powComplex, imaginary part

Percentage Accurate: 39.8% → 76.4%

Time: 18.4s

Alternatives: 15

Speedup: 2.2×

• 35.7% 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: 39.8% 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: 76.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_2 := t\_1 \cdot y.im\\ t_3 := \sqrt{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\\ \mathbf{if}\;y.re \leq -18:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin t\_0}}{-1}}\\ \mathbf{elif}\;y.re \leq 0.00092:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, t\_0\right)\right) \cdot t\_3\right) \cdot \frac{t\_3}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_2 \cdot \tan^{-1}_* \frac{x.im}{x.re}, y.re, \sin t\_2\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\mathsf{fma}\left(y.im, t\_1, t\_0\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re))
        (t_1 (log (hypot x.im x.re)))
        (t_2 (* t_1 y.im))
        (t_3 (sqrt (pow (hypot x.re x.im) y.re))))
   (if (<= y.re -18.0)
     (/ -1.0 (/ (/ (pow (hypot x.re x.im) (- y.re)) (sin t_0)) -1.0))
     (if (<= y.re 0.00092)
       (*
        (* (sin (fma (log (hypot x.re x.im)) y.im t_0)) t_3)
        (/ t_3 (pow (exp y.im) (atan2 x.im x.re))))
       (if (<= y.re 5.4e+104)
         (*
          (fma (* (cos t_2) (atan2 x.im x.re)) y.re (sin t_2))
          (exp
           (-
            (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
            (* (atan2 x.im x.re) y.im))))
         (/
          1.0
          (/
           1.0
           (* (sin (fma y.im t_1 t_0)) (pow (hypot x.im x.re) y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = log(hypot(x_46_im, x_46_re));
	double t_2 = t_1 * y_46_im;
	double t_3 = sqrt(pow(hypot(x_46_re, x_46_im), y_46_re));
	double tmp;
	if (y_46_re <= -18.0) {
		tmp = -1.0 / ((pow(hypot(x_46_re, x_46_im), -y_46_re) / sin(t_0)) / -1.0);
	} else if (y_46_re <= 0.00092) {
		tmp = (sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * t_3) * (t_3 / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 5.4e+104) {
		tmp = fma((cos(t_2) * atan2(x_46_im, x_46_re)), y_46_re, sin(t_2)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = 1.0 / (1.0 / (sin(fma(y_46_im, t_1, t_0)) * pow(hypot(x_46_im, x_46_re), y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = log(hypot(x_46_im, x_46_re))
	t_2 = Float64(t_1 * y_46_im)
	t_3 = sqrt((hypot(x_46_re, x_46_im) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -18.0)
		tmp = Float64(-1.0 / Float64(Float64((hypot(x_46_re, x_46_im) ^ Float64(-y_46_re)) / sin(t_0)) / -1.0));
	elseif (y_46_re <= 0.00092)
		tmp = Float64(Float64(sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, t_0)) * t_3) * Float64(t_3 / (exp(y_46_im) ^ atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 5.4e+104)
		tmp = Float64(fma(Float64(cos(t_2) * atan(x_46_im, x_46_re)), y_46_re, sin(t_2)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(sin(fma(y_46_im, t_1, t_0)) * (hypot(x_46_im, x_46_re) ^ y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -18.0], N[(-1.0 / N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], (-y$46$re)], $MachinePrecision] / N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.00092], N[(N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision] * t$95$3), $MachinePrecision] * N[(t$95$3 / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.4e+104], N[(N[(N[(N[Cos[t$95$2], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re + N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[Sin[N[(y$46$im * t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -18

