powComplex, imaginary part

Percentage Accurate: 39.5% → 79.4%

Time: 29.8s

Alternatives: 20

Speedup: 2.7×

• 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: 39.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 79.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ e^{\log \left({\left(e^{\sqrt[3]{{t_0}^{2}}}\right)}^{\left(\sqrt[3]{t_0}\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left({\left(\sqrt[3]{\mathsf{hypot}\left(x.re, x.im\right)}\right)}^{3}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (*
    (exp
     (-
      (* (log (pow (exp (cbrt (pow t_0 2.0))) (cbrt t_0))) y.re)
      (* (atan2 x.im x.re) y.im)))
    (sin
     (+
      (* y.im (log (pow (cbrt (hypot x.re x.im)) 3.0)))
      (* y.re (atan2 x.im x.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	return exp(((log(pow(exp(cbrt(pow(t_0, 2.0))), cbrt(t_0))) * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((y_46_im * log(pow(cbrt(hypot(x_46_re, x_46_im)), 3.0))) + (y_46_re * atan2(x_46_im, x_46_re))));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.log(Math.hypot(x_46_re, x_46_im));
	return Math.exp(((Math.log(Math.pow(Math.exp(Math.cbrt(Math.pow(t_0, 2.0))), Math.cbrt(t_0))) * y_46_re) - (Math.atan2(x_46_im, x_46_re) * y_46_im))) * Math.sin(((y_46_im * Math.log(Math.pow(Math.cbrt(Math.hypot(x_46_re, x_46_im)), 3.0))) + (y_46_re * Math.atan2(x_46_im, x_46_re))));
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	return Float64(exp(Float64(Float64(log((exp(cbrt((t_0 ^ 2.0))) ^ cbrt(t_0))) * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(Float64(y_46_im * log((cbrt(hypot(x_46_re, x_46_im)) ^ 3.0))) + Float64(y_46_re * atan(x_46_im, x_46_re)))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, N[(N[Exp[N[(N[(N[Log[N[Power[N[Exp[N[Power[N[Power[t$95$0, 2.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], N[Power[t$95$0, 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(y$46$im * N[Log[N[Power[N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 39.9%

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

   1. hypot-udef 58.9%

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

   2. add-cube-cbrt 58.9%

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

   3. pow3 58.9%

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

3. Applied egg-rr 58.9%

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

   1. add-exp-log 58.9%

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

   2. hypot-udef 78.8%

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

   3. add-cube-cbrt 78.8%

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

   4. exp-prod 78.8%

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

   5. cbrt-unprod 78.8%

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

   6. pow2 78.8%

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

5. Applied egg-rr 78.8%

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

6. Final simplification 78.8%

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

Alternative 2: 79.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 39.9%

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

3. Simplified 78.8%

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

4. Final simplification 78.8%

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

Alternative 3: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+204}:\\ \;\;\;\;\left|t_1\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+165}:\\ \;\;\;\;e^{t_0 \cdot y.re - t_2} \cdot \sin \left(t_1 + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{+15} \lor \neg \left(y.im \leq 3 \cdot 10^{+105}\right):\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_0, y.im, t_1\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{t_2 + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* (atan2 x.im x.re) y.im)))
   (if (<= y.im -3e+204)
     (* (fabs t_1) (exp (* y.im (- (atan2 x.im x.re)))))
     (if (<= y.im -4.5e+165)
       (* (exp (- (* t_0 y.re) t_2)) (sin (+ t_1 (* y.im (log x.im)))))
       (if (or (<= y.im -7e+15) (not (<= y.im 3e+105)))
         (* (atan2 x.im x.re) (/ y.re (pow (exp y.im) (atan2 x.im x.re))))
         (*
          (sin (fma t_0 y.im t_1))
          (/ (pow (hypot x.re x.im) y.re) (+ t_2 1.0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_im <= -3e+204) {
		tmp = fabs(t_1) * exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= -4.5e+165) {
		tmp = exp(((t_0 * y_46_re) - t_2)) * sin((t_1 + (y_46_im * log(x_46_im))));
	} else if ((y_46_im <= -7e+15) || !(y_46_im <= 3e+105)) {
		tmp = atan2(x_46_im, x_46_re) * (y_46_re / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	} else {
		tmp = sin(fma(t_0, y_46_im, t_1)) * (pow(hypot(x_46_re, x_46_im), y_46_re) / (t_2 + 1.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (y_46_im <= -3e+204)
		tmp = Float64(abs(t_1) * exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))));
	elseif (y_46_im <= -4.5e+165)
		tmp = Float64(exp(Float64(Float64(t_0 * y_46_re) - t_2)) * sin(Float64(t_1 + Float64(y_46_im * log(x_46_im)))));
	elseif ((y_46_im <= -7e+15) || !(y_46_im <= 3e+105))
		tmp = Float64(atan(x_46_im, x_46_re) * Float64(y_46_re / (exp(y_46_im) ^ atan(x_46_im, x_46_re))));
	else
		tmp = Float64(sin(fma(t_0, y_46_im, t_1)) * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(t_2 + 1.0)));
	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$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3e+204], N[(N[Abs[t$95$1], $MachinePrecision] * N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.5e+165], N[(N[Exp[N[(N[(t$95$0 * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$im, -7e+15], N[Not[LessEqual[y$46$im, 3e+105]], $MachinePrecision]], N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$0 * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision] * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.99999999999999983e204

   1. Initial program 47.1%

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

   2. Taylor expanded in y.im around 0 71.0%

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

   3. Taylor expanded in y.re around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. distribute-lft-neg-in 82.4%

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

      3. *-commutative 82.4%

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

   5. Simplified 82.4%

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

      1. *-commutative 82.4%

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

      2. add-sqr-sqrt 52.9%

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

      3. sqrt-unprod 64.7%

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

      4. pow2 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. *-commutative 64.7%

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

      2. unpow2 64.7%

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

      3. rem-sqrt-square 94.1%

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

   9. Simplified 94.1%

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

   if -2.99999999999999983e204 < y.im < -4.4999999999999996e165

   1. Initial program 22.2%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x.re around 0 57.9%

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

   if -4.4999999999999996e165 < y.im < -7e15 or 3.0000000000000001e105 < y.im

   1. Initial program 32.5%

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

   2. Taylor expanded in y.im around 0 58.2%

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

   3. Taylor expanded in y.re around 0 64.2%

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

      1. associate-*r* 64.2%

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

      2. exp-neg 64.2%

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

      3. associate-*l/ 64.2%

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

      4. *-lft-identity 64.2%

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

      5. exp-prod 66.5%

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

   5. Simplified 66.5%

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

   if -7e15 < y.im < 3.0000000000000001e105

   1. Initial program 44.4%

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

      1. exp-diff 42.3%

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

      2. +-rgt-identity 42.3%

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

      3. +-rgt-identity 42.3%

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

      4. exp-to-pow 42.3%

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

      5. hypot-def 42.3%

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

      6. exp-prod 42.3%

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

      7. fma-def 42.3%

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

      8. hypot-def 80.4%

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

      9. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in y.im around 0 82.7%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{+204}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+165}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{+15} \lor \neg \left(y.im \leq 3 \cdot 10^{+105}\right):\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \end{array} \]

Alternative 4: 72.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (log (hypot x.re x.im)))
        (t_2 (* y.re (atan2 x.im x.re))))
   (if (<= x.re -1e-310)
     (*
      (sin (fma t_1 y.im t_2))
      (exp (- (* (log (/ -1.0 x.re)) (- y.re)) t_0)))
     (* (exp (- (* t_1 y.re) t_0)) (sin (+ t_2 (* y.im (log x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_re <= -1e-310) {
		tmp = sin(fma(t_1, y_46_im, t_2)) * exp(((log((-1.0 / x_46_re)) * -y_46_re) - t_0));
	} else {
		tmp = exp(((t_1 * y_46_re) - t_0)) * sin((t_2 + (y_46_im * log(x_46_re))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = log(hypot(x_46_re, x_46_im))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_re <= -1e-310)
		tmp = Float64(sin(fma(t_1, y_46_im, t_2)) * exp(Float64(Float64(log(Float64(-1.0 / x_46_re)) * Float64(-y_46_re)) - t_0)));
	else
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_0)) * sin(Float64(t_2 + Float64(y_46_im * log(x_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -1e-310], N[(N[Sin[N[(t$95$1 * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision] * (-y$46$re)), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$2 + N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

