powComplex, imaginary part

Percentage Accurate: 40.4% → 70.6%

Time: 22.5s

Alternatives: 17

Speedup: 2.6×

• 35.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    t_0 = log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))
    code = exp(((t_0 * y_46re) - (atan2(x_46im, x_46re) * y_46im))) * sin(((t_0 * y_46im) + (atan2(x_46im, x_46re) * y_46re)))
end function

Java

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

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))
	return math.exp(((t_0 * y_46_re) - (math.atan2(x_46_im, x_46_re) * y_46_im))) * math.sin(((t_0 * y_46_im) + (math.atan2(x_46_im, x_46_re) * y_46_re)))

Julia

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

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))));
	tmp = exp(((t_0 * y_46_re) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(((t_0 * y_46_im) + (atan2(x_46_im, x_46_re) * y_46_re)));
end

Wolfram

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

Initial Program: 40.4% 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: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := \sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\\ t_4 := \sqrt[3]{t_0}\\ \mathbf{if}\;x.im \leq -1.62 \cdot 10^{-208}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_2} \cdot \sin \left(\mathsf{fma}\left(y.im, t_0, t_1\right)\right)\\ \mathbf{elif}\;x.im \leq 1.25 \cdot 10^{+127}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2} \cdot \sin \left(y.im \cdot \log \left({\left({\left(e^{\sqrt[3]{{t_3}^{4}}}\right)}^{\left(\sqrt[3]{{t_3}^{2}}\right)}\right)}^{t_3}\right) + t_1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(t_4 \cdot {t_4}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (cbrt (log (hypot x.re x.im))))
        (t_4 (cbrt t_0)))
   (if (<= x.im -1.62e-208)
     (* (exp (- (* (log (- x.im)) y.re) t_2)) (sin (fma y.im t_0 t_1)))
     (if (<= x.im 1.25e+127)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))
        (sin
         (+
          (*
           y.im
           (log
            (pow (pow (exp (cbrt (pow t_3 4.0))) (cbrt (pow t_3 2.0))) t_3)))
          t_1)))
       (*
        (exp (- (* y.re (log x.im)) t_2))
        (sin (fma y.re (atan2 x.im x.re) (* y.im (* t_4 (pow t_4 2.0))))))))))

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_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 = cbrt(log(hypot(x_46_re, x_46_im)));
	double t_4 = cbrt(t_0);
	double tmp;
	if (x_46_im <= -1.62e-208) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, t_1));
	} else if (x_46_im <= 1.25e+127) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * sin(((y_46_im * log(pow(pow(exp(cbrt(pow(t_3, 4.0))), cbrt(pow(t_3, 2.0))), t_3))) + t_1));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * (t_4 * pow(t_4, 2.0)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(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 = cbrt(log(hypot(x_46_re, x_46_im)))
	t_4 = cbrt(t_0)
	tmp = 0.0
	if (x_46_im <= -1.62e-208)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, t_1)));
	elseif (x_46_im <= 1.25e+127)
		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(Float64(Float64(y_46_im * log(((exp(cbrt((t_3 ^ 4.0))) ^ cbrt((t_3 ^ 2.0))) ^ t_3))) + t_1)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * Float64(t_4 * (t_4 ^ 2.0))))));
	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$46$im ^ 2 + x$46$re ^ 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]}, Block[{t$95$3 = N[Power[N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$4 = N[Power[t$95$0, 1/3], $MachinePrecision]}, If[LessEqual[x$46$im, -1.62e-208], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.25e+127], 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[N[(N[(y$46$im * N[Log[N[Power[N[Power[N[Exp[N[Power[N[Power[t$95$3, 4.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], N[Power[N[Power[t$95$3, 2.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision], t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * N[(t$95$4 * N[Power[t$95$4, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.6200000000000001e-208

   1. Initial program 37.8%

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

   2. Taylor expanded in x.im around -inf 36.1%

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

      1. mul-1-neg 36.1%

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

   4. Simplified 36.1%

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

   5. Taylor expanded in y.im around inf 36.1%

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

      1. fma-def 36.1%

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

      2. unpow2 36.1%

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

      3. unpow2 36.1%

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

      4. hypot-def 77.5%

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

   7. Simplified 77.5%

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

   if -1.6200000000000001e-208 < x.im < 1.2500000000000001e127

   1. Initial program 55.8%

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

      1. add-exp-log 55.8%

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

      2. add-cube-cbrt 58.5%

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

      3. exp-prod 58.5%

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

      4. pow2 58.5%

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

      5. hypot-def 58.5%

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

      6. hypot-def 77.5%

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

   3. Applied egg-rr 77.5%

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

      1. add-cube-cbrt 75.5%

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

      2. exp-prod 75.4%

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

      3. cbrt-unprod 78.3%

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

      4. pow-prod-up 78.5%

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

      5. metadata-eval 78.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left({\left({\left(e^{\sqrt[3]{{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{\color{blue}{4}}}}\right)}^{\left(\sqrt[3]{{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   5. Applied egg-rr 78.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\log \left({\color{blue}{\left({\left(e^{\sqrt[3]{{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{4}}}\right)}^{\left(\sqrt[3]{{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{2}}\right)}\right)}}^{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}\right) \cdot y.im + \tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \]

   if 1.2500000000000001e127 < x.im

   1. Initial program 10.5%

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

      1. add-exp-log 10.5%

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

      2. add-cube-cbrt 13.2%

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

      3. exp-prod 13.2%

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

      4. pow2 13.2%

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

      5. hypot-def 13.2%

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

      6. hypot-def 56.3%

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

   3. Applied egg-rr 56.3%

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

   4. Taylor expanded in y.im around inf 53.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)} \]
   5. Step-by-step derivation

      1. fma-def 53.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)\right)} \]

      2. *-commutative 53.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.im \cdot \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right)}\right)\right) \]

      3. log-pow 53.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \color{blue}{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}\right)\right) \]

      4. unpow1/3 51.0%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\color{blue}{\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      5. hypot-def 10.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      6. unpow2 10.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      7. unpow2 10.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      8. +-commutative 10.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      9. unpow2 10.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      10. unpow2 10.5%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      11. hypot-def 51.0%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

   6. Simplified 56.3%

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

   7. Taylor expanded in x.re around 0 91.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.62 \cdot 10^{-208}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 1.25 \cdot 10^{+127}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left({\left({\left(e^{\sqrt[3]{{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{4}}}\right)}^{\left(\sqrt[3]{{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}^{2}}\right)}\right)}^{\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}\right)}\right) + y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 2: 71.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sqrt[3]{t_0}\\ t_2 := \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(t_1 \cdot {t_1}^{2}\right)\right)\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -3.2 \cdot 10^{-209}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_3} \cdot \sin \left(\mathsf{fma}\left(y.im, t_0, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 2.05 \cdot 10^{-16}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_3} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_3} \cdot t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (cbrt t_0))
        (t_2 (sin (fma y.re (atan2 x.im x.re) (* y.im (* t_1 (pow t_1 2.0))))))
        (t_3 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -3.2e-209)
     (*
      (exp (- (* (log (- x.im)) y.re) t_3))
      (sin (fma y.im t_0 (* y.re (atan2 x.im x.re)))))
     (if (<= x.im 2.05e-16)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_3))
        t_2)
       (* (exp (- (* y.re (log x.im)) t_3)) t_2)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_im, x_46_re));
	double t_1 = cbrt(t_0);
	double t_2 = sin(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * (t_1 * pow(t_1, 2.0)))));
	double t_3 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -3.2e-209) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_3)) * sin(fma(y_46_im, t_0, (y_46_re * atan2(x_46_im, x_46_re))));
	} else if (x_46_im <= 2.05e-16) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3)) * t_2;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_3)) * t_2;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = cbrt(t_0)
	t_2 = sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * Float64(t_1 * (t_1 ^ 2.0)))))
	t_3 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -3.2e-209)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_3)) * sin(fma(y_46_im, t_0, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	elseif (x_46_im <= 2.05e-16)
		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)) * t_2);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_3)) * t_2);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -3.2e-209], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$0 + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.05e-16], 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] * t$95$2), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.2000000000000001e-209

