * t$95$6), $MachinePrecision] * y$46$im + t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -1.99999999999999998e-159

   1. Initial program 69.9%

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

      1. *-commutative N/A

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

      2. pow1/2 N/A

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

      3. log-pow N/A

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

      4. associate-*r* N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      11. atan2-lowering-atan2.f64 69.9

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

   4. Applied egg-rr 69.9%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

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

      5. unpow2 N/A

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

      6. *-lowering-*.f64 63.6

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

   7. Simplified 63.6%

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

   if -1.99999999999999998e-159 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y.re around 0

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

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

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

      2. atan2-lowering-atan2.f64 100.0

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

   8. Simplified 100.0%

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

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 80.8%

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

      1. flip3-+ N/A

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

      2. clear-num N/A

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

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

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

   4. Applied egg-rr 83.0%

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

   5. Taylor expanded in y.im around 0

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

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

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

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

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

      3. unpow2 N/A

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

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

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      6. *-lowering-*.f64 68.8

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   7. Simplified 68.8%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 0.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.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 39.7

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 39.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 41.9

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 41.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 43.7

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 43.7%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. rem-sqrt-square N/A

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. fabs-lowering-fabs.f64 58.9

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied egg-rr 58.9%

       \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq \infty:\\ \;\;\;\;\sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\\ t_2 := {\left(\sqrt{t\_1}\right)}^{y.re}\\ t_3 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_4 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_5 := e^{t\_4 \cdot y.re - t\_0}\\ t_6 := t\_5 \cdot \sin \left(t\_4 \cdot y.im + t\_3\right)\\ \mathbf{if}\;t\_6 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), t\_3\right)\right) \cdot t\_2\\ \mathbf{elif}\;t\_6 \leq 0:\\ \;\;\;\;t\_5 \cdot t\_3\\ \mathbf{elif}\;t\_6 \leq 1:\\ \;\;\;\;t\_2 \cdot \sin \left(\mathsf{fma}\left(\log t\_1, -0.5, \frac{t\_3}{0 - y.im}\right) \cdot \left(0 - y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (fma x.im x.im (* x.re x.re)))
        (t_2 (pow (sqrt t_1) y.re))
        (t_3 (* y.re (atan2 x.im x.re)))
        (t_4 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_5 (exp (- (* t_4 y.re) t_0)))
        (t_6 (* t_5 (sin (+ (* t_4 y.im) t_3)))))
   (if (<= t_6 -2e-159)
     (* (sin (fma (* y.im 0.5) (log (fma x.re x.re (* x.im x.im))) t_3)) t_2)
     (if (<= t_6 0.0)
       (* t_5 t_3)
       (if (<= t_6 1.0)
         (*
          t_2
          (sin (* (fma (log t_1) -0.5 (/ t_3 (- 0.0 y.im))) (- 0.0 y.im))))
         (* (exp (- (* y.re (log (fabs x.re))) 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 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = fma(x_46_im, x_46_im, (x_46_re * x_46_re));
	double t_2 = pow(sqrt(t_1), y_46_re);
	double t_3 = y_46_re * atan2(x_46_im, x_46_re);
	double t_4 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_5 = exp(((t_4 * y_46_re) - t_0));
	double t_6 = t_5 * sin(((t_4 * y_46_im) + t_3));
	double tmp;
	if (t_6 <= -2e-159) {
		tmp = sin(fma((y_46_im * 0.5), log(fma(x_46_re, x_46_re, (x_46_im * x_46_im))), t_3)) * t_2;
	} else if (t_6 <= 0.0) {
		tmp = t_5 * t_3;
	} else if (t_6 <= 1.0) {
		tmp = t_2 * sin((fma(log(t_1), -0.5, (t_3 / (0.0 - y_46_im))) * (0.0 - y_46_im)));
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_3;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))
	t_2 = sqrt(t_1) ^ y_46_re
	t_3 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_4 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_5 = exp(Float64(Float64(t_4 * y_46_re) - t_0))
	t_6 = Float64(t_5 * sin(Float64(Float64(t_4 * y_46_im) + t_3)))
	tmp = 0.0
	if (t_6 <= -2e-159)
		tmp = Float64(sin(fma(Float64(y_46_im * 0.5), log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))), t_3)) * t_2);
	elseif (t_6 <= 0.0)
		tmp = Float64(t_5 * t_3);
	elseif (t_6 <= 1.0)
		tmp = Float64(t_2 * sin(Float64(fma(log(t_1), -0.5, Float64(t_3 / Float64(0.0 - y_46_im))) * Float64(0.0 - y_46_im))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sqrt[t$95$1], $MachinePrecision], y$46$re], $MachinePrecision]}, Block[{t$95$3 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Exp[N[(N[(t$95$4 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$5 * N[Sin[N[(N[(t$95$4 * y$46$im), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$6, -2e-159], N[(N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$6, 0.0], N[(t$95$5 * t$95$3), $MachinePrecision], If[LessEqual[t$95$6, 1.0], N[(t$95$2 * N[Sin[N[(N[(N[Log[t$95$1], $MachinePrecision] * -0.5 + N[(t$95$3 / N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.0 - y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -1.99999999999999998e-159