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

   1. Initial program 37.4%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in x.re around -inf 75.0%

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

      1. mul-1-neg 75.0%

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

      2. *-commutative 75.0%

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

      3. distribute-rgt-neg-in 75.0%

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

   6. Simplified 75.0%

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

   if -9.999999999999969e-311 < x.re

   1. Initial program 42.4%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in x.im around 0 77.4%

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

4. Final simplification 76.2%

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

Alternative 5: 73.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := \left|t_1\right|\\ t_4 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+201}:\\ \;\;\;\;t_3 \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+165}:\\ \;\;\;\;e^{t_4 \cdot y.re - t_2} \cdot \sin \left(t_1 + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -1.12 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 175000000000:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_4, y.im, t_1\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+139}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2} \cdot \sin t_3\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) (/ y.re (pow (exp y.im) (atan2 x.im x.re)))))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (fabs t_1))
        (t_4 (log (hypot x.re x.im))))
   (if (<= y.im -2.5e+201)
     (* t_3 (exp (* y.im (- (atan2 x.im x.re)))))
     (if (<= y.im -4.5e+165)
       (* (exp (- (* t_4 y.re) t_2)) (sin (+ t_1 (* y.im (log x.im)))))
       (if (<= y.im -1.12e+15)
         t_0
         (if (<= y.im 175000000000.0)
           (* (sin (fma t_4 y.im t_1)) (pow (hypot x.re x.im) y.re))
           (if (<= y.im 2e+139)
             (*
              (exp
               (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))
              (sin t_3))
             t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * (y_46_re / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = fabs(t_1);
	double t_4 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_im <= -2.5e+201) {
		tmp = t_3 * exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= -4.5e+165) {
		tmp = exp(((t_4 * y_46_re) - t_2)) * sin((t_1 + (y_46_im * log(x_46_im))));
	} else if (y_46_im <= -1.12e+15) {
		tmp = t_0;
	} else if (y_46_im <= 175000000000.0) {
		tmp = sin(fma(t_4, y_46_im, t_1)) * pow(hypot(x_46_re, x_46_im), y_46_re);
	} else if (y_46_im <= 2e+139) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * sin(t_3);
	} else {
		tmp = 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) * Float64(y_46_re / (exp(y_46_im) ^ atan(x_46_im, x_46_re))))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_3 = abs(t_1)
	t_4 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_im <= -2.5e+201)
		tmp = Float64(t_3 * exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))));
	elseif (y_46_im <= -4.5e+165)
		tmp = Float64(exp(Float64(Float64(t_4 * y_46_re) - t_2)) * sin(Float64(t_1 + Float64(y_46_im * log(x_46_im)))));
	elseif (y_46_im <= -1.12e+15)
		tmp = t_0;
	elseif (y_46_im <= 175000000000.0)
		tmp = Float64(sin(fma(t_4, y_46_im, t_1)) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	elseif (y_46_im <= 2e+139)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_2)) * sin(t_3));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$1], $MachinePrecision]}, Block[{t$95$4 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -2.5e+201], N[(t$95$3 * N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.5e+165], N[(N[Exp[N[(N[(t$95$4 * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.12e+15], t$95$0, If[LessEqual[y$46$im, 175000000000.0], N[(N[Sin[N[(t$95$4 * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2e+139], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$3], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.4999999999999998e201

   1. Initial program 47.1%

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

   2. Taylor expanded in y.im around 0 71.0%

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

   3. Taylor expanded in y.re around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. distribute-lft-neg-in 82.4%

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

      3. *-commutative 82.4%

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

   5. Simplified 82.4%

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

      1. *-commutative 82.4%

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

      2. add-sqr-sqrt 52.9%

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

      3. sqrt-unprod 64.7%

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

      4. pow2 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. *-commutative 64.7%

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

      2. unpow2 64.7%

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

      3. rem-sqrt-square 94.1%

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

   9. Simplified 94.1%

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

   if -2.4999999999999998e201 < y.im < -4.4999999999999996e165

   1. Initial program 22.2%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x.re around 0 57.9%

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

   if -4.4999999999999996e165 < y.im < -1.12e15 or 2.00000000000000007e139 < y.im

   1. Initial program 30.4%

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

   2. Taylor expanded in y.im around 0 57.4%

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

   3. Taylor expanded in y.re around 0 63.6%

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

      1. associate-*r* 63.6%

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

      2. exp-neg 63.6%

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

      3. associate-*l/ 63.6%

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

      4. *-lft-identity 63.6%

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

      5. exp-prod 66.0%

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

   5. Simplified 66.0%

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

   if -1.12e15 < y.im < 1.75e11

   1. Initial program 43.2%

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

      1. exp-diff 43.2%

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

      2. +-rgt-identity 43.2%

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

      3. +-rgt-identity 43.2%

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

      4. exp-to-pow 43.2%

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

      5. hypot-def 43.2%

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

      6. exp-prod 43.2%

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

      7. fma-def 43.2%

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

      8. hypot-def 85.5%

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

      9. *-commutative 85.5%

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

   3. Simplified 85.5%

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

   4. Taylor expanded in y.im around 0 84.7%

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

   if 1.75e11 < y.im < 2.00000000000000007e139

   1. Initial program 57.1%

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

   2. Taylor expanded in y.im around 0 61.9%

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

      1. *-commutative 38.6%

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

      2. add-sqr-sqrt 19.3%

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

      3. sqrt-unprod 43.3%

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

      4. pow2 43.3%

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

   4. Applied egg-rr 33.3%

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

      1. *-commutative 43.3%

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

      2. unpow2 43.3%

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

      3. rem-sqrt-square 48.2%

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

   6. Simplified 81.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+201}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+165}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -1.12 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 175000000000:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+139}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right|\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 6: 73.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+201}:\\ \;\;\;\;\left|t_2\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{+165}:\\ \;\;\;\;e^{t_1 \cdot y.re - t_3} \cdot \sin \left(t_2 + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 0.0076:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_1, y.im, t_2\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+164}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_3} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) (/ y.re (pow (exp y.im) (atan2 x.im x.re)))))
        (t_1 (log (hypot x.re x.im)))
        (t_2 (* y.re (atan2 x.im x.re)))
        (t_3 (* (atan2 x.im x.re) y.im)))
   (if (<= y.im -2.5e+201)
     (* (fabs t_2) (exp (* y.im (- (atan2 x.im x.re)))))
     (if (<= y.im -2.9e+165)
       (* (exp (- (* t_1 y.re) t_3)) (sin (+ t_2 (* y.im (log x.im)))))
       (if (<= y.im -4e+15)
         t_0
         (if (<= y.im 0.0076)
           (* (sin (fma t_1 y.im t_2)) (pow (hypot x.re x.im) y.re))
           (if (<= y.im 2e+164)
             (*
              (exp
               (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_3))
              (sin (* y.im (log (hypot x.im x.re)))))
             t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * (y_46_re / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = y_46_re * atan2(x_46_im, x_46_re);
	double t_3 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_im <= -2.5e+201) {
		tmp = fabs(t_2) * exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= -2.9e+165) {
		tmp = exp(((t_1 * y_46_re) - t_3)) * sin((t_2 + (y_46_im * log(x_46_im))));
	} else if (y_46_im <= -4e+15) {
		tmp = t_0;
	} else if (y_46_im <= 0.0076) {
		tmp = sin(fma(t_1, y_46_im, t_2)) * pow(hypot(x_46_re, x_46_im), y_46_re);
	} else if (y_46_im <= 2e+164) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * Float64(y_46_re / (exp(y_46_im) ^ atan(x_46_im, x_46_re))))
	t_1 = log(hypot(x_46_re, x_46_im))
	t_2 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_3 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5e+201)
		tmp = Float64(abs(t_2) * exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))));
	elseif (y_46_im <= -2.9e+165)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_3)) * sin(Float64(t_2 + Float64(y_46_im * log(x_46_im)))));
	elseif (y_46_im <= -4e+15)
		tmp = t_0;
	elseif (y_46_im <= 0.0076)
		tmp = Float64(sin(fma(t_1, y_46_im, t_2)) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	elseif (y_46_im <= 2e+164)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_3)) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.5e+201], N[(N[Abs[t$95$2], $MachinePrecision] * N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.9e+165], N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$2 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4e+15], t$95$0, If[LessEqual[y$46$im, 0.0076], N[(N[Sin[N[(t$95$1 * y$46$im + t$95$2), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2e+164], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.4999999999999998e201

   1. Initial program 47.1%

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

   2. Taylor expanded in y.im around 0 71.0%

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

   3. Taylor expanded in y.re around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. distribute-lft-neg-in 82.4%

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

      3. *-commutative 82.4%

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

   5. Simplified 82.4%

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

      1. *-commutative 82.4%

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

      2. add-sqr-sqrt 52.9%

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

      3. sqrt-unprod 64.7%

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

      4. pow2 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. *-commutative 64.7%