   1. Initial program 37.8%

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

   2. Taylor expanded in x.im around -inf 36.1%

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

      1. mul-1-neg 36.1%

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

   4. Simplified 36.1%

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

   5. Taylor expanded in y.im around inf 36.1%

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

      1. fma-def 36.1%

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

      2. unpow2 36.1%

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

      3. unpow2 36.1%

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

      4. hypot-def 77.5%

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

   7. Simplified 77.5%

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

   if -3.2000000000000001e-209 < x.im < 2.05000000000000003e-16

   1. Initial program 54.4%

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

      1. add-exp-log 54.4%

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

      2. add-cube-cbrt 56.7%

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

      3. exp-prod 56.7%

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

      4. pow2 56.7%

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

      5. hypot-def 56.8%

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

      6. hypot-def 78.7%

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

   3. Applied egg-rr 78.7%

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

   4. Taylor expanded in y.im around inf 45.7%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)} \]
   5. Step-by-step derivation

      1. fma-def 45.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)\right)} \]

      2. *-commutative 45.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.im \cdot \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right)}\right)\right) \]

      3. log-pow 45.7%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \color{blue}{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}\right)\right) \]

      4. unpow1/3 45.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\color{blue}{\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      5. hypot-def 28.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      6. unpow2 28.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      7. unpow2 28.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      8. +-commutative 28.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      9. unpow2 28.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      10. unpow2 28.3%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      11. hypot-def 45.8%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

   6. Simplified 78.7%

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

   if 2.05000000000000003e-16 < x.im

   1. Initial program 29.4%

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

      1. add-exp-log 29.4%

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

      2. add-cube-cbrt 32.5%

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

      3. exp-prod 32.5%

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

      4. pow2 32.5%

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

      5. hypot-def 32.5%

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

      6. hypot-def 62.7%

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

   3. Applied egg-rr 62.7%

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

   4. Taylor expanded in y.im around inf 61.1%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)} \]
   5. Step-by-step derivation

      1. fma-def 61.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)\right)} \]

      2. *-commutative 61.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.im \cdot \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right)}\right)\right) \]

      3. log-pow 61.1%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \color{blue}{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}\right)\right) \]

      4. unpow1/3 59.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\color{blue}{\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      5. hypot-def 29.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      6. unpow2 29.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      7. unpow2 29.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      8. +-commutative 29.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      9. unpow2 29.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      10. unpow2 29.3%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      11. hypot-def 59.5%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

   6. Simplified 62.7%

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

   7. Taylor expanded in x.re around 0 84.8%

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

4. Final simplification 79.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.2 \cdot 10^{-209}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 2.05 \cdot 10^{-16}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 3: 71.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := \sqrt[3]{t_0}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -3.75 \cdot 10^{-208}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_2} \cdot \sin \left(\mathsf{fma}\left(y.im, t_0, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 1.2 \cdot 10^{+74}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(t_1 \cdot {t_1}^{2}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (cbrt t_0))
        (t_2 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -3.75e-208)
     (*
      (exp (- (* (log (- x.im)) y.re) t_2))
      (sin (fma y.im t_0 (* y.re (atan2 x.im x.re)))))
     (if (<= x.im 1.2e+74)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))
        (sin (fma y.re (atan2 x.im x.re) (* y.im t_0))))
       (*
        (exp (- (* y.re (log x.im)) t_2))
        (sin (fma y.re (atan2 x.im x.re) (* y.im (* t_1 (pow t_1 2.0))))))))))

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_im, x_46_re));
	double t_1 = cbrt(t_0);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -3.75e-208) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, (y_46_re * atan2(x_46_im, x_46_re))));
	} else if (x_46_im <= 1.2e+74) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * t_0)));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * (t_1 * pow(t_1, 2.0)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = cbrt(t_0)
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -3.75e-208)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	elseif (x_46_im <= 1.2e+74)
		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(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * t_0))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * Float64(t_1 * (t_1 ^ 2.0))))));
	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$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[t$95$0, 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -3.75e-208], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$0 + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.2e+74], 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[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * N[(t$95$1 * N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.7499999999999999e-208

   1. Initial program 37.8%

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

   2. Taylor expanded in x.im around -inf 36.1%

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

      1. mul-1-neg 36.1%

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

   4. Simplified 36.1%

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

   5. Taylor expanded in y.im around inf 36.1%

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

      1. fma-def 36.1%

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

      2. unpow2 36.1%

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

      3. unpow2 36.1%

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

      4. hypot-def 77.5%

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

   7. Simplified 77.5%

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

   if -3.7499999999999999e-208 < x.im < 1.20000000000000004e74

   1. Initial program 53.3%

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

      1. add-exp-log 53.3%

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

      2. add-cube-cbrt 55.3%

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

      3. exp-prod 55.3%

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

      4. pow2 55.3%

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

      5. hypot-def 55.3%

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

      6. hypot-def 76.8%

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

   3. Applied egg-rr 76.8%

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

   4. Taylor expanded in y.im around inf 47.9%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)} \]
   5. Step-by-step derivation

      1. fma-def 47.9%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)\right)} \]

      2. *-commutative 47.9%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.im \cdot \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right)}\right)\right) \]

      3. log-pow 47.9%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \color{blue}{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}\right)\right) \]

      4. unpow1/3 46.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\color{blue}{\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      5. hypot-def 28.2%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      6. unpow2 28.2%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      7. unpow2 28.2%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      8. +-commutative 28.2%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      9. unpow2 28.2%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      10. unpow2 28.2%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      11. hypot-def 46.8%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

   6. Simplified 76.8%

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

   7. Taylor expanded in y.re around inf 74.8%

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

   if 1.20000000000000004e74 < x.im

   1. Initial program 25.9%

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

      1. add-exp-log 25.9%

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

      2. add-cube-cbrt 29.9%

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

      3. exp-prod 29.9%

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

      4. pow2 29.9%

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

      5. hypot-def 29.9%

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

      6. hypot-def 62.7%

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

   3. Applied egg-rr 62.7%

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

   4. Taylor expanded in y.im around inf 60.5%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)} \]
   5. Step-by-step derivation

      1. fma-def 60.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)\right)} \]

      2. *-commutative 60.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.im \cdot \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right)}\right)\right) \]

      3. log-pow 60.5%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \color{blue}{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}\right)\right) \]

      4. unpow1/3 60.6%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\color{blue}{\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      5. hypot-def 29.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      6. unpow2 29.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      7. unpow2 29.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      8. +-commutative 29.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      9. unpow2 29.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      10. unpow2 29.8%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      11. hypot-def 60.6%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

   6. Simplified 62.7%

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

   7. Taylor expanded in x.re around 0 89.6%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.75 \cdot 10^{-208}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 1.2 \cdot 10^{+74}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot {\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{2}\right)\right)\right)\\ \end{array} \]

Alternative 4: 69.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -3.7 \cdot 10^{-209}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_2} \cdot \sin \left(\mathsf{fma}\left(y.im, t_0, t_1\right)\right)\\ \mathbf{elif}\;x.im \leq 2.1 \cdot 10^{+126}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot \sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -3.7e-209)
     (* (exp (- (* (log (- x.im)) y.re) t_2)) (sin (fma y.im t_0 t_1)))
     (if (<= x.im 2.1e+126)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))
        (sin (fma y.re (atan2 x.im x.re) (* y.im t_0))))
       (* (exp (- (* y.re (log x.im)) t_2)) (sin (pow (cbrt t_1) 3.0)))))))

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_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 tmp;
	if (x_46_im <= -3.7e-209) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, t_1));
	} else if (x_46_im <= 2.1e+126) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * sin(fma(y_46_re, atan2(x_46_im, x_46_re), (y_46_im * t_0)));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(pow(cbrt(t_1), 3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(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 (x_46_im <= -3.7e-209)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, t_1)));
	elseif (x_46_im <= 2.1e+126)
		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(fma(y_46_re, atan(x_46_im, x_46_re), Float64(y_46_im * t_0))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin((cbrt(t_1) ^ 3.0)));
	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$46$im ^ 2 + x$46$re ^ 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[x$46$im, -3.7e-209], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.1e+126], 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[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -3.6999999999999998e-209