   1. Initial program 69.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. pow1/2 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \color{blue}{\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. log-pow N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \color{blue}{\left(\frac{1}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. associate-*r* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(y.im \cdot \frac{1}{2}\right) \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{y.im \cdot \frac{1}{2}}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)\right) \]

      11. atan2-lowering-atan2.f64 69.9

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right)\right) \]

   4. Applied egg-rr 69.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]
   6. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-lowering-*.f64 63.6

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   7. Simplified 63.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   if -1.99999999999999998e-159 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 100.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 100.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 100.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   if -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 87.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right)} \]

      2. clear-num N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

   4. Applied egg-rr 87.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right)} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]
   6. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      6. *-lowering-*.f64 85.2

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   7. Simplified 85.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   8. Taylor expanded in y.im around -inf

      \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \color{blue}{\left(-1 \cdot \left(y.im \cdot \left(-1 \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} + \frac{-1}{2} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \color{blue}{\left(\mathsf{neg}\left(y.im \cdot \left(-1 \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} + \frac{-1}{2} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)\right)} \]

      2. *-commutative N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(\mathsf{neg}\left(\color{blue}{\left(-1 \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} + \frac{-1}{2} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right) \cdot y.im}\right)\right) \]

      3. distribute-rgt-neg-in N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \color{blue}{\left(\left(-1 \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} + \frac{-1}{2} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right) \cdot \left(\mathsf{neg}\left(y.im\right)\right)\right)} \]

      4. neg-mul-1 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(\left(-1 \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} + \frac{-1}{2} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right) \cdot \color{blue}{\left(-1 \cdot y.im\right)}\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \color{blue}{\left(\left(-1 \cdot \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{y.im} + \frac{-1}{2} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right) \cdot \left(-1 \cdot y.im\right)\right)} \]

   10. Simplified 85.3%

       \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right), -0.5, \frac{y.re \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}{y.im}\right) \cdot \left(0 - y.im\right)\right)} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 10.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 44.6

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 44.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 44.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 44.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 43.2

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 43.2%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. rem-sqrt-square N/A

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. fabs-lowering-fabs.f64 58.0

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied egg-rr 58.0%

       \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right), -0.5, \frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{0 - y.im}\right) \cdot \left(0 - y.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0}\\ t_4 := t\_3 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right)\\ t_5 := \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), t\_1\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;t\_4 \leq -2 \cdot 10^{-159}:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;t\_3 \cdot t\_1\\ \mathbf{elif}\;t\_4 \leq 1:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (exp (- (* t_2 y.re) t_0)))
        (t_4 (* t_3 (sin (+ (* t_2 y.im) t_1))))
        (t_5
         (*
          (sin (fma (* y.im 0.5) (log (fma x.re x.re (* x.im x.im))) t_1))
          (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= t_4 -2e-159)
     t_5
     (if (<= t_4 0.0)
       (* t_3 t_1)
       (if (<= t_4 1.0)
         t_5
         (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double t_4 = t_3 * sin(((t_2 * y_46_im) + t_1));
	double t_5 = sin(fma((y_46_im * 0.5), log(fma(x_46_re, x_46_re, (x_46_im * x_46_im))), t_1)) * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (t_4 <= -2e-159) {
		tmp = t_5;
	} else if (t_4 <= 0.0) {
		tmp = t_3 * t_1;
	} else if (t_4 <= 1.0) {
		tmp = t_5;
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	t_4 = Float64(t_3 * sin(Float64(Float64(t_2 * y_46_im) + t_1)))
	t_5 = Float64(sin(fma(Float64(y_46_im * 0.5), log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))), t_1)) * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re))
	tmp = 0.0
	if (t_4 <= -2e-159)
		tmp = t_5;
	elseif (t_4 <= 0.0)
		tmp = Float64(t_3 * t_1);
	elseif (t_4 <= 1.0)
		tmp = t_5;
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $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[t$95$4, -2e-159], t$95$5, If[LessEqual[t$95$4, 0.0], N[(t$95$3 * t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1.0], t$95$5, N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -1.99999999999999998e-159 or -0.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < 1

   1. Initial program 76.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. pow1/2 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \color{blue}{\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. log-pow N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \color{blue}{\left(\frac{1}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. associate-*r* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(y.im \cdot \frac{1}{2}\right) \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{y.im \cdot \frac{1}{2}}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)\right) \]

      11. atan2-lowering-atan2.f64 76.3

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right)\right) \]

   4. Applied egg-rr 76.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]
   6. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      6. *-lowering-*.f64 71.3

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   7. Simplified 71.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

   if -1.99999999999999998e-159 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < -0.0