   1. Initial program 46.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 84.7

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

   7. Applied rewrites 84.7%

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

   9. Applied rewrites 89.3%

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

   if -18 < y.re < 9.2000000000000003e-4

   1. Initial program 34.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 84.4%

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

   6. Applied rewrites 84.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. associate-/r/ N/A

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

   8. Applied rewrites 84.5%

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

   if 9.2000000000000003e-4 < y.re < 5.39999999999999969e104

   1. Initial program 23.9%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 66.8%

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

   if 5.39999999999999969e104 < y.re

   1. Initial program 45.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 71.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -18:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 0.00092:\\ \;\;\;\;\left(\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot \sqrt{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}\right) \cdot \frac{\sqrt{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, y.re, \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_2 := t\_0 \cdot y.im\\ t_3 := \sin \left(\mathsf{fma}\left(y.im, t\_0, t\_1\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -18:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin t\_1}}{-1}}\\ \mathbf{elif}\;y.re \leq 0.00092:\\ \;\;\;\;\frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{t\_3}}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\cos t\_2 \cdot \tan^{-1}_* \frac{x.im}{x.re}, y.re, \sin t\_2\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* (atan2 x.im x.re) y.re))
        (t_2 (* t_0 y.im))
        (t_3 (* (sin (fma y.im t_0 t_1)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -18.0)
     (/ -1.0 (/ (/ (pow (hypot x.re x.im) (- y.re)) (sin t_1)) -1.0))
     (if (<= y.re 0.00092)
       (/ 1.0 (/ (pow (exp y.im) (atan2 x.im x.re)) t_3))
       (if (<= y.re 5.4e+104)
         (*
          (fma (* (cos t_2) (atan2 x.im x.re)) y.re (sin t_2))
          (exp
           (-
            (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
            (* (atan2 x.im x.re) y.im))))
         (/ 1.0 (/ 1.0 t_3)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_im, x_46_re));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_2 = t_0 * y_46_im;
	double t_3 = sin(fma(y_46_im, t_0, t_1)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -18.0) {
		tmp = -1.0 / ((pow(hypot(x_46_re, x_46_im), -y_46_re) / sin(t_1)) / -1.0);
	} else if (y_46_re <= 0.00092) {
		tmp = 1.0 / (pow(exp(y_46_im), atan2(x_46_im, x_46_re)) / t_3);
	} else if (y_46_re <= 5.4e+104) {
		tmp = fma((cos(t_2) * atan2(x_46_im, x_46_re)), y_46_re, sin(t_2)) * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else {
		tmp = 1.0 / (1.0 / t_3);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_2 = Float64(t_0 * y_46_im)
	t_3 = Float64(sin(fma(y_46_im, t_0, t_1)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -18.0)
		tmp = Float64(-1.0 / Float64(Float64((hypot(x_46_re, x_46_im) ^ Float64(-y_46_re)) / sin(t_1)) / -1.0));
	elseif (y_46_re <= 0.00092)
		tmp = Float64(1.0 / Float64((exp(y_46_im) ^ atan(x_46_im, x_46_re)) / t_3));
	elseif (y_46_re <= 5.4e+104)
		tmp = Float64(fma(Float64(cos(t_2) * atan(x_46_im, x_46_re)), y_46_re, sin(t_2)) * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	else
		tmp = Float64(1.0 / Float64(1.0 / 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[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sin[N[(y$46$im * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -18.0], N[(-1.0 / N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], (-y$46$re)], $MachinePrecision] / N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 0.00092], N[(1.0 / N[(N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.4e+104], N[(N[(N[(N[Cos[t$95$2], $MachinePrecision] * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * y$46$re + N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$3), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -18