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

      2. unpow2 64.7%

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

      3. rem-sqrt-square 94.1%

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

   9. Simplified 94.1%

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

   if -2.4999999999999998e201 < y.im < -2.90000000000000006e165

   1. Initial program 22.2%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x.re around 0 57.9%

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

   if -2.90000000000000006e165 < y.im < -4e15 or 2e164 < y.im

   1. Initial program 30.0%

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

   2. Taylor expanded in y.im around 0 56.2%

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

   3. Taylor expanded in y.re around 0 64.6%

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

      1. associate-*r* 64.6%

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

      2. exp-neg 64.6%

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

      3. associate-*l/ 64.6%

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

      4. *-lft-identity 64.6%

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

      5. exp-prod 67.3%

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

   5. Simplified 67.3%

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

   if -4e15 < y.im < 0.00759999999999999998

   1. Initial program 42.8%

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

      1. exp-diff 42.8%

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

      2. +-rgt-identity 42.8%

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

      3. +-rgt-identity 42.8%

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

      4. exp-to-pow 42.8%

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

      5. hypot-def 42.8%

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

      6. exp-prod 42.8%

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

      7. fma-def 42.8%

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

      8. hypot-def 85.6%

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

      9. *-commutative 85.6%

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

   3. Simplified 85.6%

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

   4. Taylor expanded in y.im around 0 86.1%

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

   if 0.00759999999999999998 < y.im < 2e164

   1. Initial program 51.0%

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

   2. Taylor expanded in y.re around 0 51.0%

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

      1. unpow2 51.0%

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

      2. unpow2 51.0%

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

      3. hypot-def 71.7%

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

   4. Simplified 71.7%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+201}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{+165}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 0.0076:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+164}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 7: 71.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+201}:\\ \;\;\;\;\left|t_1\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+165}:\\ \;\;\;\;e^{t_0 \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(t_1 + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{+17} \lor \neg \left(y.im \leq 3.1 \cdot 10^{+105}\right):\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_0, y.im, t_1\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))) (t_1 (* y.re (atan2 x.im x.re))))
   (if (<= y.im -2.5e+201)
     (* (fabs t_1) (exp (* y.im (- (atan2 x.im x.re)))))
     (if (<= y.im -4.5e+165)
       (*
        (exp (- (* t_0 y.re) (* (atan2 x.im x.re) y.im)))
        (sin (+ t_1 (* y.im (log x.im)))))
       (if (or (<= y.im -2.2e+17) (not (<= y.im 3.1e+105)))
         (* (atan2 x.im x.re) (/ y.re (pow (exp y.im) (atan2 x.im x.re))))
         (* (sin (fma t_0 y.im t_1)) (pow (hypot x.re x.im) y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= -2.5e+201) {
		tmp = fabs(t_1) * exp((y_46_im * -atan2(x_46_im, x_46_re)));
	} else if (y_46_im <= -4.5e+165) {
		tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin((t_1 + (y_46_im * log(x_46_im))));
	} else if ((y_46_im <= -2.2e+17) || !(y_46_im <= 3.1e+105)) {
		tmp = atan2(x_46_im, x_46_re) * (y_46_re / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	} else {
		tmp = sin(fma(t_0, y_46_im, t_1)) * pow(hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= -2.5e+201)
		tmp = Float64(abs(t_1) * exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))));
	elseif (y_46_im <= -4.5e+165)
		tmp = Float64(exp(Float64(Float64(t_0 * y_46_re) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(Float64(t_1 + Float64(y_46_im * log(x_46_im)))));
	elseif ((y_46_im <= -2.2e+17) || !(y_46_im <= 3.1e+105))
		tmp = Float64(atan(x_46_im, x_46_re) * Float64(y_46_re / (exp(y_46_im) ^ atan(x_46_im, x_46_re))));
	else
		tmp = Float64(sin(fma(t_0, y_46_im, t_1)) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.5e+201], N[(N[Abs[t$95$1], $MachinePrecision] * N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.5e+165], 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[(t$95$1 + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y$46$im, -2.2e+17], N[Not[LessEqual[y$46$im, 3.1e+105]], $MachinePrecision]], N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(t$95$0 * y$46$im + t$95$1), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.4999999999999998e201

   1. Initial program 47.1%

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

   2. Taylor expanded in y.im around 0 71.0%

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

   3. Taylor expanded in y.re around 0 82.4%

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

      1. *-commutative 82.4%

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

      2. distribute-lft-neg-in 82.4%

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

      3. *-commutative 82.4%

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

   5. Simplified 82.4%

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

      1. *-commutative 82.4%

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

      2. add-sqr-sqrt 52.9%

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

      3. sqrt-unprod 64.7%

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

      4. pow2 64.7%

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

   7. Applied egg-rr 64.7%

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

      1. *-commutative 64.7%

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

      2. unpow2 64.7%

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

      3. rem-sqrt-square 94.1%

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

   9. Simplified 94.1%

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

   if -2.4999999999999998e201 < y.im < -4.4999999999999996e165

   1. Initial program 22.2%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in x.re around 0 57.9%

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

   if -4.4999999999999996e165 < y.im < -2.2e17 or 3.10000000000000004e105 < y.im

   1. Initial program 32.5%

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

   2. Taylor expanded in y.im around 0 58.2%

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

   3. Taylor expanded in y.re around 0 64.2%

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

      1. associate-*r* 64.2%

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

      2. exp-neg 64.2%

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

      3. associate-*l/ 64.2%

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

      4. *-lft-identity 64.2%

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

      5. exp-prod 66.5%

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

   5. Simplified 66.5%

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

   if -2.2e17 < y.im < 3.10000000000000004e105

   1. Initial program 44.4%

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

      1. exp-diff 42.3%

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

      2. +-rgt-identity 42.3%

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

      3. +-rgt-identity 42.3%

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

      4. exp-to-pow 42.3%

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

      5. hypot-def 42.3%

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

      6. exp-prod 42.3%

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

      7. fma-def 42.3%

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

      8. hypot-def 80.4%

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

      9. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in y.im around 0 80.4%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+201}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+165}:\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{+17} \lor \neg \left(y.im \leq 3.1 \cdot 10^{+105}\right):\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 8: 72.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+14} \lor \neg \left(y.im \leq 3.2 \cdot 10^{+105}\right):\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.35e+14) (not (<= y.im 3.2e+105)))
   (* (atan2 x.im x.re) (/ y.re (pow (exp y.im) (atan2 x.im x.re))))
   (*
    (sin (fma (log (hypot x.re x.im)) y.im (* y.re (atan2 x.im x.re))))
    (pow (hypot x.re x.im) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.35e+14) || !(y_46_im <= 3.2e+105)) {
		tmp = atan2(x_46_im, x_46_re) * (y_46_re / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	} else {
		tmp = sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, (y_46_re * atan2(x_46_im, x_46_re)))) * pow(hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.35e+14) || !(y_46_im <= 3.2e+105))
		tmp = Float64(atan(x_46_im, x_46_re) * Float64(y_46_re / (exp(y_46_im) ^ atan(x_46_im, x_46_re))));
	else
		tmp = Float64(sin(fma(log(hypot(x_46_re, x_46_im)), y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.35e+14], N[Not[LessEqual[y$46$im, 3.2e+105]], $MachinePrecision]], N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.35e14 or 3.2e105 < y.im

   1. Initial program 33.9%

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

   2. Taylor expanded in y.im around 0 57.3%

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

   3. Taylor expanded in y.re around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. exp-neg 64.5%

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

      3. associate-*l/ 64.5%

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

      4. *-lft-identity 64.5%

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

      5. exp-prod 66.3%

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

   5. Simplified 66.3%

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

   if -1.35e14 < y.im < 3.2e105

   1. Initial program 44.4%

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

      1. exp-diff 42.3%

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

      2. +-rgt-identity 42.3%

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

      3. +-rgt-identity 42.3%

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

      4. exp-to-pow 42.3%

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

      5. hypot-def 42.3%

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

      6. exp-prod 42.3%

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

      7. fma-def 42.3%

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

      8. hypot-def 80.4%

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

      9. *-commutative 80.4%

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

   3. Simplified 80.4%

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

   4. Taylor expanded in y.im around 0 80.4%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+14} \lor \neg \left(y.im \leq 3.2 \cdot 10^{+105}\right):\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]

Alternative 9: 65.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-272}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{-133}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (*
          (/ (pow (hypot x.re x.im) y.re) (+ (* (atan2 x.im x.re) y.im) 1.0))
          (sin (* y.im (log (hypot x.im x.re))))))
        (t_1
         (* (atan2 x.im x.re) (/ y.re (pow (exp y.im) (atan2 x.im x.re))))))
   (if (<= y.im -2.4e+15)
     t_1
     (if (<= y.im -1.2e-272)
       t_0
       (if (<= y.im 9.6e-133)
         (* (sin (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
         (if (<= y.im 6.8e+105) 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 = (pow(hypot(x_46_re, x_46_im), y_46_re) / ((atan2(x_46_im, x_46_re) * y_46_im) + 1.0)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double t_1 = atan2(x_46_im, x_46_re) * (y_46_re / pow(exp(y_46_im), atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.4e+15) {
		tmp = t_1;
	} else if (y_46_im <= -1.2e-272) {
		tmp = t_0;
	} else if (y_46_im <= 9.6e-133) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 6.8e+105) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / ((Math.atan2(x_46_im, x_46_re) * y_46_im) + 1.0)) * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double t_1 = Math.atan2(x_46_im, x_46_re) * (y_46_re / Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (y_46_im <= -2.4e+15) {
		tmp = t_1;
	} else if (y_46_im <= -1.2e-272) {
		tmp = t_0;
	} else if (y_46_im <= 9.6e-133) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 6.8e+105) {
		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.pow(math.hypot(x_46_re, x_46_im), y_46_re) / ((math.atan2(x_46_im, x_46_re) * y_46_im) + 1.0)) * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	t_1 = math.atan2(x_46_im, x_46_re) * (y_46_re / math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if y_46_im <= -2.4e+15:
		tmp = t_1
	elif y_46_im <= -1.2e-272:
		tmp = t_0
	elif y_46_im <= 9.6e-133:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_im <= 6.8e+105:
		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(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(Float64(atan(x_46_im, x_46_re) * y_46_im) + 1.0)) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))))
	t_1 = Float64(atan(x_46_im, x_46_re) * Float64(y_46_re / (exp(y_46_im) ^ atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (y_46_im <= -2.4e+15)
		tmp = t_1;
	elseif (y_46_im <= -1.2e-272)
		tmp = t_0;
	elseif (y_46_im <= 9.6e-133)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 6.8e+105)
		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 = ((hypot(x_46_re, x_46_im) ^ y_46_re) / ((atan2(x_46_im, x_46_re) * y_46_im) + 1.0)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	t_1 = atan2(x_46_im, x_46_re) * (y_46_re / (exp(y_46_im) ^ atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (y_46_im <= -2.4e+15)
		tmp = t_1;
	elseif (y_46_im <= -1.2e-272)
		tmp = t_0;
	elseif (y_46_im <= 9.6e-133)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif (y_46_im <= 6.8e+105)
		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[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$re / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.4e+15], t$95$1, If[LessEqual[y$46$im, -1.2e-272], t$95$0, If[LessEqual[y$46$im, 9.6e-133], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 6.8e+105], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.4e15 or 6.7999999999999999e105 < y.im

   1. Initial program 33.9%

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

   2. Taylor expanded in y.im around 0 57.3%

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

   3. Taylor expanded in y.re around 0 64.5%

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

      1. associate-*r* 64.5%

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

      2. exp-neg 64.5%

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

      3. associate-*l/ 64.5%

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

      4. *-lft-identity 64.5%

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

      5. exp-prod 66.3%

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

   5. Simplified 66.3%

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

   if -2.4e15 < y.im < -1.19999999999999995e-272 or 9.6e-133 < y.im < 6.7999999999999999e105