   1. Initial program 100.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 100.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 100.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 100.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 100.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   if 1 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 10.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 44.6

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 44.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 44.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 44.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 43.2

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 43.2%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. rem-sqrt-square N/A

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. fabs-lowering-fabs.f64 58.0

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied egg-rr 58.0%

       \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq -2 \cdot 10^{-159}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 0:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq 1:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 67.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0}\\ \mathbf{if}\;t\_3 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right) \leq \infty:\\ \;\;\;\;t\_3 \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, t\_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (exp (- (* t_2 y.re) t_0))))
   (if (<= (* t_3 (sin (+ (* t_2 y.im) t_1))) INFINITY)
     (*
      t_3
      (sin
       (/
        1.0
        (/ 1.0 (fma (* 0.5 (log (fma x.re x.re (* x.im x.im)))) y.im t_1)))))
     (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double tmp;
	if ((t_3 * sin(((t_2 * y_46_im) + t_1))) <= ((double) INFINITY)) {
		tmp = t_3 * sin((1.0 / (1.0 / fma((0.5 * log(fma(x_46_re, x_46_re, (x_46_im * x_46_im)))), y_46_im, t_1))));
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	tmp = 0.0
	if (Float64(t_3 * sin(Float64(Float64(t_2 * y_46_im) + t_1))) <= Inf)
		tmp = Float64(t_3 * sin(Float64(1.0 / Float64(1.0 / fma(Float64(0.5 * log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)))), y_46_im, t_1)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$3 * N[Sin[N[(1.0 / N[(1.0 / N[(N[(0.5 * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$im + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 82.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right)} \]

      2. clear-num N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

   4. Applied egg-rr 83.4%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right)} \]

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 0.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.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 39.7

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 39.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 41.9

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 41.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 43.7

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 43.7%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. rem-sqrt-square N/A

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. fabs-lowering-fabs.f64 58.9

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied egg-rr 58.9%

       \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq \infty:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)\\ t_3 := e^{t\_2 \cdot y.re - t\_0}\\ \mathbf{if}\;t\_3 \cdot \sin \left(t\_2 \cdot y.im + t\_1\right) \leq \infty:\\ \;\;\;\;t\_3 \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), t\_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_0} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (t_3 (exp (- (* t_2 y.re) t_0))))
   (if (<= (* t_3 (sin (+ (* t_2 y.im) t_1))) INFINITY)
     (* t_3 (sin (fma (* y.im 0.5) (log (fma x.re x.re (* x.im x.im))) t_1)))
     (* (exp (- (* y.re (log (fabs x.re))) t_0)) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	double t_3 = exp(((t_2 * y_46_re) - t_0));
	double tmp;
	if ((t_3 * sin(((t_2 * y_46_im) + t_1))) <= ((double) INFINITY)) {
		tmp = t_3 * sin(fma((y_46_im * 0.5), log(fma(x_46_re, x_46_re, (x_46_im * x_46_im))), t_1));
	} else {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_0)) * t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))
	t_3 = exp(Float64(Float64(t_2 * y_46_re) - t_0))
	tmp = 0.0
	if (Float64(t_3 * sin(Float64(Float64(t_2 * y_46_im) + t_1))) <= Inf)
		tmp = Float64(t_3 * sin(fma(Float64(y_46_im * 0.5), log(fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im))), t_1)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_0)) * t_1);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(t$95$3 * N[Sin[N[(N[(t$95$2 * y$46$im), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$3 * N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re)))) < +inf.0

   1. Initial program 82.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. pow1/2 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \color{blue}{\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. log-pow N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \color{blue}{\left(\frac{1}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. associate-*r* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(y.im \cdot \frac{1}{2}\right) \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{y.im \cdot \frac{1}{2}}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)\right) \]

      11. atan2-lowering-atan2.f64 82.6

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right)\right) \]

   4. Applied egg-rr 82.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   if +inf.0 < (*.f64 (exp.f64 (-.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.re) (*.f64 (atan2.f64 x.im x.re) y.im))) (sin.f64 (+.f64 (*.f64 (log.f64 (sqrt.f64 (+.f64 (*.f64 x.re x.re) (*.f64 x.im x.im)))) y.im) (*.f64 (atan2.f64 x.im x.re) y.re))))

   1. Initial program 0.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.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 39.7

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 39.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 41.9

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 41.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 43.7

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 43.7%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. rem-sqrt-square N/A

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. fabs-lowering-fabs.f64 58.9