   1. Initial program 46.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 84.7

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

   7. Applied rewrites 84.7%

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

   9. Applied rewrites 89.3%

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

   if -18 < y.re < 9.2000000000000003e-4

   1. Initial program 34.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 84.4%

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

   if 9.2000000000000003e-4 < y.re < 5.39999999999999969e104

   1. Initial program 23.9%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

   5. Applied rewrites 66.8%

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

   if 5.39999999999999969e104 < y.re

   1. Initial program 45.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 71.6%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -18:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 0.00092:\\ \;\;\;\;\frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+104}:\\ \;\;\;\;\mathsf{fma}\left(\cos \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}, y.re, \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -18:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{t\_1}}{-1}}\\ \mathbf{elif}\;y.re \leq 4.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), t\_0\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)) (t_1 (sin t_0)))
   (if (<= y.re -18.0)
     (/ -1.0 (/ (/ (pow (hypot x.re x.im) (- y.re)) t_1) -1.0))
     (if (<= y.re 4.35e-7)
       (/
        1.0
        (/
         (pow (exp y.im) (atan2 x.im x.re))
         (*
          (sin (fma y.im (log (hypot x.im x.re)) t_0))
          (pow (hypot x.im x.re) y.re))))
       (*
        t_1
        (exp
         (-
          (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
          (* (atan2 x.im x.re) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = sin(t_0);
	double tmp;
	if (y_46_re <= -18.0) {
		tmp = -1.0 / ((pow(hypot(x_46_re, x_46_im), -y_46_re) / t_1) / -1.0);
	} else if (y_46_re <= 4.35e-7) {
		tmp = 1.0 / (pow(exp(y_46_im), atan2(x_46_im, x_46_re)) / (sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0)) * pow(hypot(x_46_im, x_46_re), y_46_re)));
	} else {
		tmp = t_1 * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -18.0)
		tmp = Float64(-1.0 / Float64(Float64((hypot(x_46_re, x_46_im) ^ Float64(-y_46_re)) / t_1) / -1.0));
	elseif (y_46_re <= 4.35e-7)
		tmp = Float64(1.0 / Float64((exp(y_46_im) ^ atan(x_46_im, x_46_re)) / Float64(sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0)) * (hypot(x_46_im, x_46_re) ^ y_46_re))));
	else
		tmp = Float64(t_1 * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -18.0], N[(-1.0 / N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], (-y$46$re)], $MachinePrecision] / t$95$1), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.35e-7], N[(1.0 / N[(N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision] / N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -18

   1. Initial program 46.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 73.8%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 84.7

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

   7. Applied rewrites 84.7%

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

   9. Applied rewrites 89.3%

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

   if -18 < y.re < 4.35000000000000017e-7

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 84.3%

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

   if 4.35000000000000017e-7 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.6

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

   5. Applied rewrites 61.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -18:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 4.35 \cdot 10^{-7}:\\ \;\;\;\;\frac{1}{\frac{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}{\sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -0.0265:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{t\_0}}{-1}}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -0.0265)
     (/ -1.0 (/ (/ (pow (hypot x.re x.im) (- y.re)) t_0) -1.0))
     (if (<= y.re 3.7e-7)
       (*
        (sin
         (*
          (fma y.re (/ (atan2 x.im x.re) y.im) (log (hypot x.im x.re)))
          y.im))
        (exp (* (- y.im) (atan2 x.im x.re))))
       (*
        t_0
        (exp
         (-
          (* (log (sqrt (+ (* x.im x.im) (* x.re x.re)))) y.re)
          (* (atan2 x.im x.re) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -0.0265) {
		tmp = -1.0 / ((pow(hypot(x_46_re, x_46_im), -y_46_re) / t_0) / -1.0);
	} else if (y_46_re <= 3.7e-7) {
		tmp = sin((fma(y_46_re, (atan2(x_46_im, x_46_re) / y_46_im), log(hypot(x_46_im, x_46_re))) * y_46_im)) * exp((-y_46_im * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0 * exp(((log(sqrt(((x_46_im * x_46_im) + (x_46_re * x_46_re)))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -0.0265)
		tmp = Float64(-1.0 / Float64(Float64((hypot(x_46_re, x_46_im) ^ Float64(-y_46_re)) / t_0) / -1.0));
	elseif (y_46_re <= 3.7e-7)
		tmp = Float64(sin(Float64(fma(y_46_re, Float64(atan(x_46_im, x_46_re) / y_46_im), log(hypot(x_46_im, x_46_re))) * y_46_im)) * exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(t_0 * exp(Float64(Float64(log(sqrt(Float64(Float64(x_46_im * x_46_im) + Float64(x_46_re * x_46_re)))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -0.0265], N[(-1.0 / N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], (-y$46$re)], $MachinePrecision] / t$95$0), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.7e-7], N[(N[Sin[N[(N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] / y$46$im), $MachinePrecision] + N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision] * N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[N[(N[(N[Log[N[Sqrt[N[(N[(x$46$im * x$46$im), $MachinePrecision] + N[(x$46$re * x$46$re), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.0264999999999999993

   1. Initial program 45.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 89.5%

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

   if -0.0264999999999999993 < y.re < 3.70000000000000004e-7

   1. Initial program 34.4%

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

   3. Taylor expanded in y.im around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-atan2.f64 N/A

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

      8. lower-log.f64 N/A

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

      9. unpow2 N/A

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

      10. unpow2 N/A

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

      11. lower-hypot.f64 42.8

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in y.re around 0

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

      1. neg-mul-1 N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-atan2.f64 83.6