   1. Initial program 43.1%

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

      1. exp-diff 40.0%

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

      2. +-rgt-identity 40.0%

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

      3. +-rgt-identity 40.0%

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

      4. exp-to-pow 40.0%

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

      5. hypot-def 40.0%

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

      6. exp-prod 40.0%

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

      7. fma-def 40.0%

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

      8. hypot-def 75.7%

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

      9. *-commutative 75.7%

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

   3. Simplified 75.7%

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

   4. Taylor expanded in y.im around 0 79.2%

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

   5. Taylor expanded in y.im around inf 39.6%

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

      1. unpow2 39.6%

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

      2. unpow2 39.6%

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

      3. hypot-def 71.4%

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

   7. Simplified 71.4%

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

   if -1.19999999999999995e-272 < y.im < 9.6e-133

   1. Initial program 46.9%

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

   2. Taylor expanded in y.im around 0 64.0%

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

   3. Taylor expanded in y.im around 0 64.0%

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

      1. unpow2 64.0%

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

      2. unpow2 64.0%

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

      3. hypot-def 75.6%

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

   5. Simplified 75.6%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+15}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-272}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{-133}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.re}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \end{array} \]

Alternative 10: 59.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t_1\\ \mathbf{if}\;x.im \leq -6.2 \cdot 10^{-220}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq 4.9 \cdot 10^{-306}:\\ \;\;\;\;t_2 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 8.4 \cdot 10^{-222} \lor \neg \left(x.im \leq 9 \cdot 10^{-18}\right):\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.im - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (sin t_1)))
   (if (<= x.im -6.2e-220)
     (* t_2 (exp (- (* y.re (log (- x.im))) t_0)))
     (if (<= x.im 4.9e-306)
       (* t_2 (pow (hypot x.im x.re) y.re))
       (if (or (<= x.im 8.4e-222) (not (<= x.im 9e-18)))
         (*
          (sin (* y.im (log (hypot x.im x.re))))
          (exp (- (* y.re (log x.im)) t_0)))
         (*
          t_1
          (exp
           (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.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_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(t_1);
	double tmp;
	if (x_46_im <= -6.2e-220) {
		tmp = t_2 * exp(((y_46_re * log(-x_46_im)) - t_0));
	} else if (x_46_im <= 4.9e-306) {
		tmp = t_2 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if ((x_46_im <= 8.4e-222) || !(x_46_im <= 9e-18)) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_im)) - t_0));
	} else {
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (x_46_im <= -6.2e-220) {
		tmp = t_2 * Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	} else if (x_46_im <= 4.9e-306) {
		tmp = t_2 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if ((x_46_im <= 8.4e-222) || !(x_46_im <= 9e-18)) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	} else {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.sin(t_1)
	tmp = 0
	if x_46_im <= -6.2e-220:
		tmp = t_2 * math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	elif x_46_im <= 4.9e-306:
		tmp = t_2 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif (x_46_im <= 8.4e-222) or not (x_46_im <= 9e-18):
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	else:
		tmp = t_1 * math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = sin(t_1)
	tmp = 0.0
	if (x_46_im <= -6.2e-220)
		tmp = Float64(t_2 * exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0)));
	elseif (x_46_im <= 4.9e-306)
		tmp = Float64(t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif ((x_46_im <= 8.4e-222) || !(x_46_im <= 9e-18))
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0)));
	else
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - 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_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = sin(t_1);
	tmp = 0.0;
	if (x_46_im <= -6.2e-220)
		tmp = t_2 * exp(((y_46_re * log(-x_46_im)) - t_0));
	elseif (x_46_im <= 4.9e-306)
		tmp = t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re);
	elseif ((x_46_im <= 8.4e-222) || ~((x_46_im <= 9e-18)))
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_im)) - t_0));
	else
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[x$46$im, -6.2e-220], N[(t$95$2 * N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 4.9e-306], N[(t$95$2 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$im, 8.4e-222], N[Not[LessEqual[x$46$im, 9e-18]], $MachinePrecision]], N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -6.20000000000000023e-220

   1. Initial program 36.1%

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

   2. Taylor expanded in y.im around 0 57.1%

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

   3. Taylor expanded in x.im around -inf 68.1%

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

      1. mul-1-neg 68.1%

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

   5. Simplified 68.1%

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

   if -6.20000000000000023e-220 < x.im < 4.90000000000000025e-306

   1. Initial program 61.8%

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

   2. Taylor expanded in y.im around 0 53.3%

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

   3. Taylor expanded in y.im around 0 48.7%

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

      1. unpow2 48.7%

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

      2. unpow2 48.7%

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

      3. hypot-def 49.9%

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

   5. Simplified 49.9%

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

   if 4.90000000000000025e-306 < x.im < 8.3999999999999996e-222 or 8.99999999999999987e-18 < x.im

   1. Initial program 32.9%

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

      1. hypot-udef 44.7%

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

      2. add-cube-cbrt 44.7%

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

      3. pow3 44.7%

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

   3. Applied egg-rr 44.7%

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

      1. add-exp-log 44.7%

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

      2. hypot-udef 74.0%

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

      3. add-cube-cbrt 74.0%

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

      4. exp-prod 74.0%

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

      5. cbrt-unprod 74.0%

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

      6. pow2 74.0%

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

   5. Applied egg-rr 74.0%

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

   6. Taylor expanded in y.re around 0 32.6%

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

      1. unpow2 32.6%

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

      2. unpow2 32.6%

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

      3. hypot-def 69.3%

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

   8. Simplified 69.3%

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

   9. Taylor expanded in x.re around 0 66.9%

      \[\leadsto e^{\log \color{blue}{\left({x.im}^{\left({1}^{0.3333333333333333}\right)}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]
   10. Step-by-step derivation

       1. pow-base-1 66.9%

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

       2. unpow1 66.9%

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

   11. Simplified 66.9%

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

   if 8.3999999999999996e-222 < x.im < 8.99999999999999987e-18

   1. Initial program 52.2%

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

   2. Taylor expanded in y.im around 0 68.6%

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

   3. Taylor expanded in y.re around 0 68.6%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -6.2 \cdot 10^{-220}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 4.9 \cdot 10^{-306}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;x.im \leq 8.4 \cdot 10^{-222} \lor \neg \left(x.im \leq 9 \cdot 10^{-18}\right):\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 11: 60.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{if}\;x.re \leq -1.65 \cdot 10^{-47}:\\ \;\;\;\;\sin t_1 \cdot e^{y.re \cdot \log \left(-0.5 \cdot \frac{x.im}{\frac{x.re}{x.im}} - x.re\right) - t_0}\\ \mathbf{elif}\;x.re \leq -2.1 \cdot 10^{-112}:\\ \;\;\;\;\left|t_1\right| \cdot t_2\\ \mathbf{elif}\;x.re \leq -1.06 \cdot 10^{-223}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;x.re \leq 45000:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.re - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (exp (* y.im (- (atan2 x.im x.re))))))
   (if (<= x.re -1.65e-47)
     (*
      (sin t_1)
      (exp (- (* y.re (log (- (* -0.5 (/ x.im (/ x.re x.im))) x.re))) t_0)))
     (if (<= x.re -2.1e-112)
       (* (fabs t_1) t_2)
       (if (<= x.re -1.06e-223)
         (* t_1 t_2)
         (if (<= x.re 45000.0)
           (*
            t_1
            (exp
             (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0)))
           (*
            (sin (* y.im (log (hypot x.im x.re))))
            (exp (- (* y.re (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_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -1.65e-47) {
		tmp = sin(t_1) * exp(((y_46_re * log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0));
	} else if (x_46_re <= -2.1e-112) {
		tmp = fabs(t_1) * t_2;
	} else if (x_46_re <= -1.06e-223) {
		tmp = t_1 * t_2;
	} else if (x_46_re <= 45000.0) {
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	} else {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_re)) - 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_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -1.65e-47) {
		tmp = Math.sin(t_1) * Math.exp(((y_46_re * Math.log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0));
	} else if (x_46_re <= -2.1e-112) {
		tmp = Math.abs(t_1) * t_2;
	} else if (x_46_re <= -1.06e-223) {
		tmp = t_1 * t_2;
	} else if (x_46_re <= 45000.0) {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	} else {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp(((y_46_re * Math.log(x_46_re)) - 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_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_re <= -1.65e-47:
		tmp = math.sin(t_1) * math.exp(((y_46_re * math.log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0))
	elif x_46_re <= -2.1e-112:
		tmp = math.fabs(t_1) * t_2
	elif x_46_re <= -1.06e-223:
		tmp = t_1 * t_2
	elif x_46_re <= 45000.0:
		tmp = t_1 * math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	else:
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp(((y_46_re * math.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_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (x_46_re <= -1.65e-47)
		tmp = Float64(sin(t_1) * exp(Float64(Float64(y_46_re * log(Float64(Float64(-0.5 * Float64(x_46_im / Float64(x_46_re / x_46_im))) - x_46_re))) - t_0)));
	elseif (x_46_re <= -2.1e-112)
		tmp = Float64(abs(t_1) * t_2);
	elseif (x_46_re <= -1.06e-223)
		tmp = Float64(t_1 * t_2);
	elseif (x_46_re <= 45000.0)
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)));
	else
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(Float64(y_46_re * log(x_46_re)) - 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_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_re <= -1.65e-47)
		tmp = sin(t_1) * exp(((y_46_re * log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0));
	elseif (x_46_re <= -2.1e-112)
		tmp = abs(t_1) * t_2;
	elseif (x_46_re <= -1.06e-223)
		tmp = t_1 * t_2;
	elseif (x_46_re <= 45000.0)
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	else
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_re)) - 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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.65e-47], N[(N[Sin[t$95$1], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[N[(N[(-0.5 * N[(x$46$im / N[(x$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -2.1e-112], N[(N[Abs[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[x$46$re, -1.06e-223], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[x$46$re, 45000.0], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.re < -1.65000000000000002e-47