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied egg-rr 58.9%

       \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \leq \infty:\\ \;\;\;\;e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 62.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq -0.01:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - t\_1} \cdot t\_0\\ \mathbf{elif}\;x.re \leq 3.2 \cdot 10^{-308}:\\ \;\;\;\;t\_0 \cdot \frac{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(y.re \cdot 0.5\right)}}{e^{t\_1}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - t\_1} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log x.re\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re -0.01)
     (* (exp (- (* y.re (log (fabs x.re))) t_1)) t_0)
     (if (<= x.re 3.2e-308)
       (* t_0 (/ (pow (fma x.re x.re (* x.im x.im)) (* y.re 0.5)) (exp t_1)))
       (*
        (exp (- (* y.re (log x.re)) t_1))
        (sin (fma y.re (atan2 x.im x.re) (* y.im (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 t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= -0.01) {
		tmp = exp(((y_46_re * log(fabs(x_46_re))) - t_1)) * t_0;
	} else if (x_46_re <= 3.2e-308) {
		tmp = t_0 * (pow(fma(x_46_re, x_46_re, (x_46_im * x_46_im)), (y_46_re * 0.5)) / exp(t_1));
	} else {
		tmp = exp(((y_46_re * log(x_46_re)) - t_1)) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * 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))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= -0.01)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(abs(x_46_re))) - t_1)) * t_0);
	elseif (x_46_re <= 3.2e-308)
		tmp = Float64(t_0 * Float64((fma(x_46_re, x_46_re, Float64(x_46_im * x_46_im)) ^ Float64(y_46_re * 0.5)) / exp(t_1)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_re)) - t_1)) * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * 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]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, -0.01], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Abs[x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[x$46$re, 3.2e-308], N[(t$95$0 * N[(N[Power[N[(x$46$re * x$46$re + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision], N[(y$46$re * 0.5), $MachinePrecision]], $MachinePrecision] / N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -0.0100000000000000002

   1. Initial program 21.8%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 47.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 47.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 47.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 47.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 54.0

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 54.0%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   12. Step-by-step derivation

       1. rem-sqrt-square N/A

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. fabs-lowering-fabs.f64 70.9

          \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   13. Applied egg-rr 70.9%

       \[\leadsto e^{\log \color{blue}{\left(\left|x.re\right|\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -0.0100000000000000002 < x.re < 3.2000000000000001e-308

   1. Initial program 54.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 56.2

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 56.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 58.9

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 58.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   9. Step-by-step derivation

      1. exp-diff N/A

         \[\leadsto \color{blue}{\frac{e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. +-commutative N/A

         \[\leadsto \frac{e^{\log \left(\sqrt{\color{blue}{x.im \cdot x.im + x.re \cdot x.re}}\right) \cdot y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. pow-to-exp N/A

         \[\leadsto \frac{\color{blue}{{\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\right)}{\mathsf{neg}\left(e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left({\left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right)}^{y.re}\right)}{\mathsf{neg}\left(e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\right)}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   10. Applied egg-rr 60.2%

       \[\leadsto \color{blue}{\frac{0 - {\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(y.re \cdot 0.5\right)}}{0 - e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if 3.2000000000000001e-308 < x.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. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{y.im \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. pow1/2 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \color{blue}{\left({\left(x.re \cdot x.re + x.im \cdot x.im\right)}^{\frac{1}{2}}\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. log-pow N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \color{blue}{\left(\frac{1}{2} \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. associate-*r* N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\color{blue}{\left(y.im \cdot \frac{1}{2}\right) \cdot \log \left(x.re \cdot x.re + x.im \cdot x.im\right)} + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{y.im \cdot \frac{1}{2}}, \log \left(x.re \cdot x.re + x.im \cdot x.im\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      7. log-lowering-log.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \color{blue}{\log \left(x.re \cdot x.re + x.im \cdot x.im\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      8. accelerator-lowering-fma.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \color{blue}{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, \color{blue}{x.im \cdot x.im}\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot \frac{1}{2}, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re}\right)\right) \]

      11. atan2-lowering-atan2.f64 37.6

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot y.re\right)\right) \]

   4. Applied egg-rr 37.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.im \cdot 0.5, \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \]

   5. Taylor expanded in x.re around inf

      \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{e^{-1 \cdot \left(y.re \cdot \log \left(\frac{1}{x.re}\right)\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(-1 \cdot \left(y.im \cdot \log \left(\frac{1}{x.re}\right)\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   7. Simplified 67.3%

      \[\leadsto \color{blue}{e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log x.re\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -0.01:\\ \;\;\;\;e^{y.re \cdot \log \left(\left|x.re\right|\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.re \leq 3.2 \cdot 10^{-308}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right)}^{\left(y.re \cdot 0.5\right)}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log x.re\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -210000.0)
     t_1
     (if (<= y.re 46000000000000.0)
       (* (exp (* y.im (- 0.0 (atan2 x.im x.re)))) t_0)
       t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = t_0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = t_1;
	} else if (y_46_re <= 46000000000000.0) {
		tmp = exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re)))) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(t_0 * (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 <= -210000.0)
		tmp = t_1;
	elseif (y_46_re <= 46000000000000.0)
		tmp = Float64(exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re)))) * 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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[N[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, -210000.0], t$95$1, If[LessEqual[y$46$re, 46000000000000.0], N[(N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.1e5 or 4.6e13 < y.re