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

   8. Applied rewrites 83.6%

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

   if 3.70000000000000004e-7 < y.re

   1. Initial program 38.6%

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

   3. Taylor expanded in y.re around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-atan2.f64 61.6

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

   5. Applied rewrites 61.6%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.0265:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 3.7 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(y.re, \frac{\tan^{-1}_* \frac{x.im}{x.re}}{y.im}, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot y.im\right) \cdot e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot e^{\log \left(\sqrt{x.im \cdot x.im + x.re \cdot x.re}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{if}\;y.re \leq -0.0265:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{t\_0}}{-1}}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-39}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))))
   (if (<= y.re -0.0265)
     (/ -1.0 (/ (/ (pow (hypot x.re x.im) (- y.re)) t_0) -1.0))
     (if (<= y.re 2.1e-39)
       (*
        (exp (* (- y.im) (atan2 x.im x.re)))
        (sin (* (log (hypot x.im x.re)) y.im)))
       (* t_0 (pow (hypot 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 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -0.0265) {
		tmp = -1.0 / ((pow(hypot(x_46_re, x_46_im), -y_46_re) / t_0) / -1.0);
	} else if (y_46_re <= 2.1e-39) {
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log(hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double tmp;
	if (y_46_re <= -0.0265) {
		tmp = -1.0 / ((Math.pow(Math.hypot(x_46_re, x_46_im), -y_46_re) / t_0) / -1.0);
	} else if (y_46_re <= 2.1e-39) {
		tmp = Math.exp((-y_46_im * Math.atan2(x_46_im, x_46_re))) * Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im));
	} else {
		tmp = t_0 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	tmp = 0
	if y_46_re <= -0.0265:
		tmp = -1.0 / ((math.pow(math.hypot(x_46_re, x_46_im), -y_46_re) / t_0) / -1.0)
	elif y_46_re <= 2.1e-39:
		tmp = math.exp((-y_46_im * math.atan2(x_46_im, x_46_re))) * math.sin((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im))
	else:
		tmp = t_0 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	tmp = 0.0
	if (y_46_re <= -0.0265)
		tmp = Float64(-1.0 / Float64(Float64((hypot(x_46_re, x_46_im) ^ Float64(-y_46_re)) / t_0) / -1.0));
	elseif (y_46_re <= 2.1e-39)
		tmp = Float64(exp(Float64(Float64(-y_46_im) * atan(x_46_im, x_46_re))) * sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = Float64(t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	tmp = 0.0;
	if (y_46_re <= -0.0265)
		tmp = -1.0 / (((hypot(x_46_re, x_46_im) ^ -y_46_re) / t_0) / -1.0);
	elseif (y_46_re <= 2.1e-39)
		tmp = exp((-y_46_im * atan2(x_46_im, x_46_re))) * sin((log(hypot(x_46_im, x_46_re)) * y_46_im));
	else
		tmp = t_0 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -0.0265], N[(-1.0 / N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], (-y$46$re)], $MachinePrecision] / t$95$0), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.1e-39], N[(N[Exp[N[((-y$46$im) * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -0.0264999999999999993

   1. Initial program 45.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 74.2%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 84.9

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

   7. Applied rewrites 84.9%

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

   9. Applied rewrites 89.5%

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

   if -0.0264999999999999993 < y.re < 2.09999999999999993e-39

   1. Initial program 35.0%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-log.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 N/A

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

      10. lower-exp.f64 N/A

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

      11. neg-mul-1 N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. neg-mul-1 N/A