   1. Initial program 29.6%

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

   2. Taylor expanded in y.im around 0 53.5%

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

   3. Taylor expanded in x.re around -inf 64.1%

      \[\leadsto e^{\log \color{blue}{\left(-0.5 \cdot \frac{{x.im}^{2}}{x.re} + -1 \cdot x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg 64.1%

         \[\leadsto e^{\log \left(-0.5 \cdot \frac{{x.im}^{2}}{x.re} + \color{blue}{\left(-x.re\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. unsub-neg 64.1%

         \[\leadsto e^{\log \color{blue}{\left(-0.5 \cdot \frac{{x.im}^{2}}{x.re} - x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. unpow2 64.1%

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

      4. associate-/l* 66.8%

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

   5. Simplified 66.8%

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

   if -1.65000000000000002e-47 < x.re < -2.1000000000000001e-112

   1. Initial program 49.9%

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

   2. Taylor expanded in y.im around 0 41.2%

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

   3. Taylor expanded in y.re around 0 36.3%

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

      1. *-commutative 36.3%

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

      2. distribute-lft-neg-in 36.3%

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

      3. *-commutative 36.3%

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

   5. Simplified 36.3%

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

      1. *-commutative 36.3%

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

      2. add-sqr-sqrt 16.1%

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

      3. sqrt-unprod 50.0%

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

      4. pow2 50.0%

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

   7. Applied egg-rr 50.0%

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

      1. *-commutative 50.0%

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

      2. unpow2 50.0%

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

      3. rem-sqrt-square 61.4%

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

   9. Simplified 61.4%

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

   if -2.1000000000000001e-112 < x.re < -1.05999999999999994e-223

   1. Initial program 52.6%

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

   2. Taylor expanded in y.im around 0 62.9%

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

   3. Taylor expanded in y.re around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. distribute-lft-neg-in 68.1%

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

      3. *-commutative 68.1%

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

   5. Simplified 68.1%

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

   if -1.05999999999999994e-223 < x.re < 45000

   1. Initial program 48.7%

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

   2. Taylor expanded in y.im around 0 64.5%

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

   3. Taylor expanded in y.re around 0 63.2%

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

   if 45000 < x.re

   1. Initial program 33.9%

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

      1. hypot-udef 52.3%

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

      2. add-cube-cbrt 52.4%

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

      3. pow3 52.4%

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

   3. Applied egg-rr 52.4%

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

      1. add-exp-log 52.4%

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

      2. hypot-udef 76.2%

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

      3. add-cube-cbrt 76.3%

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

      4. exp-prod 76.3%

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

      5. cbrt-unprod 76.3%

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

      6. pow2 76.3%

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

   5. Applied egg-rr 76.3%

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

   6. Taylor expanded in y.re around 0 27.3%

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

      1. unpow2 27.3%

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

      2. unpow2 27.3%

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

      3. hypot-def 72.9%

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

   8. Simplified 72.9%

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

   9. Taylor expanded in x.im around 0 72.9%

      \[\leadsto e^{\log \color{blue}{\left({x.re}^{\left({1}^{0.3333333333333333}\right)}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \]
   10. Step-by-step derivation

       1. pow-base-1 72.9%

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

       2. unpow1 72.9%

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

   11. Simplified 72.9%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.65 \cdot 10^{-47}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-0.5 \cdot \frac{x.im}{\frac{x.re}{x.im}} - x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq -2.1 \cdot 10^{-112}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq -1.06 \cdot 10^{-223}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq 45000:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 12: 59.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{if}\;x.re \leq -1.8 \cdot 10^{-48}:\\ \;\;\;\;\sin t_1 \cdot e^{y.re \cdot \log \left(-0.5 \cdot \frac{x.im}{\frac{x.re}{x.im}} - x.re\right) - t_0}\\ \mathbf{elif}\;x.re \leq -2.6 \cdot 10^{-114}:\\ \;\;\;\;\left|t_1\right| \cdot t_2\\ \mathbf{elif}\;x.re \leq -8.2 \cdot 10^{-224}:\\ \;\;\;\;t_1 \cdot t_2\\ \mathbf{elif}\;x.re \leq 2.45 \cdot 10^{-61}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0.5 \cdot \frac{y.re}{\frac{x.re}{x.im} \cdot \frac{x.re}{x.im}} + y.re \cdot \log x.re\right) - t_0} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (exp (* y.im (- (atan2 x.im x.re))))))
   (if (<= x.re -1.8e-48)
     (*
      (sin t_1)
      (exp (- (* y.re (log (- (* -0.5 (/ x.im (/ x.re x.im))) x.re))) t_0)))
     (if (<= x.re -2.6e-114)
       (* (fabs t_1) t_2)
       (if (<= x.re -8.2e-224)
         (* t_1 t_2)
         (if (<= x.re 2.45e-61)
           (*
            t_1
            (exp
             (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0)))
           (*
            (exp
             (-
              (+
               (* 0.5 (/ y.re (* (/ x.re x.im) (/ x.re x.im))))
               (* y.re (log x.re)))
              t_0))
            (sin (* y.im (log x.re))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -1.8e-48) {
		tmp = sin(t_1) * exp(((y_46_re * log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0));
	} else if (x_46_re <= -2.6e-114) {
		tmp = fabs(t_1) * t_2;
	} else if (x_46_re <= -8.2e-224) {
		tmp = t_1 * t_2;
	} else if (x_46_re <= 2.45e-61) {
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	} else {
		tmp = exp((((0.5 * (y_46_re / ((x_46_re / x_46_im) * (x_46_re / x_46_im)))) + (y_46_re * log(x_46_re))) - t_0)) * sin((y_46_im * log(x_46_re)));
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = atan2(x_46im, x_46re) * y_46im
    t_1 = y_46re * atan2(x_46im, x_46re)
    t_2 = exp((y_46im * -atan2(x_46im, x_46re)))
    if (x_46re <= (-1.8d-48)) then
        tmp = sin(t_1) * exp(((y_46re * log((((-0.5d0) * (x_46im / (x_46re / x_46im))) - x_46re))) - t_0))
    else if (x_46re <= (-2.6d-114)) then
        tmp = abs(t_1) * t_2
    else if (x_46re <= (-8.2d-224)) then
        tmp = t_1 * t_2
    else if (x_46re <= 2.45d-61) then
        tmp = t_1 * exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - t_0))
    else
        tmp = exp((((0.5d0 * (y_46re / ((x_46re / x_46im) * (x_46re / x_46im)))) + (y_46re * log(x_46re))) - t_0)) * sin((y_46im * log(x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_re <= -1.8e-48) {
		tmp = Math.sin(t_1) * Math.exp(((y_46_re * Math.log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0));
	} else if (x_46_re <= -2.6e-114) {
		tmp = Math.abs(t_1) * t_2;
	} else if (x_46_re <= -8.2e-224) {
		tmp = t_1 * t_2;
	} else if (x_46_re <= 2.45e-61) {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	} else {
		tmp = Math.exp((((0.5 * (y_46_re / ((x_46_re / x_46_im) * (x_46_re / x_46_im)))) + (y_46_re * Math.log(x_46_re))) - t_0)) * Math.sin((y_46_im * Math.log(x_46_re)));
	}
	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_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_re <= -1.8e-48:
		tmp = math.sin(t_1) * math.exp(((y_46_re * math.log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0))
	elif x_46_re <= -2.6e-114:
		tmp = math.fabs(t_1) * t_2
	elif x_46_re <= -8.2e-224:
		tmp = t_1 * t_2
	elif x_46_re <= 2.45e-61:
		tmp = t_1 * math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	else:
		tmp = math.exp((((0.5 * (y_46_re / ((x_46_re / x_46_im) * (x_46_re / x_46_im)))) + (y_46_re * math.log(x_46_re))) - t_0)) * math.sin((y_46_im * math.log(x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re))))
	tmp = 0.0
	if (x_46_re <= -1.8e-48)
		tmp = Float64(sin(t_1) * exp(Float64(Float64(y_46_re * log(Float64(Float64(-0.5 * Float64(x_46_im / Float64(x_46_re / x_46_im))) - x_46_re))) - t_0)));
	elseif (x_46_re <= -2.6e-114)
		tmp = Float64(abs(t_1) * t_2);
	elseif (x_46_re <= -8.2e-224)
		tmp = Float64(t_1 * t_2);
	elseif (x_46_re <= 2.45e-61)
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)));
	else
		tmp = Float64(exp(Float64(Float64(Float64(0.5 * Float64(y_46_re / Float64(Float64(x_46_re / x_46_im) * Float64(x_46_re / x_46_im)))) + Float64(y_46_re * log(x_46_re))) - t_0)) * sin(Float64(y_46_im * log(x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = exp((y_46_im * -atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_re <= -1.8e-48)
		tmp = sin(t_1) * exp(((y_46_re * log(((-0.5 * (x_46_im / (x_46_re / x_46_im))) - x_46_re))) - t_0));
	elseif (x_46_re <= -2.6e-114)
		tmp = abs(t_1) * t_2;
	elseif (x_46_re <= -8.2e-224)
		tmp = t_1 * t_2;
	elseif (x_46_re <= 2.45e-61)
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	else
		tmp = exp((((0.5 * (y_46_re / ((x_46_re / x_46_im) * (x_46_re / x_46_im)))) + (y_46_re * log(x_46_re))) - t_0)) * sin((y_46_im * log(x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -1.8e-48], N[(N[Sin[t$95$1], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[N[(N[(-0.5 * N[(x$46$im / N[(x$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, -2.6e-114], N[(N[Abs[t$95$1], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[x$46$re, -8.2e-224], N[(t$95$1 * t$95$2), $MachinePrecision], If[LessEqual[x$46$re, 2.45e-61], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(N[(0.5 * N[(y$46$re / N[(N[(x$46$re / x$46$im), $MachinePrecision] * N[(x$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.re < -1.8000000000000001e-48