   1. Initial program 33.9%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 71.5

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 71.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. *-lowering-*.f64 70.6

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. Simplified 70.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -2.1e5 < y.re < 4.6e13

   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 35.2

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 35.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   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-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. atan2-lowering-atan2.f64 54.0

         \[\leadsto e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. Simplified 54.0%

      \[\leadsto \color{blue}{e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -6.1 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -6.1e-15)
     t_0
     (if (<= y.re 46000000000000.0)
       (* y.re (* (atan2 x.im x.re) (exp (* y.im (- 0.0 (atan2 x.im x.re))))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 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 <= -6.1e-15) {
		tmp = t_0;
	} else if (y_46_re <= 46000000000000.0) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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 <= -6.1e-15)
		tmp = t_0;
	elseif (y_46_re <= 46000000000000.0)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re))))));
	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, -6.1e-15], t$95$0, If[LessEqual[y$46$re, 46000000000000.0], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.09999999999999972e-15 or 4.6e13 < y.re

   1. Initial program 34.2%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 71.9

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 71.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   7. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      6. *-lowering-*.f64 70.2

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. Simplified 70.2%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -6.09999999999999972e-15 < y.re < 4.6e13

   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 33.6

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 33.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{y.re \cdot \left(e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   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. *-commutative N/A

         \[\leadsto y.re \cdot \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto y.re \cdot \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto y.re \cdot \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \]

      6. neg-sub0 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

      9. atan2-lowering-atan2.f64 53.2

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

   8. Simplified 53.2%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.1 \cdot 10^{-15}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 46000000000000:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -6.1 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (* y.re (atan2 x.im x.re))
          (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))))
   (if (<= y.re -6.1e-15)
     t_0
     (if (<= y.re 4.8e+23)
       (* y.re (* (atan2 x.im x.re) (exp (* y.im (- 0.0 (atan2 x.im x.re))))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (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 <= -6.1e-15) {
		tmp = t_0;
	} else if (y_46_re <= 4.8e+23) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((y_46_im * (0.0 - atan2(x_46_im, x_46_re)))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(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 <= -6.1e-15)
		tmp = t_0;
	elseif (y_46_re <= 4.8e+23)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(y_46_im * Float64(0.0 - atan(x_46_im, x_46_re))))));
	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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $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, -6.1e-15], t$95$0, If[LessEqual[y$46$re, 4.8e+23], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(y$46$im * N[(0.0 - N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.09999999999999972e-15 or 4.8e23 < y.re

   1. Initial program 33.6%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 71.6

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 71.6%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 68.2

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 68.2%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       3. unpow2 N/A

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       5. unpow2 N/A

          \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       6. *-lowering-*.f64 67.4

          \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 67.4%

       \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -6.09999999999999972e-15 < y.re < 4.8e23

   1. Initial program 42.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 34.1

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 34.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   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. *-commutative N/A

         \[\leadsto y.re \cdot \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto y.re \cdot \color{blue}{\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)} \]

      4. atan2-lowering-atan2.f64 N/A

         \[\leadsto y.re \cdot \left(\color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \cdot e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right) \]

      5. exp-lowering-exp.f64 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{e^{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}}\right) \]

      6. neg-sub0 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

      9. atan2-lowering-atan2.f64 52.8

         \[\leadsto y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}\right) \]

   8. Simplified 52.8%

      \[\leadsto \color{blue}{y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.1 \cdot 10^{-15}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+23}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{y.im \cdot \left(0 - \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.1% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.re \leq 3.2 \cdot 10^{-92}:\\ \;\;\;\;t\_0 \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot {x.re}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= x.re 3.2e-92)
     (* t_0 (pow (sqrt (fma x.im x.im (* x.re x.re))) y.re))
     (* (sin t_0) (pow x.re y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= 3.2e-92) {
		tmp = t_0 * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
	} else {
		tmp = sin(t_0) * pow(x_46_re, y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= 3.2e-92)
		tmp = Float64(t_0 * (sqrt(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re))) ^ y_46_re));
	else
		tmp = Float64(sin(t_0) * (x_46_re ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, 3.2e-92], 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], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.1999999999999997e-92

   1. Initial program 40.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 52.5

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 52.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 53.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 53.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 N/A

          \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       3. unpow2 N/A

          \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       4. accelerator-lowering-fma.f64 N/A

          \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       5. unpow2 N/A

          \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       6. *-lowering-*.f64 42.8

          \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 42.8%

       \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if 3.1999999999999997e-92 < x.re

   1. Initial program 33.0%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   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 47.9

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 47.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. *-lowering-*.f64 49.1

         \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   8. Simplified 49.1%

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   9. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. pow-lowering-pow.f64 41.9