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

      15. lower-neg.f64 N/A

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

      16. lower-atan2.f64 63.6

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

   5. Applied rewrites 63.6%

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

   if 2.09999999999999993e-39 < y.re

   1. Initial program 37.1%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 60.0

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

   5. Applied rewrites 60.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.0265:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{-39}:\\ \;\;\;\;e^{\left(-y.im\right) \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \sin t\_0\\ \mathbf{if}\;y.re \leq -3.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{t\_1}}{-1}}\\ \mathbf{elif}\;y.re \leq 1.66 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{\frac{1}{1 \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), t\_0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)) (t_1 (sin t_0)))
   (if (<= y.re -3.8e-11)
     (/ -1.0 (/ (/ (pow (hypot x.re x.im) (- y.re)) t_1) -1.0))
     (if (<= y.re 1.66e-41)
       (/ 1.0 (/ 1.0 (* 1.0 (sin (fma y.im (log (hypot x.im x.re)) t_0)))))
       (/ 1.0 (/ 1.0 (* t_1 (pow (hypot x.im x.re) y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = sin(t_0);
	double tmp;
	if (y_46_re <= -3.8e-11) {
		tmp = -1.0 / ((pow(hypot(x_46_re, x_46_im), -y_46_re) / t_1) / -1.0);
	} else if (y_46_re <= 1.66e-41) {
		tmp = 1.0 / (1.0 / (1.0 * sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0))));
	} else {
		tmp = 1.0 / (1.0 / (t_1 * pow(hypot(x_46_im, x_46_re), y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = sin(t_0)
	tmp = 0.0
	if (y_46_re <= -3.8e-11)
		tmp = Float64(-1.0 / Float64(Float64((hypot(x_46_re, x_46_im) ^ Float64(-y_46_re)) / t_1) / -1.0));
	elseif (y_46_re <= 1.66e-41)
		tmp = Float64(1.0 / Float64(1.0 / Float64(1.0 * sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0)))));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, If[LessEqual[y$46$re, -3.8e-11], N[(-1.0 / N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], (-y$46$re)], $MachinePrecision] / t$95$1), $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.66e-41], N[(1.0 / N[(1.0 / N[(1.0 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(t$95$1 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.7999999999999998e-11

   1. Initial program 45.6%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 84.0

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

   7. Applied rewrites 84.0%

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

   9. Applied rewrites 88.4%

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

   if -3.7999999999999998e-11 < y.re < 1.65999999999999993e-41

   1. Initial program 34.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 57.9%

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

   if 1.65999999999999993e-41 < y.re

   1. Initial program 37.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 59.7

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

   7. Applied rewrites 59.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{\frac{\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{\left(-y.re\right)}}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}}{-1}}\\ \mathbf{elif}\;y.re \leq 1.66 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{\frac{1}{1 \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := \sin t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -3.8 \cdot 10^{-11}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.66 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{\frac{1}{1 \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), t\_0\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{t\_1}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re))
        (t_1 (* (sin t_0) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -3.8e-11)
     t_1
     (if (<= y.re 1.66e-41)
       (/ 1.0 (/ 1.0 (* 1.0 (sin (fma y.im (log (hypot x.im x.re)) t_0)))))
       (/ 1.0 (/ 1.0 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_re;
	double t_1 = sin(t_0) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -3.8e-11) {
		tmp = t_1;
	} else if (y_46_re <= 1.66e-41) {
		tmp = 1.0 / (1.0 / (1.0 * sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0))));
	} else {
		tmp = 1.0 / (1.0 / 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_re)
	t_1 = Float64(sin(t_0) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -3.8e-11)
		tmp = t_1;
	elseif (y_46_re <= 1.66e-41)
		tmp = Float64(1.0 / Float64(1.0 / Float64(1.0 * sin(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0)))));
	else
		tmp = Float64(1.0 / Float64(1.0 / 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$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.8e-11], t$95$1, If[LessEqual[y$46$re, 1.66e-41], N[(1.0 / N[(1.0 / N[(1.0 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.7999999999999998e-11

   1. Initial program 45.6%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -3.7999999999999998e-11 < y.re < 1.65999999999999993e-41

   1. Initial program 34.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 84.6%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in y.re around 0

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

   10. Applied rewrites 57.9%

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

   if 1.65999999999999993e-41 < y.re

   1. Initial program 37.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in y.im around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-atan2.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. unpow2 N/A