   1. Initial program 29.6%

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

   2. Taylor expanded in y.im around 0 53.5%

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

   3. Taylor expanded in x.re around -inf 64.1%

      \[\leadsto e^{\log \color{blue}{\left(-0.5 \cdot \frac{{x.im}^{2}}{x.re} + -1 \cdot x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
   4. Step-by-step derivation

      1. mul-1-neg 64.1%

         \[\leadsto e^{\log \left(-0.5 \cdot \frac{{x.im}^{2}}{x.re} + \color{blue}{\left(-x.re\right)}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      2. unsub-neg 64.1%

         \[\leadsto e^{\log \color{blue}{\left(-0.5 \cdot \frac{{x.im}^{2}}{x.re} - x.re\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]

      3. unpow2 64.1%

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

      4. associate-/l* 66.8%

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

   5. Simplified 66.8%

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

   if -1.8000000000000001e-48 < x.re < -2.60000000000000013e-114

   1. Initial program 49.9%

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

   2. Taylor expanded in y.im around 0 41.2%

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

   3. Taylor expanded in y.re around 0 36.3%

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

      1. *-commutative 36.3%

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

      2. distribute-lft-neg-in 36.3%

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

      3. *-commutative 36.3%

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

   5. Simplified 36.3%

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

      1. *-commutative 36.3%

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

      2. add-sqr-sqrt 16.1%

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

      3. sqrt-unprod 50.0%

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

      4. pow2 50.0%

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

   7. Applied egg-rr 50.0%

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

      1. *-commutative 50.0%

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

      2. unpow2 50.0%

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

      3. rem-sqrt-square 61.4%

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

   9. Simplified 61.4%

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

   if -2.60000000000000013e-114 < x.re < -8.19999999999999972e-224

   1. Initial program 52.6%

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

   2. Taylor expanded in y.im around 0 62.9%

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

   3. Taylor expanded in y.re around 0 68.1%

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

      1. *-commutative 68.1%

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

      2. distribute-lft-neg-in 68.1%

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

      3. *-commutative 68.1%

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

   5. Simplified 68.1%

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

   if -8.19999999999999972e-224 < x.re < 2.45000000000000001e-61

   1. Initial program 47.6%

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

   2. Taylor expanded in y.im around 0 65.7%

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

   3. Taylor expanded in y.re around 0 64.1%

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

   if 2.45000000000000001e-61 < x.re

   1. Initial program 38.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x.re around inf 62.9%

      \[\leadsto e^{\color{blue}{\left(0.5 \cdot \frac{y.re \cdot {x.im}^{2}}{{x.re}^{2}} + -1 \cdot \left(\log \left(\frac{1}{x.re}\right) \cdot y.re\right)\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]
   5. Step-by-step derivation

      1. mul-1-neg 62.9%

         \[\leadsto e^{\left(0.5 \cdot \frac{y.re \cdot {x.im}^{2}}{{x.re}^{2}} + \color{blue}{\left(-\log \left(\frac{1}{x.re}\right) \cdot y.re\right)}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      2. unsub-neg 62.9%

         \[\leadsto e^{\color{blue}{\left(0.5 \cdot \frac{y.re \cdot {x.im}^{2}}{{x.re}^{2}} - \log \left(\frac{1}{x.re}\right) \cdot y.re\right)} - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      3. associate-/l* 64.1%

         \[\leadsto e^{\left(0.5 \cdot \color{blue}{\frac{y.re}{\frac{{x.re}^{2}}{{x.im}^{2}}}} - \log \left(\frac{1}{x.re}\right) \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      4. unpow2 64.1%

         \[\leadsto e^{\left(0.5 \cdot \frac{y.re}{\frac{\color{blue}{x.re \cdot x.re}}{{x.im}^{2}}} - \log \left(\frac{1}{x.re}\right) \cdot y.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \]

      5. unpow2 64.1%

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

      6. times-frac 67.7%

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

      7. log-rec 67.7%

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

      8. distribute-lft-neg-out 67.7%

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

      9. *-commutative 67.7%

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

   6. Simplified 67.7%

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

   7. Taylor expanded in x.im around 0 67.2%

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

   8. Taylor expanded in y.re around 0 61.2%

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

      1. *-commutative 61.2%

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

   10. Simplified 61.2%

       \[\leadsto e^{\left(0.5 \cdot \frac{y.re}{\frac{x.re}{x.im} \cdot \frac{x.re}{x.im}} - \left(-y.re \cdot \log x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(\log x.re \cdot y.im\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.8 \cdot 10^{-48}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-0.5 \cdot \frac{x.im}{\frac{x.re}{x.im}} - x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq -2.6 \cdot 10^{-114}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq -8.2 \cdot 10^{-224}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq 2.45 \cdot 10^{-61}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(0.5 \cdot \frac{y.re}{\frac{x.re}{x.im} \cdot \frac{x.re}{x.im}} + y.re \cdot \log x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log x.re\right)\\ \end{array} \]

Alternative 13: 58.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t_1\\ \mathbf{if}\;x.im \leq -4.2 \cdot 10^{-219}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{-283}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{t_0 + 1} \cdot t_2\\ \mathbf{elif}\;x.im \leq 3.8 \cdot 10^{-218} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-17}\right):\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log x.im - t_0}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (sin t_1)))
   (if (<= x.im -4.2e-219)
     (* t_2 (exp (- (* y.re (log (- x.im))) t_0)))
     (if (<= x.im 5e-283)
       (* (/ (pow (hypot x.re x.im) y.re) (+ t_0 1.0)) t_2)
       (if (or (<= x.im 3.8e-218) (not (<= x.im 1.4e-17)))
         (* t_2 (exp (- (* y.re (log x.im)) t_0)))
         (*
          t_1
          (exp
           (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.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_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = sin(t_1);
	double tmp;
	if (x_46_im <= -4.2e-219) {
		tmp = t_2 * exp(((y_46_re * log(-x_46_im)) - t_0));
	} else if (x_46_im <= 5e-283) {
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / (t_0 + 1.0)) * t_2;
	} else if ((x_46_im <= 3.8e-218) || !(x_46_im <= 1.4e-17)) {
		tmp = t_2 * exp(((y_46_re * log(x_46_im)) - t_0));
	} else {
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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_im;
	double t_1 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_2 = Math.sin(t_1);
	double tmp;
	if (x_46_im <= -4.2e-219) {
		tmp = t_2 * Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	} else if (x_46_im <= 5e-283) {
		tmp = (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / (t_0 + 1.0)) * t_2;
	} else if ((x_46_im <= 3.8e-218) || !(x_46_im <= 1.4e-17)) {
		tmp = t_2 * Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	} else {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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_im
	t_1 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_2 = math.sin(t_1)
	tmp = 0
	if x_46_im <= -4.2e-219:
		tmp = t_2 * math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	elif x_46_im <= 5e-283:
		tmp = (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / (t_0 + 1.0)) * t_2
	elif (x_46_im <= 3.8e-218) or not (x_46_im <= 1.4e-17):
		tmp = t_2 * math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	else:
		tmp = t_1 * math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = sin(t_1)
	tmp = 0.0
	if (x_46_im <= -4.2e-219)
		tmp = Float64(t_2 * exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0)));
	elseif (x_46_im <= 5e-283)
		tmp = Float64(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(t_0 + 1.0)) * t_2);
	elseif ((x_46_im <= 3.8e-218) || !(x_46_im <= 1.4e-17))
		tmp = Float64(t_2 * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0)));
	else
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - 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_im;
	t_1 = y_46_re * atan2(x_46_im, x_46_re);
	t_2 = sin(t_1);
	tmp = 0.0;
	if (x_46_im <= -4.2e-219)
		tmp = t_2 * exp(((y_46_re * log(-x_46_im)) - t_0));
	elseif (x_46_im <= 5e-283)
		tmp = ((hypot(x_46_re, x_46_im) ^ y_46_re) / (t_0 + 1.0)) * t_2;
	elseif ((x_46_im <= 3.8e-218) || ~((x_46_im <= 1.4e-17)))
		tmp = t_2 * exp(((y_46_re * log(x_46_im)) - t_0));
	else
		tmp = t_1 * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - 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[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[x$46$im, -4.2e-219], N[(t$95$2 * N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 5e-283], N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], If[Or[LessEqual[x$46$im, 3.8e-218], N[Not[LessEqual[x$46$im, 1.4e-17]], $MachinePrecision]], N[(t$95$2 * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -4.20000000000000001e-219

   1. Initial program 36.1%

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

   2. Taylor expanded in y.im around 0 57.1%

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

   3. Taylor expanded in x.im around -inf 68.1%

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

      1. mul-1-neg 68.1%

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

   5. Simplified 68.1%

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

   if -4.20000000000000001e-219 < x.im < 5.0000000000000001e-283

   1. Initial program 57.6%

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

      1. exp-diff 49.9%

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

      2. +-rgt-identity 49.9%

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

      3. +-rgt-identity 49.9%

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

      4. exp-to-pow 49.9%

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

      5. hypot-def 49.9%

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

      6. exp-prod 49.9%

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

      7. fma-def 49.9%

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

      8. hypot-def 81.5%

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

      9. *-commutative 81.5%

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

   3. Simplified 81.5%

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

   4. Taylor expanded in y.im around 0 77.8%

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

   5. Taylor expanded in y.im around 0 44.3%

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

      1. *-commutative 44.3%

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

   7. Simplified 44.3%

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

   if 5.0000000000000001e-283 < x.im < 3.7999999999999999e-218 or 1.3999999999999999e-17 < x.im