          \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 41.9%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.2 \cdot 10^{-92}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t\_0 \cdot {x.re}^{y.re}\\ \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-165}:\\ \;\;\;\;\sin \left(\frac{-1}{\frac{-1}{t\_0}}\right)\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-115}:\\ \;\;\;\;\sin \left(\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;\sin t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (* t_0 (pow x.re y.re))))
   (if (<= y.re -210000.0)
     t_1
     (if (<= y.re -1.2e-165)
       (sin (/ -1.0 (/ -1.0 t_0)))
       (if (<= y.re 9.2e-115)
         (sin (* (* y.im 0.5) (log (fma x.im x.im (* x.re x.re)))))
         (if (<= y.re 2.15e+30) (sin 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = t_0 * pow(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = t_1;
	} else if (y_46_re <= -1.2e-165) {
		tmp = sin((-1.0 / (-1.0 / t_0)));
	} else if (y_46_re <= 9.2e-115) {
		tmp = sin(((y_46_im * 0.5) * log(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))));
	} else if (y_46_re <= 2.15e+30) {
		tmp = sin(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 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (x_46_re ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -210000.0)
		tmp = t_1;
	elseif (y_46_re <= -1.2e-165)
		tmp = sin(Float64(-1.0 / Float64(-1.0 / t_0)));
	elseif (y_46_re <= 9.2e-115)
		tmp = sin(Float64(Float64(y_46_im * 0.5) * log(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))));
	elseif (y_46_re <= 2.15e+30)
		tmp = sin(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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -210000.0], t$95$1, If[LessEqual[y$46$re, -1.2e-165], N[Sin[N[(-1.0 / N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 9.2e-115], N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 2.15e+30], N[Sin[t$95$0], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.1e5 or 2.15e30 < y.re

   1. Initial program 32.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. 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 71.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 71.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 68.3

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 68.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 58.5

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 58.5%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   12. Taylor expanded in y.im around 0

       \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   13. Step-by-step derivation

       1. pow-lowering-pow.f64 55.8

          \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   14. Simplified 55.8%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -2.1e5 < y.re < -1.2000000000000001e-165

   1. Initial program 40.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right)} \]

      2. clear-num N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

   4. Applied egg-rr 40.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right)} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]
   6. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      6. *-lowering-*.f64 27.6

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   7. Simplified 27.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{1} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]
   9. Step-by-step derivation

   10. Simplified 24.9%

       \[\leadsto \color{blue}{1} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   11. Taylor expanded in y.im around 0

       \[\leadsto 1 \cdot \sin \left(\frac{1}{\color{blue}{\frac{1}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\right) \]
   12. Step-by-step derivation

       1. /-lowering-/.f64 N/A

          \[\leadsto 1 \cdot \sin \left(\frac{1}{\color{blue}{\frac{1}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\right) \]

       2. *-lowering-*.f64 N/A

          \[\leadsto 1 \cdot \sin \left(\frac{1}{\frac{1}{\color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\right) \]

       3. atan2-lowering-atan2.f64 37.8

          \[\leadsto 1 \cdot \sin \left(\frac{1}{\frac{1}{y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}}}\right) \]

   13. Simplified 37.8%

       \[\leadsto 1 \cdot \sin \left(\frac{1}{\color{blue}{\frac{1}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}}\right) \]

   if -1.2000000000000001e-165 < y.re < 9.19999999999999938e-115

   1. Initial program 46.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right)} \]

      2. clear-num N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

   4. Applied egg-rr 44.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right)} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]
   6. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      6. *-lowering-*.f64 32.5

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   7. Simplified 32.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \left(y.im \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \left(y.im \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{1}{2} \cdot y.im\right)} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \color{blue}{\log \left({x.im}^{2} + {x.re}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left(\color{blue}{x.im \cdot x.im} + {x.re}^{2}\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 25.7

         \[\leadsto \sin \left(\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)\right)\right) \]

   10. Simplified 25.7%

       \[\leadsto \color{blue}{\sin \left(\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)\right)} \]

   if 9.19999999999999938e-115 < y.re < 2.15e30

   1. Initial program 38.5%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. neg-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. atan2-lowering-atan2.f64 32.6

         \[\leadsto e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Simplified 32.6%

      \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      3. atan2-lowering-atan2.f64 22.8