   1. Initial program 37.8%

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

   2. Taylor expanded in x.im around -inf 36.1%

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

      1. mul-1-neg 36.1%

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

   4. Simplified 36.1%

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

   5. Taylor expanded in y.im around inf 36.1%

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

      1. fma-def 36.1%

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

      2. unpow2 36.1%

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

      3. unpow2 36.1%

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

      4. hypot-def 77.5%

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

   7. Simplified 77.5%

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

   if -3.6999999999999998e-209 < x.im < 2.0999999999999999e126

   1. Initial program 56.3%

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

      1. add-exp-log 56.3%

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

      2. add-cube-cbrt 59.0%

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

      3. exp-prod 59.0%

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

      4. pow2 59.0%

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

      5. hypot-def 59.1%

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

      6. hypot-def 78.3%

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

   3. Applied egg-rr 78.3%

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

   4. Taylor expanded in y.im around inf 52.3%

      \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \color{blue}{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)} \]
   5. Step-by-step derivation

      1. fma-def 52.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right) \cdot y.im\right)\right)} \]

      2. *-commutative 52.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, \color{blue}{y.im \cdot \log \left({\left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)}^{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}\right)}\right)\right) \]

      3. log-pow 52.3%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \color{blue}{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)}\right)\right) \]

      4. unpow1/3 51.4%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\color{blue}{\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      5. hypot-def 34.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      6. unpow2 34.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.re}^{2}} + x.im \cdot x.im}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      7. unpow2 34.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{{x.re}^{2} + \color{blue}{{x.im}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      8. +-commutative 34.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{{x.im}^{2} + {x.re}^{2}}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      9. unpow2 34.8%

         \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{\color{blue}{x.im \cdot x.im} + {x.re}^{2}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      10. unpow2 34.8%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \left(\sqrt{x.im \cdot x.im + \color{blue}{x.re \cdot x.re}}\right)} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

      11. hypot-def 51.4%

          \[\leadsto e^{\log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \left(\sqrt[3]{\log \color{blue}{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}} \cdot \log \left(e^{{\left({\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{0.3333333333333333}\right)}^{2}}\right)\right)\right)\right) \]

   6. Simplified 78.3%

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

   7. Taylor expanded in y.re around inf 75.5%

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

   if 2.0999999999999999e126 < x.im

   1. Initial program 10.3%

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

   2. Taylor expanded in y.im around 0 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 x.re around 0 75.1%

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

      1. *-commutative 75.1%

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

      2. add-cube-cbrt 77.6%

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

      3. pow3 80.2%

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

   5. Applied egg-rr 80.2%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3.7 \cdot 10^{-209}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 2.1 \cdot 10^{+126}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.re, \tan^{-1}_* \frac{x.im}{x.re}, y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \end{array} \]

Alternative 5: 66.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -4.4 \cdot 10^{-209}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_2} \cdot \sin \left(\mathsf{fma}\left(y.im, t_0, t_1\right)\right)\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{+78}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2} \cdot \sin \left(y.im \cdot t_0\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot \sin \left({\left(\sqrt[3]{t_1}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re)))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -4.4e-209)
     (* (exp (- (* (log (- x.im)) y.re) t_2)) (sin (fma y.im t_0 t_1)))
     (if (<= x.im 2.6e+78)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))
        (sin (* y.im t_0)))
       (* (exp (- (* y.re (log x.im)) t_2)) (sin (pow (cbrt t_1) 3.0)))))))

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_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 tmp;
	if (x_46_im <= -4.4e-209) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, t_1));
	} else if (x_46_im <= 2.6e+78) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * sin((y_46_im * t_0));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(pow(cbrt(t_1), 3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_im, x_46_re))
	t_1 = Float64(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 (x_46_im <= -4.4e-209)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_2)) * sin(fma(y_46_im, t_0, t_1)));
	elseif (x_46_im <= 2.6e+78)
		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(Float64(y_46_im * t_0)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin((cbrt(t_1) ^ 3.0)));
	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$46$im ^ 2 + x$46$re ^ 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[x$46$im, -4.4e-209], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * t$95$0 + t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 2.6e+78], 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[N[(y$46$im * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$1, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -4.40000000000000019e-209

   1. Initial program 37.8%

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

   2. Taylor expanded in x.im around -inf 36.1%

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

      1. mul-1-neg 36.1%

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

   4. Simplified 36.1%

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

   5. Taylor expanded in y.im around inf 36.1%

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

      1. fma-def 36.1%

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

      2. unpow2 36.1%

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

      3. unpow2 36.1%

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

      4. hypot-def 77.5%

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

   7. Simplified 77.5%

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

   if -4.40000000000000019e-209 < x.im < 2.6e78

   1. Initial program 53.8%

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

   2. Taylor expanded in y.re around 0 48.8%

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

      1. unpow2 6.7%

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

      2. unpow2 6.7%

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

      3. hypot-def 8.0%

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

   4. Simplified 69.0%

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

   if 2.6e78 < x.im

   1. Initial program 24.4%

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

   2. Taylor expanded in y.im around 0 60.1%

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

   3. Taylor expanded in x.re around 0 74.2%

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

      1. *-commutative 74.2%

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

      2. add-cube-cbrt 76.3%

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

      3. pow3 78.3%

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

   5. Applied egg-rr 78.3%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.4 \cdot 10^{-209}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{elif}\;x.im \leq 2.6 \cdot 10^{+78}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \end{array} \]

Alternative 6: 65.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-x.im\right)\\ t_2 := \sin \left(t_0 + t_1 \cdot y.im\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_4 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_3}\\ \mathbf{if}\;x.im \leq -51000000000:\\ \;\;\;\;e^{t_1 \cdot y.re - t_3} \cdot t_2\\ \mathbf{elif}\;x.im \leq -6.8 \cdot 10^{-90}:\\ \;\;\;\;t_4 \cdot t_0\\ \mathbf{elif}\;x.im \leq -5.9 \cdot 10^{-180}:\\ \;\;\;\;t_4 \cdot t_2\\ \mathbf{elif}\;x.im \leq 1.02 \cdot 10^{+85}:\\ \;\;\;\;t_4 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_3} \cdot \sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (- x.im)))
        (t_2 (sin (+ t_0 (* t_1 y.im))))
        (t_3 (* (atan2 x.im x.re) y.im))
        (t_4
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_3))))
   (if (<= x.im -51000000000.0)
     (* (exp (- (* t_1 y.re) t_3)) t_2)
     (if (<= x.im -6.8e-90)
       (* t_4 t_0)
       (if (<= x.im -5.9e-180)
         (* t_4 t_2)
         (if (<= x.im 1.02e+85)
           (* t_4 (sin (* y.im (log (hypot x.im x.re)))))
           (*
            (exp (- (* y.re (log x.im)) t_3))
            (sin (pow (cbrt t_0) 3.0)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(-x_46_im);
	double t_2 = sin((t_0 + (t_1 * y_46_im)));
	double t_3 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_4 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3));
	double tmp;
	if (x_46_im <= -51000000000.0) {
		tmp = exp(((t_1 * y_46_re) - t_3)) * t_2;
	} else if (x_46_im <= -6.8e-90) {
		tmp = t_4 * t_0;
	} else if (x_46_im <= -5.9e-180) {
		tmp = t_4 * t_2;
	} else if (x_46_im <= 1.02e+85) {
		tmp = t_4 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_3)) * sin(pow(cbrt(t_0), 3.0));
	}
	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 t_1 = Math.log(-x_46_im);
	double t_2 = Math.sin((t_0 + (t_1 * y_46_im)));
	double t_3 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_4 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3));
	double tmp;
	if (x_46_im <= -51000000000.0) {
		tmp = Math.exp(((t_1 * y_46_re) - t_3)) * t_2;
	} else if (x_46_im <= -6.8e-90) {
		tmp = t_4 * t_0;
	} else if (x_46_im <= -5.9e-180) {
		tmp = t_4 * t_2;
	} else if (x_46_im <= 1.02e+85) {
		tmp = t_4 * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_3)) * Math.sin(Math.pow(Math.cbrt(t_0), 3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-x_46_im))
	t_2 = sin(Float64(t_0 + Float64(t_1 * y_46_im)))
	t_3 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_4 = 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))
	tmp = 0.0
	if (x_46_im <= -51000000000.0)
		tmp = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_3)) * t_2);
	elseif (x_46_im <= -6.8e-90)
		tmp = Float64(t_4 * t_0);
	elseif (x_46_im <= -5.9e-180)
		tmp = Float64(t_4 * t_2);
	elseif (x_46_im <= 1.02e+85)
		tmp = Float64(t_4 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_3)) * sin((cbrt(t_0) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(t$95$0 + N[(t$95$1 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[x$46$im, -51000000000.0], N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[x$46$im, -6.8e-90], N[(t$95$4 * t$95$0), $MachinePrecision], If[LessEqual[x$46$im, -5.9e-180], N[(t$95$4 * t$95$2), $MachinePrecision], If[LessEqual[x$46$im, 1.02e+85], N[(t$95$4 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if x.im < -5.1e10