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

      8. unpow2 N/A

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

      9. lower-hypot.f64 59.7

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

   7. Applied rewrites 59.7%

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

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{-11}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 1.66 \cdot 10^{-41}:\\ \;\;\;\;\frac{1}{\frac{1}{1 \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{if}\;x.re \leq 3.8 \cdot 10^{-186}:\\ \;\;\;\;\sin t\_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{x.re}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, t\_0\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)))
   (if (<= x.re 3.8e-186)
     (* (sin t_0) (pow (hypot x.im x.re) y.re))
     (/ 1.0 (/ 1.0 (* (pow x.re y.re) (sin (fma y.im (log x.re) t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_re;
	double tmp;
	if (x_46_re <= 3.8e-186) {
		tmp = sin(t_0) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = 1.0 / (1.0 / (pow(x_46_re, y_46_re) * sin(fma(y_46_im, log(x_46_re), t_0))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_re)
	tmp = 0.0
	if (x_46_re <= 3.8e-186)
		tmp = Float64(sin(t_0) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64((x_46_re ^ y_46_re) * sin(fma(y_46_im, log(x_46_re), t_0)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, If[LessEqual[x$46$re, 3.8e-186], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.re < 3.79999999999999974e-186

   1. Initial program 38.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if 3.79999999999999974e-186 < x.re

   1. Initial program 38.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 74.0%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in x.im around 0

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

      1. lower-*.f64 N/A

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

      2. lower-sin.f64 N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-log.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-atan2.f64 N/A

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

      7. lower-pow.f64 60.3

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

   10. Applied rewrites 60.3%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.8 \cdot 10^{-186}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{{x.re}^{y.re} \cdot \sin \left(\mathsf{fma}\left(y.im, \log x.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.85 \cdot 10^{-168}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (* (sin (* (atan2 x.im x.re) y.re)) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -2.85e-168)
     t_0
     (if (<= y.re 2.6e-150)
       (/ 1.0 (/ 1.0 (sin (* (log (hypot x.im x.re)) y.im))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.85e-168) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-150) {
		tmp = 1.0 / (1.0 / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re)) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.85e-168) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-150) {
		tmp = 1.0 / (1.0 / Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re)) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -2.85e-168:
		tmp = t_0
	elif y_46_re <= 2.6e-150:
		tmp = 1.0 / (1.0 / math.sin((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(sin(Float64(atan(x_46_im, x_46_re) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.85e-168)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-150)
		tmp = Float64(1.0 / Float64(1.0 / sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re)) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.85e-168)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-150)
		tmp = 1.0 / (1.0 / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.85e-168], t$95$0, If[LessEqual[y$46$re, 2.6e-150], N[(1.0 / N[(1.0 / N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.85000000000000004e-168 or 2.5999999999999998e-150 < y.re

   1. Initial program 40.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 58.8

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

   5. Applied rewrites 58.8%

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

   if -2.85000000000000004e-168 < y.re < 2.5999999999999998e-150

   1. Initial program 31.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 54.9

         \[\leadsto \frac{1}{\frac{1}{\sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}} \]

   10. Applied rewrites 54.9%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.85 \cdot 10^{-168}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.85 \cdot 10^{-168}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (* (atan2 x.im x.re) y.re) (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -2.85e-168)
     t_0
     (if (<= y.re 2.6e-150)
       (/ 1.0 (/ 1.0 (sin (* (log (hypot x.im x.re)) y.im))))
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (atan2(x_46_im, x_46_re) * y_46_re) * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.85e-168) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-150) {
		tmp = 1.0 / (1.0 / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (Math.atan2(x_46_im, x_46_re) * y_46_re) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.85e-168) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-150) {
		tmp = 1.0 / (1.0 / Math.sin((Math.log(Math.hypot(x_46_im, x_46_re)) * y_46_im)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (math.atan2(x_46_im, x_46_re) * y_46_re) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -2.85e-168:
		tmp = t_0
	elif y_46_re <= 2.6e-150:
		tmp = 1.0 / (1.0 / math.sin((math.log(math.hypot(x_46_im, x_46_re)) * y_46_im)))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(atan(x_46_im, x_46_re) * y_46_re) * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -2.85e-168)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-150)
		tmp = Float64(1.0 / Float64(1.0 / sin(Float64(log(hypot(x_46_im, x_46_re)) * y_46_im))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (atan2(x_46_im, x_46_re) * y_46_re) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2.85e-168)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-150)
		tmp = 1.0 / (1.0 / sin((log(hypot(x_46_im, x_46_re)) * y_46_im)));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.85e-168], t$95$0, If[LessEqual[y$46$re, 2.6e-150], N[(1.0 / N[(1.0 / N[Sin[N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.85000000000000004e-168 or 2.5999999999999998e-150 < y.re