         \[\leadsto \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 22.8%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-165}:\\ \;\;\;\;\sin \left(\frac{-1}{\frac{-1}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}}\right)\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-115}:\\ \;\;\;\;\sin \left(\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t\_0\\ t_2 := t\_0 \cdot {x.re}^{y.re}\\ \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-158}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-115}:\\ \;\;\;\;\sin \left(\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 (* t_0 (pow x.re y.re))))
   (if (<= y.re -210000.0)
     t_2
     (if (<= y.re -1.75e-158)
       t_1
       (if (<= y.re 9.2e-115)
         (sin (* (* y.im 0.5) (log (fma x.im x.im (* x.re x.re)))))
         (if (<= y.re 2.15e+30) t_1 t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = t_0 * pow(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = t_2;
	} else if (y_46_re <= -1.75e-158) {
		tmp = t_1;
	} else if (y_46_re <= 9.2e-115) {
		tmp = sin(((y_46_im * 0.5) * log(fma(x_46_im, x_46_im, (x_46_re * x_46_re)))));
	} else if (y_46_re <= 2.15e+30) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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(t_0 * (x_46_re ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -210000.0)
		tmp = t_2;
	elseif (y_46_re <= -1.75e-158)
		tmp = t_1;
	elseif (y_46_re <= 9.2e-115)
		tmp = sin(Float64(Float64(y_46_im * 0.5) * log(fma(x_46_im, x_46_im, Float64(x_46_re * x_46_re)))));
	elseif (y_46_re <= 2.15e+30)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -210000.0], t$95$2, If[LessEqual[y$46$re, -1.75e-158], t$95$1, If[LessEqual[y$46$re, 9.2e-115], N[Sin[N[(N[(y$46$im * 0.5), $MachinePrecision] * N[Log[N[(x$46$im * x$46$im + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[y$46$re, 2.15e+30], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.1e5 or 2.15e30 < y.re

   1. Initial program 32.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. 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 71.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 71.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 68.3

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 68.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 58.5

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 58.5%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   12. Taylor expanded in y.im around 0

       \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   13. Step-by-step derivation

       1. pow-lowering-pow.f64 55.8

          \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   14. Simplified 55.8%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -2.1e5 < y.re < -1.75000000000000006e-158 or 9.19999999999999938e-115 < y.re < 2.15e30

   1. Initial program 39.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. neg-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. atan2-lowering-atan2.f64 36.8

         \[\leadsto e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Simplified 36.8%

      \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      3. atan2-lowering-atan2.f64 30.9

         \[\leadsto \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 30.9%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   if -1.75000000000000006e-158 < y.re < 9.19999999999999938e-115

   1. Initial program 46.3%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. flip3-+ N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}\right)} \]

      2. clear-num N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) + \left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) - \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}{{\left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im\right)}^{3} + {\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}}}\right)} \]

   4. Applied egg-rr 44.8%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right)} \]

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]
   6. Step-by-step derivation

      1. pow-lowering-pow.f64 N/A

         \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      3. unpow2 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      5. unpow2 N/A

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(\frac{1}{2} \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

      6. *-lowering-*.f64 32.5

         \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   7. Simplified 32.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \sin \left(\frac{1}{\frac{1}{\mathsf{fma}\left(0.5 \cdot \log \left(\mathsf{fma}\left(x.re, x.re, x.im \cdot x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}\right) \]

   8. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \left(y.im \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)} \]
   9. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(\frac{1}{2} \cdot \left(y.im \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)\right)} \]

      2. associate-*r* N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \sin \left(\color{blue}{\left(\frac{1}{2} \cdot y.im\right)} \cdot \log \left({x.im}^{2} + {x.re}^{2}\right)\right) \]

      5. log-lowering-log.f64 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \color{blue}{\log \left({x.im}^{2} + {x.re}^{2}\right)}\right) \]

      6. unpow2 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left(\color{blue}{x.im \cdot x.im} + {x.re}^{2}\right)\right) \]

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \color{blue}{\left(\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)\right)}\right) \]

      8. unpow2 N/A

         \[\leadsto \sin \left(\left(\frac{1}{2} \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)\right)\right) \]

      9. *-lowering-*.f64 25.7

         \[\leadsto \sin \left(\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)\right)\right) \]

   10. Simplified 25.7%

       \[\leadsto \color{blue}{\sin \left(\left(0.5 \cdot y.im\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-158}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-115}:\\ \;\;\;\;\sin \left(\left(y.im \cdot 0.5\right) \cdot \log \left(\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)\right)\right)\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (*
  (* y.re (atan2 x.im x.re))
  (pow (sqrt (fma x.im x.im (* x.re x.re))) y.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)) * pow(sqrt(fma(x_46_im, x_46_im, (x_46_re * x_46_re))), y_46_re);
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(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))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $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]

Derivation

1. Initial program 38.4%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing

3. Taylor expanded in y.im around 0

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
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 51.1

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

5. Simplified 51.1%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

6. Taylor expanded in y.re around 0

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   2. atan2-lowering-atan2.f64 49.5

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

8. Simplified 49.5%

   \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

9. Taylor expanded in y.im around 0

   \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
10. Step-by-step derivation

    1. pow-lowering-pow.f64 N/A

       \[\leadsto \color{blue}{{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

    2. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto {\color{blue}{\left(\sqrt{{x.im}^{2} + {x.re}^{2}}\right)}}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

    3. unpow2 N/A

       \[\leadsto {\left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

    4. accelerator-lowering-fma.f64 N/A

       \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{fma}\left(x.im, x.im, {x.re}^{2}\right)}}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

    5. unpow2 N/A

       \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

    6. *-lowering-*.f64 40.9

       \[\leadsto {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, \color{blue}{x.re \cdot x.re}\right)}\right)}^{y.re} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