   1. Initial program 31.2%

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

   2. Taylor expanded in x.im around -inf 31.2%

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

      1. mul-1-neg 31.2%

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

   4. Simplified 31.2%

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

   5. Taylor expanded in x.im around -inf 86.5%

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

      1. mul-1-neg 31.2%

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

   7. Simplified 86.5%

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

   if -5.1e10 < x.im < -6.79999999999999988e-90

   1. Initial program 42.2%

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

   2. Taylor expanded in y.im around 0 50.6%

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

   3. Taylor expanded in y.re around 0 54.4%

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

   if -6.79999999999999988e-90 < x.im < -5.9000000000000003e-180

   1. Initial program 63.9%

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

   2. Taylor expanded in x.im around -inf 74.3%

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

      1. mul-1-neg 64.0%

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

   4. Simplified 74.3%

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

   if -5.9000000000000003e-180 < x.im < 1.02e85

   1. Initial program 51.1%

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

   2. Taylor expanded in y.re around 0 46.3%

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

      1. unpow2 6.3%

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

      2. unpow2 6.3%

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

      3. hypot-def 10.6%

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

   4. Simplified 67.6%

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

   if 1.02e85 < x.im

   1. Initial program 24.4%

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

   2. Taylor expanded in y.im around 0 60.1%

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

   3. Taylor expanded in x.re around 0 74.2%

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

      1. *-commutative 74.2%

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

      2. add-cube-cbrt 76.3%

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

      3. pow3 78.3%

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

   5. Applied egg-rr 78.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -51000000000:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq -6.8 \cdot 10^{-90}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -5.9 \cdot 10^{-180}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 1.02 \cdot 10^{+85}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \end{array} \]

Alternative 7: 64.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-x.im\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := e^{t_1 \cdot y.re - t_2} \cdot \sin \left(t_0 + t_1 \cdot y.im\right)\\ t_4 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2}\\ \mathbf{if}\;x.im \leq -22500000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq -3.2 \cdot 10^{-93}:\\ \;\;\;\;t_4 \cdot t_0\\ \mathbf{elif}\;x.im \leq -2.26 \cdot 10^{-181}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq 2.25 \cdot 10^{+84}:\\ \;\;\;\;t_4 \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot \sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (- x.im)))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (* (exp (- (* t_1 y.re) t_2)) (sin (+ t_0 (* t_1 y.im)))))
        (t_4
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))))
   (if (<= x.im -22500000000.0)
     t_3
     (if (<= x.im -3.2e-93)
       (* t_4 t_0)
       (if (<= x.im -2.26e-181)
         t_3
         (if (<= x.im 2.25e+84)
           (* t_4 (sin (* y.im (log (hypot x.im x.re)))))
           (*
            (exp (- (* y.re (log x.im)) t_2))
            (sin (pow (cbrt t_0) 3.0)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(-x_46_im);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = exp(((t_1 * y_46_re) - t_2)) * sin((t_0 + (t_1 * y_46_im)));
	double t_4 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2));
	double tmp;
	if (x_46_im <= -22500000000.0) {
		tmp = t_3;
	} else if (x_46_im <= -3.2e-93) {
		tmp = t_4 * t_0;
	} else if (x_46_im <= -2.26e-181) {
		tmp = t_3;
	} else if (x_46_im <= 2.25e+84) {
		tmp = t_4 * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(pow(cbrt(t_0), 3.0));
	}
	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 t_1 = Math.log(-x_46_im);
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = Math.exp(((t_1 * y_46_re) - t_2)) * Math.sin((t_0 + (t_1 * y_46_im)));
	double t_4 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2));
	double tmp;
	if (x_46_im <= -22500000000.0) {
		tmp = t_3;
	} else if (x_46_im <= -3.2e-93) {
		tmp = t_4 * t_0;
	} else if (x_46_im <= -2.26e-181) {
		tmp = t_3;
	} else if (x_46_im <= 2.25e+84) {
		tmp = t_4 * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_2)) * Math.sin(Math.pow(Math.cbrt(t_0), 3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-x_46_im))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_3 = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_2)) * sin(Float64(t_0 + Float64(t_1 * y_46_im))))
	t_4 = 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))
	tmp = 0.0
	if (x_46_im <= -22500000000.0)
		tmp = t_3;
	elseif (x_46_im <= -3.2e-93)
		tmp = Float64(t_4 * t_0);
	elseif (x_46_im <= -2.26e-181)
		tmp = t_3;
	elseif (x_46_im <= 2.25e+84)
		tmp = Float64(t_4 * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin((cbrt(t_0) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 + N[(t$95$1 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[x$46$im, -22500000000.0], t$95$3, If[LessEqual[x$46$im, -3.2e-93], N[(t$95$4 * t$95$0), $MachinePrecision], If[LessEqual[x$46$im, -2.26e-181], t$95$3, If[LessEqual[x$46$im, 2.25e+84], N[(t$95$4 * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -2.25e10 or -3.1999999999999999e-93 < x.im < -2.25999999999999993e-181

   1. Initial program 38.7%

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

   2. Taylor expanded in x.im around -inf 38.7%

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

      1. mul-1-neg 38.7%

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

   4. Simplified 38.7%

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

   5. Taylor expanded in x.im around -inf 83.7%

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

      1. mul-1-neg 38.7%

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

   7. Simplified 83.7%

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

   if -2.25e10 < x.im < -3.1999999999999999e-93

   1. Initial program 42.2%

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

   2. Taylor expanded in y.im around 0 50.6%

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

   3. Taylor expanded in y.re around 0 54.4%

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

   if -2.25999999999999993e-181 < x.im < 2.2499999999999999e84

   1. Initial program 51.1%

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

   2. Taylor expanded in y.re around 0 46.3%

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

      1. unpow2 6.3%

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

      2. unpow2 6.3%

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

      3. hypot-def 10.6%

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

   4. Simplified 67.6%

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

   if 2.2499999999999999e84 < x.im

   1. Initial program 24.4%

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

   2. Taylor expanded in y.im around 0 60.1%

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

   3. Taylor expanded in x.re around 0 74.2%

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

      1. *-commutative 74.2%

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

      2. add-cube-cbrt 76.3%

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

      3. pow3 78.3%

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

   5. Applied egg-rr 78.3%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -22500000000:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq -3.2 \cdot 10^{-93}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -2.26 \cdot 10^{-181}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 2.25 \cdot 10^{+84}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \end{array} \]