   1. Initial program 32.0%

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

   2. Taylor expanded in y.im around 0 36.9%

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

   3. Taylor expanded in x.re around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. *-commutative 56.7%

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

   5. Simplified 56.7%

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

   if 3.7999999999999999e-218 < x.im < 1.3999999999999999e-17

   1. Initial program 53.4%

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

   2. Taylor expanded in y.im around 0 67.9%

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

   3. Taylor expanded in y.re around 0 67.9%

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

4. Final simplification 62.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.2 \cdot 10^{-219}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 5 \cdot 10^{-283}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq 3.8 \cdot 10^{-218} \lor \neg \left(x.im \leq 1.4 \cdot 10^{-17}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 14: 56.2% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ t_2 := e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;y.re \leq -0.19:\\ \;\;\;\;t_1 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 29500000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_3}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+176}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{t_3 + 1} \cdot t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (* (exp (* y.im (- (atan2 x.im x.re)))) (log1p (expm1 t_0))))
        (t_3 (* (atan2 x.im x.re) y.im)))
   (if (<= y.re -0.19)
     (* t_1 (pow (hypot x.im x.re) y.re))
     (if (<= y.re 29500000.0)
       t_2
       (if (<= y.re 4.1e+99)
         (* t_1 (exp (- (* y.re (log (- x.im))) t_3)))
         (if (<= y.re 4.8e+176)
           t_2
           (* (/ (pow (hypot x.re x.im) y.re) (+ t_3 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 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = exp((y_46_im * -atan2(x_46_im, x_46_re))) * log1p(expm1(t_0));
	double t_3 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= -0.19) {
		tmp = t_1 * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 29500000.0) {
		tmp = t_2;
	} else if (y_46_re <= 4.1e+99) {
		tmp = t_1 * exp(((y_46_re * log(-x_46_im)) - t_3));
	} else if (y_46_re <= 4.8e+176) {
		tmp = t_2;
	} else {
		tmp = (pow(hypot(x_46_re, x_46_im), y_46_re) / (t_3 + 1.0)) * t_1;
	}
	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 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re))) * Math.log1p(Math.expm1(t_0));
	double t_3 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (y_46_re <= -0.19) {
		tmp = t_1 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_re <= 29500000.0) {
		tmp = t_2;
	} else if (y_46_re <= 4.1e+99) {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(-x_46_im)) - t_3));
	} else if (y_46_re <= 4.8e+176) {
		tmp = t_2;
	} else {
		tmp = (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / (t_3 + 1.0)) * t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin(t_0)
	t_2 = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re))) * math.log1p(math.expm1(t_0))
	t_3 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if y_46_re <= -0.19:
		tmp = t_1 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	elif y_46_re <= 29500000.0:
		tmp = t_2
	elif y_46_re <= 4.1e+99:
		tmp = t_1 * math.exp(((y_46_re * math.log(-x_46_im)) - t_3))
	elif y_46_re <= 4.8e+176:
		tmp = t_2
	else:
		tmp = (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / (t_3 + 1.0)) * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64(exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))) * log1p(expm1(t_0)))
	t_3 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (y_46_re <= -0.19)
		tmp = Float64(t_1 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_re <= 29500000.0)
		tmp = t_2;
	elseif (y_46_re <= 4.1e+99)
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_3)));
	elseif (y_46_re <= 4.8e+176)
		tmp = t_2;
	else
		tmp = Float64(Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(t_3 + 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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[y$46$re, -0.19], N[(t$95$1 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 29500000.0], t$95$2, If[LessEqual[y$46$re, 4.1e+99], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.8e+176], t$95$2, N[(N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(t$95$3 + 1.0), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -0.19

   1. Initial program 45.2%

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

   2. Taylor expanded in y.im around 0 82.4%

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

   3. Taylor expanded in y.im around 0 79.2%

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

      1. unpow2 79.2%

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

      2. unpow2 79.2%

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

      3. hypot-def 79.2%

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

   5. Simplified 79.2%

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

   if -0.19 < y.re < 2.95e7 or 4.09999999999999979e99 < y.re < 4.8000000000000003e176

   1. Initial program 39.9%

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

   2. Taylor expanded in y.im around 0 36.1%

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

   3. Taylor expanded in y.re around 0 50.2%

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

      1. *-commutative 50.2%

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

      2. distribute-lft-neg-in 50.2%

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

      3. *-commutative 50.2%

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

   5. Simplified 50.2%

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

      1. *-commutative 50.2%

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

      2. log1p-expm1-u 52.9%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \]

   7. Applied egg-rr 52.9%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \]

   if 2.95e7 < y.re < 4.09999999999999979e99

   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. Taylor expanded in y.im around 0 54.4%

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

   3. Taylor expanded in x.im around -inf 50.3%

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

      1. mul-1-neg 50.3%

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

   5. Simplified 50.3%

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

   if 4.8000000000000003e176 < y.re

   1. Initial program 32.4%

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

      1. exp-diff 20.6%

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

      2. +-rgt-identity 20.6%

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

      3. +-rgt-identity 20.6%

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

      4. exp-to-pow 20.6%

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

      5. hypot-def 20.6%

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

      6. exp-prod 20.6%

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

      7. fma-def 20.6%

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

      8. hypot-def 47.1%

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

      9. *-commutative 47.1%

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

   3. Simplified 47.1%

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

   4. Taylor expanded in y.im around 0 64.8%

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

   5. Taylor expanded in y.im around 0 64.8%

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

      1. *-commutative 64.8%

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

   7. Simplified 64.8%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.19:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 29500000:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+99}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+176}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 15: 56.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ t_2 := \sin t_0\\ t_3 := t_2 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.2:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq 26500000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+98}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (* (exp (* y.im (- (atan2 x.im x.re)))) (log1p (expm1 t_0))))
        (t_2 (sin t_0))
        (t_3 (* t_2 (pow (hypot x.im x.re) y.re))))
   (if (<= y.re -4.2)
     t_3
     (if (<= y.re 26500000.0)
       t_1
       (if (<= y.re 6.6e+98)
         (* t_2 (exp (- (* y.re (log (- x.im))) (* (atan2 x.im x.re) y.im))))
         (if (<= y.re 1.05e+160) t_1 t_3))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = exp((y_46_im * -atan2(x_46_im, x_46_re))) * log1p(expm1(t_0));
	double t_2 = sin(t_0);
	double t_3 = t_2 * pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.2) {
		tmp = t_3;
	} else if (y_46_re <= 26500000.0) {
		tmp = t_1;
	} else if (y_46_re <= 6.6e+98) {
		tmp = t_2 * exp(((y_46_re * log(-x_46_im)) - (atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 1.05e+160) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	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 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re))) * Math.log1p(Math.expm1(t_0));
	double t_2 = Math.sin(t_0);
	double t_3 = t_2 * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.2) {
		tmp = t_3;
	} else if (y_46_re <= 26500000.0) {
		tmp = t_1;
	} else if (y_46_re <= 6.6e+98) {
		tmp = t_2 * Math.exp(((y_46_re * Math.log(-x_46_im)) - (Math.atan2(x_46_im, x_46_re) * y_46_im)));
	} else if (y_46_re <= 1.05e+160) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re))) * math.log1p(math.expm1(t_0))
	t_2 = math.sin(t_0)
	t_3 = t_2 * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -4.2:
		tmp = t_3
	elif y_46_re <= 26500000.0:
		tmp = t_1
	elif y_46_re <= 6.6e+98:
		tmp = t_2 * math.exp(((y_46_re * math.log(-x_46_im)) - (math.atan2(x_46_im, x_46_re) * y_46_im)))
	elif y_46_re <= 1.05e+160:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))) * log1p(expm1(t_0)))
	t_2 = sin(t_0)
	t_3 = Float64(t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re))
	tmp = 0.0
	if (y_46_re <= -4.2)
		tmp = t_3;
	elseif (y_46_re <= 26500000.0)
		tmp = t_1;
	elseif (y_46_re <= 6.6e+98)
		tmp = Float64(t_2 * exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - Float64(atan(x_46_im, x_46_re) * y_46_im))));
	elseif (y_46_re <= 1.05e+160)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.2], t$95$3, If[LessEqual[y$46$re, 26500000.0], t$95$1, If[LessEqual[y$46$re, 6.6e+98], N[(t$95$2 * N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+160], t$95$1, t$95$3]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.20000000000000018 or 1.04999999999999998e160 < y.re

   1. Initial program 39.8%

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

   2. Taylor expanded in y.im around 0 72.6%

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

   3. Taylor expanded in y.im around 0 70.6%

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

      1. unpow2 70.6%

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

      2. unpow2 70.6%

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

      3. hypot-def 70.6%

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

   5. Simplified 70.6%

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

   if -4.20000000000000018 < y.re < 2.65e7 or 6.60000000000000056e98 < y.re < 1.04999999999999998e160

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0 35.9%

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

   3. Taylor expanded in y.re around 0 50.9%

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

      1. *-commutative 50.9%

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

      2. distribute-lft-neg-in 50.9%

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

      3. *-commutative 50.9%

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

   5. Simplified 50.9%

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

      1. *-commutative 50.9%

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

      2. log1p-expm1-u 53.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \]

   7. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \]

   if 2.65e7 < y.re < 6.60000000000000056e98

   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. Taylor expanded in y.im around 0 54.4%