   1. Initial program 40.3%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 58.8

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

   5. Applied rewrites 58.8%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 56.2%

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

   if -2.85000000000000004e-168 < y.re < 2.5999999999999998e-150

   1. Initial program 31.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-exp.f64 N/A

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

      4. lift--.f64 N/A

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

      5. exp-diff N/A

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

      6. associate-*r/ N/A

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

      7. clear-num N/A

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in y.im around 0

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

   7. Applied rewrites 65.8%

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

   8. Taylor expanded in y.re around 0

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

      1. lower-sin.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-log.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 54.9

         \[\leadsto \frac{1}{\frac{1}{\sin \left(y.im \cdot \log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}} \]

   10. Applied rewrites 54.9%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.85 \cdot 10^{-168}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{1}{\frac{1}{\sin \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) \cdot y.im\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ t_1 := {x.re}^{y.re} \cdot t\_0\\ \mathbf{if}\;x.re \leq -4 \cdot 10^{+47}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.re \leq 8 \cdot 10^{-223}:\\ \;\;\;\;{x.im}^{y.re} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* (atan2 x.im x.re) y.re))) (t_1 (* (pow x.re y.re) t_0)))
   (if (<= x.re -4e+47)
     t_1
     (if (<= x.re 8e-223) (* (pow x.im y.re) t_0) t_1))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = pow(x_46_re, y_46_re) * t_0;
	double tmp;
	if (x_46_re <= -4e+47) {
		tmp = t_1;
	} else if (x_46_re <= 8e-223) {
		tmp = pow(x_46_im, y_46_re) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((atan2(x_46im, x_46re) * y_46re))
    t_1 = (x_46re ** y_46re) * t_0
    if (x_46re <= (-4d+47)) then
        tmp = t_1
    else if (x_46re <= 8d-223) then
        tmp = (x_46im ** y_46re) * t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((Math.atan2(x_46_im, x_46_re) * y_46_re));
	double t_1 = Math.pow(x_46_re, y_46_re) * t_0;
	double tmp;
	if (x_46_re <= -4e+47) {
		tmp = t_1;
	} else if (x_46_re <= 8e-223) {
		tmp = Math.pow(x_46_im, y_46_re) * t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((math.atan2(x_46_im, x_46_re) * y_46_re))
	t_1 = math.pow(x_46_re, y_46_re) * t_0
	tmp = 0
	if x_46_re <= -4e+47:
		tmp = t_1
	elif x_46_re <= 8e-223:
		tmp = math.pow(x_46_im, y_46_re) * t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(atan(x_46_im, x_46_re) * y_46_re))
	t_1 = Float64((x_46_re ^ y_46_re) * t_0)
	tmp = 0.0
	if (x_46_re <= -4e+47)
		tmp = t_1;
	elseif (x_46_re <= 8e-223)
		tmp = Float64((x_46_im ^ y_46_re) * t_0);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((atan2(x_46_im, x_46_re) * y_46_re));
	t_1 = (x_46_re ^ y_46_re) * t_0;
	tmp = 0.0;
	if (x_46_re <= -4e+47)
		tmp = t_1;
	elseif (x_46_re <= 8e-223)
		tmp = (x_46_im ^ y_46_re) * t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x$46$re, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[LessEqual[x$46$re, -4e+47], t$95$1, If[LessEqual[x$46$re, 8e-223], N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * t$95$0), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x.re < -4.0000000000000002e47 or 7.9999999999999998e-223 < x.re