Alternative 8: 62.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-x.im\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := e^{t_1 \cdot y.re - t_2} \cdot \sin \left(t_0 + t_1 \cdot y.im\right)\\ t_4 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2}\\ \mathbf{if}\;x.im \leq -122000000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq -3.8 \cdot 10^{-90}:\\ \;\;\;\;t_4 \cdot t_0\\ \mathbf{elif}\;x.im \leq -1.45 \cdot 10^{-152}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq 6.2 \cdot 10^{-28}:\\ \;\;\;\;t_4 \cdot \sin t_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot \sin \left({\left(\sqrt[3]{t_0}\right)}^{3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (- x.im)))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (* (exp (- (* t_1 y.re) t_2)) (sin (+ t_0 (* t_1 y.im)))))
        (t_4
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))))
   (if (<= x.im -122000000000.0)
     t_3
     (if (<= x.im -3.8e-90)
       (* t_4 t_0)
       (if (<= x.im -1.45e-152)
         t_3
         (if (<= x.im 6.2e-28)
           (* t_4 (sin t_0))
           (*
            (exp (- (* y.re (log x.im)) t_2))
            (sin (pow (cbrt t_0) 3.0)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(-x_46_im);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = exp(((t_1 * y_46_re) - t_2)) * sin((t_0 + (t_1 * y_46_im)));
	double t_4 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2));
	double tmp;
	if (x_46_im <= -122000000000.0) {
		tmp = t_3;
	} else if (x_46_im <= -3.8e-90) {
		tmp = t_4 * t_0;
	} else if (x_46_im <= -1.45e-152) {
		tmp = t_3;
	} else if (x_46_im <= 6.2e-28) {
		tmp = t_4 * sin(t_0);
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(pow(cbrt(t_0), 3.0));
	}
	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 t_1 = Math.log(-x_46_im);
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = Math.exp(((t_1 * y_46_re) - t_2)) * Math.sin((t_0 + (t_1 * y_46_im)));
	double t_4 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2));
	double tmp;
	if (x_46_im <= -122000000000.0) {
		tmp = t_3;
	} else if (x_46_im <= -3.8e-90) {
		tmp = t_4 * t_0;
	} else if (x_46_im <= -1.45e-152) {
		tmp = t_3;
	} else if (x_46_im <= 6.2e-28) {
		tmp = t_4 * Math.sin(t_0);
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_2)) * Math.sin(Math.pow(Math.cbrt(t_0), 3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-x_46_im))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_3 = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_2)) * sin(Float64(t_0 + Float64(t_1 * y_46_im))))
	t_4 = 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))
	tmp = 0.0
	if (x_46_im <= -122000000000.0)
		tmp = t_3;
	elseif (x_46_im <= -3.8e-90)
		tmp = Float64(t_4 * t_0);
	elseif (x_46_im <= -1.45e-152)
		tmp = t_3;
	elseif (x_46_im <= 6.2e-28)
		tmp = Float64(t_4 * sin(t_0));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin((cbrt(t_0) ^ 3.0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 + N[(t$95$1 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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]}, If[LessEqual[x$46$im, -122000000000.0], t$95$3, If[LessEqual[x$46$im, -3.8e-90], N[(t$95$4 * t$95$0), $MachinePrecision], If[LessEqual[x$46$im, -1.45e-152], t$95$3, If[LessEqual[x$46$im, 6.2e-28], N[(t$95$4 * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[Power[N[Power[t$95$0, 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -1.22e11 or -3.8e-90 < x.im < -1.4500000000000001e-152

   1. Initial program 40.7%

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

   2. Taylor expanded in x.im around -inf 40.7%

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

      1. mul-1-neg 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in x.im around -inf 85.4%

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

      1. mul-1-neg 40.7%

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

   7. Simplified 85.4%

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

   if -1.22e11 < x.im < -3.8e-90

   1. Initial program 42.2%

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

   2. Taylor expanded in y.im around 0 50.6%

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

   3. Taylor expanded in y.re around 0 54.4%

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

   if -1.4500000000000001e-152 < x.im < 6.19999999999999984e-28

   1. Initial program 49.0%

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

   2. Taylor expanded in y.im around 0 58.7%

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

   if 6.19999999999999984e-28 < x.im

   1. Initial program 30.1%

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

   2. Taylor expanded in y.im around 0 56.6%

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

   3. Taylor expanded in x.re around 0 67.6%

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

      1. *-commutative 67.6%

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

      2. add-cube-cbrt 69.1%

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

      3. pow3 70.7%

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

   5. Applied egg-rr 70.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -122000000000:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq -3.8 \cdot 10^{-90}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -1.45 \cdot 10^{-152}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 6.2 \cdot 10^{-28}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \end{array} \]

Alternative 9: 62.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ t_2 := \log \left(-x.im\right)\\ t_3 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_4 := e^{t_2 \cdot y.re - t_3} \cdot \sin \left(t_0 + t_2 \cdot y.im\right)\\ t_5 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_3}\\ \mathbf{if}\;x.im \leq -51000000000:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.im \leq -2.2 \cdot 10^{-90}:\\ \;\;\;\;t_5 \cdot t_0\\ \mathbf{elif}\;x.im \leq -1.25 \cdot 10^{-149}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.im \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;t_5 \cdot t_1\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_3} \cdot t_1\\ \end{array} \end{array} \]

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 (log (- x.im)))
        (t_3 (* (atan2 x.im x.re) y.im))
        (t_4 (* (exp (- (* t_2 y.re) t_3)) (sin (+ t_0 (* t_2 y.im)))))
        (t_5
         (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_3))))
   (if (<= x.im -51000000000.0)
     t_4
     (if (<= x.im -2.2e-90)
       (* t_5 t_0)
       (if (<= x.im -1.25e-149)
         t_4
         (if (<= x.im 3.7e-28)
           (* t_5 t_1)
           (* (exp (- (* y.re (log x.im)) t_3)) 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 = log(-x_46_im);
	double t_3 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_4 = exp(((t_2 * y_46_re) - t_3)) * sin((t_0 + (t_2 * y_46_im)));
	double t_5 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3));
	double tmp;
	if (x_46_im <= -51000000000.0) {
		tmp = t_4;
	} else if (x_46_im <= -2.2e-90) {
		tmp = t_5 * t_0;
	} else if (x_46_im <= -1.25e-149) {
		tmp = t_4;
	} else if (x_46_im <= 3.7e-28) {
		tmp = t_5 * t_1;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_3)) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    t_1 = sin(t_0)
    t_2 = log(-x_46im)
    t_3 = atan2(x_46im, x_46re) * y_46im
    t_4 = exp(((t_2 * y_46re) - t_3)) * sin((t_0 + (t_2 * y_46im)))
    t_5 = exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - t_3))
    if (x_46im <= (-51000000000.0d0)) then
        tmp = t_4
    else if (x_46im <= (-2.2d-90)) then
        tmp = t_5 * t_0
    else if (x_46im <= (-1.25d-149)) then
        tmp = t_4
    else if (x_46im <= 3.7d-28) then
        tmp = t_5 * t_1
    else
        tmp = exp(((y_46re * log(x_46im)) - t_3)) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.log(-x_46_im);
	double t_3 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_4 = Math.exp(((t_2 * y_46_re) - t_3)) * Math.sin((t_0 + (t_2 * y_46_im)));
	double t_5 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3));
	double tmp;
	if (x_46_im <= -51000000000.0) {
		tmp = t_4;
	} else if (x_46_im <= -2.2e-90) {
		tmp = t_5 * t_0;
	} else if (x_46_im <= -1.25e-149) {
		tmp = t_4;
	} else if (x_46_im <= 3.7e-28) {
		tmp = t_5 * t_1;
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_3)) * 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.log(-x_46_im)
	t_3 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_4 = math.exp(((t_2 * y_46_re) - t_3)) * math.sin((t_0 + (t_2 * y_46_im)))
	t_5 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3))
	tmp = 0
	if x_46_im <= -51000000000.0:
		tmp = t_4
	elif x_46_im <= -2.2e-90:
		tmp = t_5 * t_0
	elif x_46_im <= -1.25e-149:
		tmp = t_4
	elif x_46_im <= 3.7e-28:
		tmp = t_5 * t_1
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_3)) * 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 = log(Float64(-x_46_im))
	t_3 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_4 = Float64(exp(Float64(Float64(t_2 * y_46_re) - t_3)) * sin(Float64(t_0 + Float64(t_2 * y_46_im))))
	t_5 = 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))
	tmp = 0.0
	if (x_46_im <= -51000000000.0)
		tmp = t_4;
	elseif (x_46_im <= -2.2e-90)
		tmp = Float64(t_5 * t_0);
	elseif (x_46_im <= -1.25e-149)
		tmp = t_4;
	elseif (x_46_im <= 3.7e-28)
		tmp = Float64(t_5 * t_1);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_3)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	t_2 = log(-x_46_im);
	t_3 = atan2(x_46_im, x_46_re) * y_46_im;
	t_4 = exp(((t_2 * y_46_re) - t_3)) * sin((t_0 + (t_2 * y_46_im)));
	t_5 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_3));
	tmp = 0.0;
	if (x_46_im <= -51000000000.0)
		tmp = t_4;
	elseif (x_46_im <= -2.2e-90)
		tmp = t_5 * t_0;
	elseif (x_46_im <= -1.25e-149)
		tmp = t_4;
	elseif (x_46_im <= 3.7e-28)
		tmp = t_5 * t_1;
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_3)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$3 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$4 = N[(N[Exp[N[(N[(t$95$2 * y$46$re), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 + N[(t$95$2 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = 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]}, If[LessEqual[x$46$im, -51000000000.0], t$95$4, If[LessEqual[x$46$im, -2.2e-90], N[(t$95$5 * t$95$0), $MachinePrecision], If[LessEqual[x$46$im, -1.25e-149], t$95$4, If[LessEqual[x$46$im, 3.7e-28], N[(t$95$5 * t$95$1), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x.im < -5.1e10 or -2.19999999999999986e-90 < x.im < -1.24999999999999992e-149