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

   3. Taylor expanded in x.im around -inf 50.3%

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

      1. mul-1-neg 50.3%

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

   5. Simplified 50.3%

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

4. Final simplification 59.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.re \leq 26500000:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{+98}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+160}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 16: 58.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -0.102 \lor \neg \left(y.re \leq 1.7\right) \land \left(y.re \leq 1.55 \cdot 10^{+98} \lor \neg \left(y.re \leq 3.9 \cdot 10^{+160}\right)\right):\\ \;\;\;\;\sin t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.re -0.102)
           (and (not (<= y.re 1.7))
                (or (<= y.re 1.55e+98) (not (<= y.re 3.9e+160)))))
     (* (sin t_0) (pow (hypot x.im x.re) y.re))
     (* (exp (* y.im (- (atan2 x.im x.re)))) (log1p (expm1 t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -0.102) || (!(y_46_re <= 1.7) && ((y_46_re <= 1.55e+98) || !(y_46_re <= 3.9e+160)))) {
		tmp = sin(t_0) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = exp((y_46_im * -atan2(x_46_im, x_46_re))) * log1p(expm1(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 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -0.102) || (!(y_46_re <= 1.7) && ((y_46_re <= 1.55e+98) || !(y_46_re <= 3.9e+160)))) {
		tmp = Math.sin(t_0) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re))) * Math.log1p(Math.expm1(t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if (y_46_re <= -0.102) or (not (y_46_re <= 1.7) and ((y_46_re <= 1.55e+98) or not (y_46_re <= 3.9e+160))):
		tmp = math.sin(t_0) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = math.exp((y_46_im * -math.atan2(x_46_im, x_46_re))) * math.log1p(math.expm1(t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_re <= -0.102) || (!(y_46_re <= 1.7) && ((y_46_re <= 1.55e+98) || !(y_46_re <= 3.9e+160))))
		tmp = Float64(sin(t_0) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))) * log1p(expm1(t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -0.102], And[N[Not[LessEqual[y$46$re, 1.7]], $MachinePrecision], Or[LessEqual[y$46$re, 1.55e+98], N[Not[LessEqual[y$46$re, 3.9e+160]], $MachinePrecision]]]], 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[(N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision] * N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.101999999999999993 or 1.69999999999999996 < y.re < 1.5500000000000001e98 or 3.90000000000000007e160 < y.re

   1. Initial program 39.3%

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

   2. Taylor expanded in y.im around 0 69.8%

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

   3. Taylor expanded in y.im around 0 65.8%

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

      1. unpow2 65.8%

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

      2. unpow2 65.8%

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

      3. hypot-def 65.8%

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

   5. Simplified 65.8%

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

   if -0.101999999999999993 < y.re < 1.69999999999999996 or 1.5500000000000001e98 < y.re < 3.90000000000000007e160

   1. Initial program 40.5%

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

   2. Taylor expanded in y.im around 0 35.1%

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

   3. Taylor expanded in y.re around 0 50.9%

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

      1. *-commutative 50.9%

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

      2. distribute-lft-neg-in 50.9%

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

      3. *-commutative 50.9%

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

   5. Simplified 50.9%

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

      1. *-commutative 50.9%

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

      2. log1p-expm1-u 53.7%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \]

   7. Applied egg-rr 53.7%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.102 \lor \neg \left(y.re \leq 1.7\right) \land \left(y.re \leq 1.55 \cdot 10^{+98} \lor \neg \left(y.re \leq 3.9 \cdot 10^{+160}\right)\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternative 17: 60.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -2.9 \lor \neg \left(y.re \leq 17\right):\\ \;\;\;\;\sin t_0 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.re -2.9) (not (<= y.re 17.0)))
     (* (sin t_0) (pow (hypot x.im x.re) y.re))
     (* t_0 (exp (* y.im (- (atan2 x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -2.9) || !(y_46_re <= 17.0)) {
		tmp = sin(t_0) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_0 * exp((y_46_im * -atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -2.9) || !(y_46_re <= 17.0)) {
		tmp = Math.sin(t_0) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = t_0 * Math.exp((y_46_im * -Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if (y_46_re <= -2.9) or not (y_46_re <= 17.0):
		tmp = math.sin(t_0) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = t_0 * math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_re <= -2.9) || !(y_46_re <= 17.0))
		tmp = Float64(sin(t_0) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(t_0 * exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if ((y_46_re <= -2.9) || ~((y_46_re <= 17.0)))
		tmp = sin(t_0) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = t_0 * exp((y_46_im * -atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -2.9], N[Not[LessEqual[y$46$re, 17.0]], $MachinePrecision]], 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[(t$95$0 * N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.89999999999999991 or 17 < y.re

   1. Initial program 35.8%

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

   2. Taylor expanded in y.im around 0 65.1%

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

   3. Taylor expanded in y.im around 0 61.5%

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

      1. unpow2 61.5%

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

      2. unpow2 61.5%

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

      3. hypot-def 61.5%

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

   5. Simplified 61.5%

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

   if -2.89999999999999991 < y.re < 17

   1. Initial program 44.7%

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

   2. Taylor expanded in y.im around 0 36.2%

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

   3. Taylor expanded in y.re around 0 53.7%

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

      1. *-commutative 53.7%

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

      2. distribute-lft-neg-in 53.7%

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

      3. *-commutative 53.7%

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

   5. Simplified 53.7%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \lor \neg \left(y.re \leq 17\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \end{array} \]

Alternative 18: 39.9% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (* y.re (atan2 x.im x.re)) (exp (* y.im (- (atan2 x.im x.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * atan2(x_46_im, x_46_re)) * exp((y_46_im * -atan2(x_46_im, x_46_re)));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = (y_46re * atan2(x_46im, x_46re)) * exp((y_46im * -atan2(x_46im, x_46re)))
end function

Java

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

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (y_46_re * math.atan2(x_46_im, x_46_re)) * math.exp((y_46_im * -math.atan2(x_46_im, x_46_re)))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * exp(Float64(y_46_im * Float64(-atan(x_46_im, x_46_re)))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (y_46_re * atan2(x_46_im, x_46_re)) * exp((y_46_im * -atan2(x_46_im, x_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(y$46$im * (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

2. Taylor expanded in y.im around 0 51.7%

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

3. Taylor expanded in y.re around 0 38.3%

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

   1. *-commutative 38.3%

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

   2. distribute-lft-neg-in 38.3%

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

   3. *-commutative 38.3%

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

5. Simplified 38.3%

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

6. Final simplification 38.3%

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

Alternative 19: 17.2% accurate, 2.7× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (* y.re (atan2 x.im x.re)) (exp (* (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) {
	return (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = (y_46re * atan2(x_46im, x_46re)) * exp((atan2(x_46im, x_46re) * y_46im))
end function

Java

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

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * exp(Float64(atan(x_46_im, x_46_re) * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (y_46_re * atan2(x_46_im, x_46_re)) * exp((atan2(x_46_im, x_46_re) * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 39.9%

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

2. Taylor expanded in y.im around 0 51.7%

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

3. Taylor expanded in y.re around 0 38.3%

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

   1. *-commutative 38.3%

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

   2. distribute-lft-neg-in 38.3%

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

   3. *-commutative 38.3%

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

5. Simplified 38.3%

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

   1. expm1-log1p-u 18.7%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)\right)}} \]

   2. expm1-udef 18.7%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} - 1}} \]

   3. add-sqr-sqrt 10.3%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(\sqrt{-y.im} \cdot \sqrt{-y.im}\right)}\right)} - 1} \]

   4. sqrt-unprod 14.5%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\sqrt{\left(-y.im\right) \cdot \left(-y.im\right)}}\right)} - 1} \]

   5. sqr-neg 14.5%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \sqrt{\color{blue}{y.im \cdot y.im}}\right)} - 1} \]

   6. sqrt-unprod 4.7%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(\sqrt{y.im} \cdot \sqrt{y.im}\right)}\right)} - 1} \]

   7. add-sqr-sqrt 9.6%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{y.im}\right)} - 1} \]

7. Applied egg-rr 9.6%

   \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 9.6%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)}} \]

   2. expm1-log1p 13.1%

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

   3. *-commutative 13.1%

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

9. Simplified 13.1%

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

10. Final simplification 13.1%

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

Alternative 20: 14.3% accurate, 8.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (* y.re (atan2 x.im x.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return y_46_re * atan2(x_46_im, x_46_re);
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = y_46re * atan2(x_46im, x_46re)
end function

Java

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

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(y_46_re * atan(x_46_im, x_46_re))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = y_46_re * atan2(x_46_im, x_46_re);
end

Wolfram

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

Derivation

1. Initial program 39.9%

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

2. Taylor expanded in y.im around 0 51.7%

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

3. Taylor expanded in y.re around 0 38.3%

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

   1. *-commutative 38.3%

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

   2. distribute-lft-neg-in 38.3%

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

   3. *-commutative 38.3%

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

5. Simplified 38.3%

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

   1. expm1-log1p-u 18.7%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)\right)}} \]

   2. expm1-udef 18.7%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} - 1}} \]

   3. add-sqr-sqrt 10.3%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(\sqrt{-y.im} \cdot \sqrt{-y.im}\right)}\right)} - 1} \]

   4. sqrt-unprod 14.5%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\sqrt{\left(-y.im\right) \cdot \left(-y.im\right)}}\right)} - 1} \]

   5. sqr-neg 14.5%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \sqrt{\color{blue}{y.im \cdot y.im}}\right)} - 1} \]

   6. sqrt-unprod 4.7%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{\left(\sqrt{y.im} \cdot \sqrt{y.im}\right)}\right)} - 1} \]

   7. add-sqr-sqrt 9.6%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot \color{blue}{y.im}\right)} - 1} \]

7. Applied egg-rr 9.6%

   \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{e^{\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)} - 1}} \]
8. Step-by-step derivation

   1. expm1-def 9.6%

      \[\leadsto \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\right)\right)}} \]

   2. expm1-log1p 13.1%

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

   3. *-commutative 13.1%

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

9. Simplified 13.1%

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

10. Taylor expanded in y.im around 0 10.4%

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

    1. *-commutative 10.4%

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

12. Simplified 10.4%

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

13. Final simplification 10.4%

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

Reproduce

herbie shell --seed 2023188 
(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)))))