   1. Initial program 33.2%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in x.im around 0

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

   8. Applied rewrites 41.0%

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

   if -4.0000000000000002e47 < x.re < 7.9999999999999998e-223

   1. Initial program 47.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 46.5%

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

Alternative 12: 36.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ t_1 := {x.im}^{y.re} \cdot \sin t\_0\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+26}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.re)) (t_1 (* (pow x.im y.re) (sin t_0))))
   (if (<= y.re -1.4e-10) t_1 (if (<= y.re 2.8e+26) 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_re;
	double t_1 = pow(x_46_im, y_46_re) * sin(t_0);
	double tmp;
	if (y_46_re <= -1.4e-10) {
		tmp = t_1;
	} else if (y_46_re <= 2.8e+26) {
		tmp = 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 = atan2(x_46im, x_46re) * y_46re
    t_1 = (x_46im ** y_46re) * sin(t_0)
    if (y_46re <= (-1.4d-10)) then
        tmp = t_1
    else if (y_46re <= 2.8d+26) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_re;
	double t_1 = Math.pow(x_46_im, y_46_re) * Math.sin(t_0);
	double tmp;
	if (y_46_re <= -1.4e-10) {
		tmp = t_1;
	} else if (y_46_re <= 2.8e+26) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_re
	t_1 = math.pow(x_46_im, y_46_re) * math.sin(t_0)
	tmp = 0
	if y_46_re <= -1.4e-10:
		tmp = t_1
	elif y_46_re <= 2.8e+26:
		tmp = 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(atan(x_46_im, x_46_re) * y_46_re)
	t_1 = Float64((x_46_im ^ y_46_re) * sin(t_0))
	tmp = 0.0
	if (y_46_re <= -1.4e-10)
		tmp = t_1;
	elseif (y_46_re <= 2.8e+26)
		tmp = 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 = atan2(x_46_im, x_46_re) * y_46_re;
	t_1 = (x_46_im ^ y_46_re) * sin(t_0);
	tmp = 0.0;
	if (y_46_re <= -1.4e-10)
		tmp = t_1;
	elseif (y_46_re <= 2.8e+26)
		tmp = 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[x$46$im, y$46$re], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.4e-10], t$95$1, If[LessEqual[y$46$re, 2.8e+26], t$95$0, t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.40000000000000008e-10 or 2.8e26 < y.re

   1. Initial program 43.7%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in x.re around 0

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

   8. Applied rewrites 58.3%

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

   if -1.40000000000000008e-10 < y.re < 2.8e26

   1. Initial program 33.4%

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

   3. Taylor expanded in y.im around 0

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

      1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower-atan2.f64 27.2

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

   5. Applied rewrites 27.2%

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

   6. Taylor expanded in y.re around 0

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

   8. Applied rewrites 24.1%

      \[\leadsto y.re \cdot \color{blue}{\tan^{-1}_* \frac{x.im}{x.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{-10}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+26}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\\ \mathbf{else}:\\ \;\;\;\;{x.im}^{y.re} \cdot \sin \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 45.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (* (atan2 x.im x.re) y.re) (pow (hypot 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) {
	return (atan2(x_46_im, x_46_re) * y_46_re) * pow(hypot(x_46_im, x_46_re), y_46_re);
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (Math.atan2(x_46_im, x_46_re) * y_46_re) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (math.atan2(x_46_im, x_46_re) * y_46_re) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(atan(x_46_im, x_46_re) * y_46_re) * (hypot(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)
	tmp = (atan2(x_46_im, x_46_re) * y_46_re) * (hypot(x_46_im, x_46_re) ^ y_46_re);
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-atan2.f64 48.6

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

5. Applied rewrites 48.6%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 46.6%

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

9. Final simplification 46.6%

   \[\leadsto \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \]
10. Add Preprocessing

Alternative 14: 13.9% accurate, 3.2× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* 1.0 (sin (* (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) {
	return 1.0 * sin((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
    code = 1.0d0 * sin((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) {
	return 1.0 * Math.sin((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):
	return 1.0 * math.sin((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)
	return Float64(1.0 * sin(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)
	tmp = 1.0 * sin((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_] := N[(1.0 * N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-atan2.f64 48.6

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

5. Applied rewrites 48.6%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 16.2%

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

Alternative 15: 13.9% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* (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) {
	return 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
    code = 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) {
	return 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):
	return 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)
	return 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)
	tmp = 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_] := N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re), $MachinePrecision]

Derivation

1. Initial program 38.2%

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

3. Taylor expanded in y.im around 0

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

   1. *-commutative 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. lower-*.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. lower-pow.f64 N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. lower-hypot.f64 N/A

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

   7. lower-sin.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lower-atan2.f64 48.6

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

5. Applied rewrites 48.6%

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

6. Taylor expanded in y.re around 0

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

8. Applied rewrites 16.2%

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

9. Final simplification 16.2%

   \[\leadsto \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024283 
(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)))))