   1. Initial program 40.7%

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

   2. Taylor expanded in x.im around -inf 40.7%

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

      1. mul-1-neg 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in x.im around -inf 85.4%

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

      1. mul-1-neg 40.7%

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

   7. Simplified 85.4%

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

   if -5.1e10 < x.im < -2.19999999999999986e-90

   1. Initial program 42.2%

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

   2. Taylor expanded in y.im around 0 50.6%

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

   3. Taylor expanded in y.re around 0 54.4%

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

   if -1.24999999999999992e-149 < x.im < 3.7000000000000002e-28

   1. Initial program 49.0%

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

   2. Taylor expanded in y.im around 0 58.7%

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

   if 3.7000000000000002e-28 < x.im

   1. Initial program 30.1%

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

   2. Taylor expanded in y.im around 0 56.6%

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

   3. Taylor expanded in x.re around 0 67.6%

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

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -51000000000:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq -2.2 \cdot 10^{-90}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -1.25 \cdot 10^{-149}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 3.7 \cdot 10^{-28}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 10: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-x.im\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_3 := e^{t_1 \cdot y.re - t_2} \cdot \sin \left(t_0 + t_1 \cdot y.im\right)\\ t_4 := e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2} \cdot t_0\\ \mathbf{if}\;x.im \leq -22500000000:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq -3.5 \cdot 10^{-93}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;x.im \leq -1.02 \cdot 10^{-149}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;x.im \leq 5.6 \cdot 10^{+124}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (- x.im)))
        (t_2 (* (atan2 x.im x.re) y.im))
        (t_3 (* (exp (- (* t_1 y.re) t_2)) (sin (+ t_0 (* t_1 y.im)))))
        (t_4
         (*
          (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))
          t_0)))
   (if (<= x.im -22500000000.0)
     t_3
     (if (<= x.im -3.5e-93)
       t_4
       (if (<= x.im -1.02e-149)
         t_3
         (if (<= x.im 5.6e+124)
           t_4
           (* (exp (- (* y.re (log x.im)) t_2)) (sin t_0))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(-x_46_im);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = exp(((t_1 * y_46_re) - t_2)) * sin((t_0 + (t_1 * y_46_im)));
	double t_4 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0;
	double tmp;
	if (x_46_im <= -22500000000.0) {
		tmp = t_3;
	} else if (x_46_im <= -3.5e-93) {
		tmp = t_4;
	} else if (x_46_im <= -1.02e-149) {
		tmp = t_3;
	} else if (x_46_im <= 5.6e+124) {
		tmp = t_4;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(t_0);
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    t_1 = log(-x_46im)
    t_2 = atan2(x_46im, x_46re) * y_46im
    t_3 = exp(((t_1 * y_46re) - t_2)) * sin((t_0 + (t_1 * y_46im)))
    t_4 = exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - t_2)) * t_0
    if (x_46im <= (-22500000000.0d0)) then
        tmp = t_3
    else if (x_46im <= (-3.5d-93)) then
        tmp = t_4
    else if (x_46im <= (-1.02d-149)) then
        tmp = t_3
    else if (x_46im <= 5.6d+124) then
        tmp = t_4
    else
        tmp = exp(((y_46re * log(x_46im)) - t_2)) * sin(t_0)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.log(-x_46_im);
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_3 = Math.exp(((t_1 * y_46_re) - t_2)) * Math.sin((t_0 + (t_1 * y_46_im)));
	double t_4 = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0;
	double tmp;
	if (x_46_im <= -22500000000.0) {
		tmp = t_3;
	} else if (x_46_im <= -3.5e-93) {
		tmp = t_4;
	} else if (x_46_im <= -1.02e-149) {
		tmp = t_3;
	} else if (x_46_im <= 5.6e+124) {
		tmp = t_4;
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_2)) * Math.sin(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)
	t_1 = math.log(-x_46_im)
	t_2 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_3 = math.exp(((t_1 * y_46_re) - t_2)) * math.sin((t_0 + (t_1 * y_46_im)))
	t_4 = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0
	tmp = 0
	if x_46_im <= -22500000000.0:
		tmp = t_3
	elif x_46_im <= -3.5e-93:
		tmp = t_4
	elif x_46_im <= -1.02e-149:
		tmp = t_3
	elif x_46_im <= 5.6e+124:
		tmp = t_4
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_2)) * math.sin(t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-x_46_im))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_3 = Float64(exp(Float64(Float64(t_1 * y_46_re) - t_2)) * sin(Float64(t_0 + Float64(t_1 * y_46_im))))
	t_4 = 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)) * t_0)
	tmp = 0.0
	if (x_46_im <= -22500000000.0)
		tmp = t_3;
	elseif (x_46_im <= -3.5e-93)
		tmp = t_4;
	elseif (x_46_im <= -1.02e-149)
		tmp = t_3;
	elseif (x_46_im <= 5.6e+124)
		tmp = t_4;
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * sin(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 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = log(-x_46_im);
	t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	t_3 = exp(((t_1 * y_46_re) - t_2)) * sin((t_0 + (t_1 * y_46_im)));
	t_4 = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0;
	tmp = 0.0;
	if (x_46_im <= -22500000000.0)
		tmp = t_3;
	elseif (x_46_im <= -3.5e-93)
		tmp = t_4;
	elseif (x_46_im <= -1.02e-149)
		tmp = t_3;
	elseif (x_46_im <= 5.6e+124)
		tmp = t_4;
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * sin(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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$3 = N[(N[Exp[N[(N[(t$95$1 * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 + N[(t$95$1 * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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] * t$95$0), $MachinePrecision]}, If[LessEqual[x$46$im, -22500000000.0], t$95$3, If[LessEqual[x$46$im, -3.5e-93], t$95$4, If[LessEqual[x$46$im, -1.02e-149], t$95$3, If[LessEqual[x$46$im, 5.6e+124], t$95$4, N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -2.25e10 or -3.5e-93 < x.im < -1.0200000000000001e-149

   1. Initial program 40.7%

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

   2. Taylor expanded in x.im around -inf 40.7%

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

      1. mul-1-neg 40.7%

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

   4. Simplified 40.7%

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

   5. Taylor expanded in x.im around -inf 85.4%

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

      1. mul-1-neg 40.7%

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

   7. Simplified 85.4%

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

   if -2.25e10 < x.im < -3.5e-93 or -1.0200000000000001e-149 < x.im < 5.59999999999999999e124