11. Simplified 40.9%

    \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

12. Final simplification 40.9%

    \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{\mathsf{fma}\left(x.im, x.im, x.re \cdot x.re\right)}\right)}^{y.re} \]
13. Add Preprocessing

Alternative 22: 35.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := t\_0 \cdot {x.re}^{y.re}\\ \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;\sin t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (* t_0 (pow x.re y.re))))
   (if (<= y.re -210000.0) t_1 (if (<= y.re 2.15e+30) (sin 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = t_0 * pow(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = t_1;
	} else if (y_46_re <= 2.15e+30) {
		tmp = sin(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    t_1 = t_0 * (x_46re ** y_46re)
    if (y_46re <= (-210000.0d0)) then
        tmp = t_1
    else if (y_46re <= 2.15d+30) then
        tmp = sin(t_0)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = t_0 * Math.pow(x_46_re, y_46_re);
	double tmp;
	if (y_46_re <= -210000.0) {
		tmp = t_1;
	} else if (y_46_re <= 2.15e+30) {
		tmp = Math.sin(t_0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = t_0 * math.pow(x_46_re, y_46_re)
	tmp = 0
	if y_46_re <= -210000.0:
		tmp = t_1
	elif y_46_re <= 2.15e+30:
		tmp = math.sin(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 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(t_0 * (x_46_re ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -210000.0)
		tmp = t_1;
	elseif (y_46_re <= 2.15e+30)
		tmp = sin(t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = t_0 * (x_46_re ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -210000.0)
		tmp = t_1;
	elseif (y_46_re <= 2.15e+30)
		tmp = sin(t_0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Power[x$46$re, y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -210000.0], t$95$1, If[LessEqual[y$46$re, 2.15e+30], N[Sin[t$95$0], $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.1e5 or 2.15e30 < y.re

   1. Initial program 32.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. 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 71.0

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   5. Simplified 71.0%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   6. Taylor expanded in y.re around 0

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. atan2-lowering-atan2.f64 68.3

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 68.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Taylor expanded in x.re around inf

      \[\leadsto e^{\log \left(\sqrt{\color{blue}{{x.re}^{2}}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   10. Step-by-step derivation

       1. unpow2 N/A

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

       2. *-lowering-*.f64 58.5

          \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   11. Simplified 58.5%

       \[\leadsto e^{\log \left(\sqrt{\color{blue}{x.re \cdot x.re}}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   12. Taylor expanded in y.im around 0

       \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   13. Step-by-step derivation

       1. pow-lowering-pow.f64 55.8

          \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   14. Simplified 55.8%

       \[\leadsto \color{blue}{{x.re}^{y.re}} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

   if -2.1e5 < y.re < 2.15e30

   1. Initial program 42.7%

      \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y.re around 0

      \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      2. neg-sub0 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

      5. atan2-lowering-atan2.f64 41.0

         \[\leadsto e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Simplified 41.0%

      \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   6. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
   7. Step-by-step derivation

      1. sin-lowering-sin.f64 N/A

         \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      3. atan2-lowering-atan2.f64 22.4

         \[\leadsto \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

   8. Simplified 22.4%

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -210000:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+30}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.re}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 13.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (sin (* y.re (atan2 x.im x.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return sin((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 = sin((y_46re * atan2(x_46im, x_46re)))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = sin((y_46_re * atan2(x_46_im, x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 38.4%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing

3. Taylor expanded in y.re around 0

   \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. neg-sub0 N/A

      \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. atan2-lowering-atan2.f64 27.2

      \[\leadsto e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

5. Simplified 27.2%

   \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

6. Taylor expanded in y.im around 0

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
7. Step-by-step derivation

   1. sin-lowering-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. atan2-lowering-atan2.f64 15.7

      \[\leadsto \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

8. Simplified 15.7%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
9. Add Preprocessing

Alternative 24: 13.4% 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 38.4%

   \[e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
2. Add Preprocessing

3. Taylor expanded in y.re around 0

   \[\leadsto e^{\color{blue}{-1 \cdot \left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto e^{\color{blue}{\mathsf{neg}\left(y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   2. neg-sub0 N/A

      \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   3. --lowering--.f64 N/A

      \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto e^{0 - \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. atan2-lowering-atan2.f64 27.2

      \[\leadsto e^{0 - y.im \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

5. Simplified 27.2%

   \[\leadsto e^{\color{blue}{0 - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot \sin \left(\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

6. Taylor expanded in y.im around 0

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]
7. Step-by-step derivation

   1. sin-lowering-sin.f64 N/A

      \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \sin \color{blue}{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   3. atan2-lowering-atan2.f64 15.7

      \[\leadsto \sin \left(y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}}\right) \]

8. Simplified 15.7%

   \[\leadsto \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \]

9. Taylor expanded in y.re around 0

   \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
10. 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 15.7

       \[\leadsto y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \]

11. Simplified 15.7%

    \[\leadsto \color{blue}{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024194 
(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)))))