   1. Initial program 50.0%

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

   2. Taylor expanded in y.im around 0 56.5%

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

   3. Taylor expanded in y.re around 0 55.1%

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

   if 5.59999999999999999e124 < x.im

   1. Initial program 10.3%

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

   2. Taylor expanded in y.im around 0 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 x.re around 0 75.1%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -22500000000:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq -3.5 \cdot 10^{-93}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq -1.02 \cdot 10^{-149}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \log \left(-x.im\right) \cdot y.im\right)\\ \mathbf{elif}\;x.im \leq 5.6 \cdot 10^{+124}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 11: 60.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1.1 \cdot 10^{-175}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_1} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 1.7 \cdot 10^{+124}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_1} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_1} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))) (t_1 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -1.1e-175)
     (*
      (exp (- (* (log (- x.im)) y.re) t_1))
      (sin (* y.im (log (hypot x.im x.re)))))
     (if (<= x.im 1.7e+124)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_1))
        t_0)
       (* (exp (- (* y.re (log x.im)) t_1)) (sin t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -1.1e-175) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_1)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else if (x_46_im <= 1.7e+124) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1)) * t_0;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_1)) * sin(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 t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -1.1e-175) {
		tmp = Math.exp(((Math.log(-x_46_im) * y_46_re) - t_1)) * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else if (x_46_im <= 1.7e+124) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1)) * t_0;
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_1)) * Math.sin(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)
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -1.1e-175:
		tmp = math.exp(((math.log(-x_46_im) * y_46_re) - t_1)) * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	elif x_46_im <= 1.7e+124:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1)) * t_0
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_1)) * math.sin(t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.1e-175)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_1)) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	elseif (x_46_im <= 1.7e+124)
		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_1)) * t_0);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_1)) * sin(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 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -1.1e-175)
		tmp = exp(((log(-x_46_im) * y_46_re) - t_1)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	elseif (x_46_im <= 1.7e+124)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_1)) * t_0;
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_1)) * sin(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[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -1.1e-175], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $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], If[LessEqual[x$46$im, 1.7e+124], 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$1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -1.1e-175

   1. Initial program 40.7%

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

   2. Taylor expanded in x.im around -inf 38.8%

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

      1. mul-1-neg 38.8%

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

   4. Simplified 38.8%

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

   5. Taylor expanded in y.re around 0 34.8%

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

      1. unpow2 34.8%

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

      2. unpow2 34.8%

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

      3. hypot-def 69.8%

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

   7. Simplified 69.8%

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

   if -1.1e-175 < x.im < 1.7e124

   1. Initial program 52.3%

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

   2. Taylor expanded in y.im around 0 58.4%

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

   3. Taylor expanded in y.re around 0 55.7%

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

   if 1.7e124 < x.im

   1. Initial program 10.3%

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

   2. Taylor expanded in y.im around 0 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 x.re around 0 75.1%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.1 \cdot 10^{-175}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 1.7 \cdot 10^{+124}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 12: 60.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -46000000000:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_2} \cdot t_1\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{+124}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_2} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_2} \cdot t_1\\ \end{array} \end{array} \]

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 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -46000000000.0)
     (* (exp (- (* (log (- x.im)) y.re) t_2)) t_1)
     (if (<= x.im 2.4e+124)
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_2))
        t_0)
       (* (exp (- (* y.re (log x.im)) t_2)) t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -46000000000.0) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_2)) * t_1;
	} else if (x_46_im <= 2.4e+124) {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    t_1 = sin(t_0)
    t_2 = atan2(x_46im, x_46re) * y_46im
    if (x_46im <= (-46000000000.0d0)) then
        tmp = exp(((log(-x_46im) * y_46re) - t_2)) * t_1
    else if (x_46im <= 2.4d+124) then
        tmp = exp(((y_46re * log(sqrt(((x_46re * x_46re) + (x_46im * x_46im))))) - t_2)) * t_0
    else
        tmp = exp(((y_46re * log(x_46im)) - t_2)) * t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -46000000000.0) {
		tmp = Math.exp(((Math.log(-x_46_im) * y_46_re) - t_2)) * t_1;
	} else if (x_46_im <= 2.4e+124) {
		tmp = Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0;
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_2)) * 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.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -46000000000.0:
		tmp = math.exp(((math.log(-x_46_im) * y_46_re) - t_2)) * t_1
	elif x_46_im <= 2.4e+124:
		tmp = math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_2)) * t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -46000000000.0)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_2)) * t_1);
	elseif (x_46_im <= 2.4e+124)
		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)) * t_0);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_2)) * t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -46000000000.0)
		tmp = exp(((log(-x_46_im) * y_46_re) - t_2)) * t_1;
	elseif (x_46_im <= 2.4e+124)
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_2)) * t_0;
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_2)) * t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -46000000000.0], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[x$46$im, 2.4e+124], 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] * t$95$0), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -4.6e10

   1. Initial program 31.2%

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

   2. Taylor expanded in y.im around 0 46.7%

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

   3. Taylor expanded in x.im around -inf 62.1%

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

      1. mul-1-neg 31.2%

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

   5. Simplified 62.1%

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

   if -4.6e10 < x.im < 2.40000000000000006e124

   1. Initial program 53.0%

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

   2. Taylor expanded in y.im around 0 55.7%

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

   3. Taylor expanded in y.re around 0 54.4%

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

   if 2.40000000000000006e124 < x.im

   1. Initial program 10.3%

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

   2. Taylor expanded in y.im around 0 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 x.re around 0 75.1%

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

4. Final simplification 59.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -46000000000:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;x.im \leq 2.4 \cdot 10^{+124}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 13: 54.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \sin t_1\\ \mathbf{if}\;x.re \leq -1.2 \cdot 10^{-212}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log \left(-x.re\right) - t_0}\\ \mathbf{elif}\;x.re \leq 2.35 \cdot 10^{-244}:\\ \;\;\;\;t_1 \cdot e^{-t_0}\\ \mathbf{elif}\;x.re \leq 1.25 \cdot 10^{-208}:\\ \;\;\;\;t_2 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot e^{y.re \cdot \log x.re - t_0}\\ \end{array} \end{array} \]

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 (sin t_1)))
   (if (<= x.re -1.2e-212)
     (* t_2 (exp (- (* y.re (log (- x.re))) t_0)))
     (if (<= x.re 2.35e-244)
       (* t_1 (exp (- t_0)))
       (if (<= x.re 1.25e-208)
         (* t_2 (pow x.im y.re))
         (* t_2 (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 = sin(t_1);
	double tmp;
	if (x_46_re <= -1.2e-212) {
		tmp = t_2 * exp(((y_46_re * log(-x_46_re)) - t_0));
	} else if (x_46_re <= 2.35e-244) {
		tmp = t_1 * exp(-t_0);
	} else if (x_46_re <= 1.25e-208) {
		tmp = t_2 * pow(x_46_im, y_46_re);
	} else {
		tmp = t_2 * exp(((y_46_re * log(x_46_re)) - t_0));
	}
	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 = sin(t_1)
    if (x_46re <= (-1.2d-212)) then
        tmp = t_2 * exp(((y_46re * log(-x_46re)) - t_0))
    else if (x_46re <= 2.35d-244) then
        tmp = t_1 * exp(-t_0)
    else if (x_46re <= 1.25d-208) then
        tmp = t_2 * (x_46im ** y_46re)
    else
        tmp = t_2 * exp(((y_46re * log(x_46re)) - t_0))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.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_re <= -1.2e-212) {
		tmp = t_2 * Math.exp(((y_46_re * Math.log(-x_46_re)) - t_0));
	} else if (x_46_re <= 2.35e-244) {
		tmp = t_1 * Math.exp(-t_0);
	} else if (x_46_re <= 1.25e-208) {
		tmp = t_2 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = t_2 * 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.sin(t_1)
	tmp = 0
	if x_46_re <= -1.2e-212:
		tmp = t_2 * math.exp(((y_46_re * math.log(-x_46_re)) - t_0))
	elif x_46_re <= 2.35e-244:
		tmp = t_1 * math.exp(-t_0)
	elif x_46_re <= 1.25e-208:
		tmp = t_2 * math.pow(x_46_im, y_46_re)
	else:
		tmp = t_2 * 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 = sin(t_1)
	tmp = 0.0
	if (x_46_re <= -1.2e-212)
		tmp = Float64(t_2 * exp(Float64(Float64(y_46_re * log(Float64(-x_46_re))) - t_0)));
	elseif (x_46_re <= 2.35e-244)
		tmp = Float64(t_1 * exp(Float64(-t_0)));
	elseif (x_46_re <= 1.25e-208)
		tmp = Float64(t_2 * (x_46_im ^ y_46_re));
	else
		tmp = Float64(t_2 * 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 = sin(t_1);
	tmp = 0.0;
	if (x_46_re <= -1.2e-212)
		tmp = t_2 * exp(((y_46_re * log(-x_46_re)) - t_0));
	elseif (x_46_re <= 2.35e-244)
		tmp = t_1 * exp(-t_0);
	elseif (x_46_re <= 1.25e-208)
		tmp = t_2 * (x_46_im ^ y_46_re);
	else
		tmp = t_2 * 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[Sin[t$95$1], $MachinePrecision]}, If[LessEqual[x$46$re, -1.2e-212], N[(t$95$2 * N[Exp[N[(N[(y$46$re * N[Log[(-x$46$re)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 2.35e-244], N[(t$95$1 * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.25e-208], N[(t$95$2 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * 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 4 regimes

2. if x.re < -1.19999999999999995e-212

   1. Initial program 43.8%

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

   2. Taylor expanded in y.im around 0 56.1%

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

   3. Taylor expanded in x.re around -inf 58.8%

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

      1. mul-1-neg 58.8%

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

   5. Simplified 58.8%

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

   if -1.19999999999999995e-212 < x.re < 2.3499999999999999e-244

   1. Initial program 48.2%

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

   2. Taylor expanded in y.im around 0 62.0%

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

   3. Taylor expanded in x.re around 0 26.2%

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

   4. Taylor expanded in y.re around 0 63.0%

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

      1. *-commutative 63.0%

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

      2. distribute-rgt-neg-in 63.0%

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

   6. Simplified 63.0%

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

   if 2.3499999999999999e-244 < x.re < 1.24999999999999991e-208

   1. Initial program 36.4%

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

   2. Taylor expanded in y.im around 0 55.4%

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

   3. Taylor expanded in x.re around 0 19.2%

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

   4. Taylor expanded in y.im around 0 55.6%

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

   if 1.24999999999999991e-208 < x.re

   1. Initial program 36.2%

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

   2. Taylor expanded in y.im around 0 48.2%

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

   3. Taylor expanded in x.re around inf 54.1%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -1.2 \cdot 10^{-212}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.re\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq 2.35 \cdot 10^{-244}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq 1.25 \cdot 10^{-208}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 14: 47.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin t_0\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.re \leq 3.8 \cdot 10^{-246}:\\ \;\;\;\;t_0 \cdot e^{-t_2}\\ \mathbf{elif}\;x.re \leq 1.06 \cdot 10^{-207}:\\ \;\;\;\;t_1 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log x.re - t_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (sin t_0))
        (t_2 (* (atan2 x.im x.re) y.im)))
   (if (<= x.re 3.8e-246)
     (* t_0 (exp (- t_2)))
     (if (<= x.re 1.06e-207)
       (* t_1 (pow x.im y.re))
       (* t_1 (exp (- (* y.re (log x.re)) t_2)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = sin(t_0);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= 3.8e-246) {
		tmp = t_0 * exp(-t_2);
	} else if (x_46_re <= 1.06e-207) {
		tmp = t_1 * pow(x_46_im, y_46_re);
	} else {
		tmp = t_1 * exp(((y_46_re * log(x_46_re)) - t_2));
	}
	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 = y_46re * atan2(x_46im, x_46re)
    t_1 = sin(t_0)
    t_2 = atan2(x_46im, x_46re) * y_46im
    if (x_46re <= 3.8d-246) then
        tmp = t_0 * exp(-t_2)
    else if (x_46re <= 1.06d-207) then
        tmp = t_1 * (x_46im ** y_46re)
    else
        tmp = t_1 * exp(((y_46re * log(x_46re)) - t_2))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin(t_0);
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_re <= 3.8e-246) {
		tmp = t_0 * Math.exp(-t_2);
	} else if (x_46_re <= 1.06e-207) {
		tmp = t_1 * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(x_46_re)) - t_2));
	}
	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.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_re <= 3.8e-246:
		tmp = t_0 * math.exp(-t_2)
	elif x_46_re <= 1.06e-207:
		tmp = t_1 * math.pow(x_46_im, y_46_re)
	else:
		tmp = t_1 * math.exp(((y_46_re * math.log(x_46_re)) - t_2))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = sin(t_0)
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_re <= 3.8e-246)
		tmp = Float64(t_0 * exp(Float64(-t_2)));
	elseif (x_46_re <= 1.06e-207)
		tmp = Float64(t_1 * (x_46_im ^ y_46_re));
	else
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(x_46_re)) - t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = sin(t_0);
	t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_re <= 3.8e-246)
		tmp = t_0 * exp(-t_2);
	elseif (x_46_re <= 1.06e-207)
		tmp = t_1 * (x_46_im ^ y_46_re);
	else
		tmp = t_1 * exp(((y_46_re * log(x_46_re)) - t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$re, 3.8e-246], N[(t$95$0 * N[Exp[(-t$95$2)], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$re, 1.06e-207], N[(t$95$1 * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < 3.79999999999999976e-246

   1. Initial program 44.8%

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

   2. Taylor expanded in y.im around 0 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 x.re around 0 21.8%

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

   4. Taylor expanded in y.re around 0 42.2%

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

      1. *-commutative 42.2%

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

      2. distribute-rgt-neg-in 42.2%

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

   6. Simplified 42.2%

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

   if 3.79999999999999976e-246 < x.re < 1.05999999999999997e-207

   1. Initial program 36.4%

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

   2. Taylor expanded in y.im around 0 55.4%

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

   3. Taylor expanded in x.re around 0 19.2%

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

   4. Taylor expanded in y.im around 0 55.6%

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

   if 1.05999999999999997e-207 < x.re

   1. Initial program 36.2%

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

   2. Taylor expanded in y.im around 0 48.2%

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

   3. Taylor expanded in x.re around inf 54.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq 3.8 \cdot 10^{-246}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.re \leq 1.06 \cdot 10^{-207}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 15: 56.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -4 \cdot 10^{-307}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - t_1} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t_1} \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -4e-307)
     (* (exp (- (* (log (- x.im)) y.re) t_1)) t_0)
     (* (exp (- (* y.re (log x.im)) t_1)) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -4e-307) {
		tmp = exp(((log(-x_46_im) * y_46_re) - t_1)) * t_0;
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_1)) * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin((y_46re * atan2(x_46im, x_46re)))
    t_1 = atan2(x_46im, x_46re) * y_46im
    if (x_46im <= (-4d-307)) then
        tmp = exp(((log(-x_46im) * y_46re) - t_1)) * t_0
    else
        tmp = exp(((y_46re * log(x_46im)) - t_1)) * t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -4e-307) {
		tmp = Math.exp(((Math.log(-x_46_im) * y_46_re) - t_1)) * t_0;
	} else {
		tmp = Math.exp(((y_46_re * Math.log(x_46_im)) - t_1)) * t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -4e-307:
		tmp = math.exp(((math.log(-x_46_im) * y_46_re) - t_1)) * t_0
	else:
		tmp = math.exp(((y_46_re * math.log(x_46_im)) - t_1)) * t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -4e-307)
		tmp = Float64(exp(Float64(Float64(log(Float64(-x_46_im)) * y_46_re) - t_1)) * t_0);
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(x_46_im)) - t_1)) * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -4e-307)
		tmp = exp(((log(-x_46_im) * y_46_re) - t_1)) * t_0;
	else
		tmp = exp(((y_46_re * log(x_46_im)) - t_1)) * 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[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -4e-307], N[(N[Exp[N[(N[(N[Log[(-x$46$im)], $MachinePrecision] * y$46$re), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.99999999999999964e-307

   1. Initial program 39.2%

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

   2. Taylor expanded in y.im around 0 50.4%

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

   3. Taylor expanded in x.im around -inf 49.8%

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

      1. mul-1-neg 35.7%

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

   5. Simplified 49.8%

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

   if -3.99999999999999964e-307 < x.im

   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. Taylor expanded in y.im around 0 57.6%

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

   3. Taylor expanded in x.re around 0 50.1%

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

4. Final simplification 50.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{-307}:\\ \;\;\;\;e^{\log \left(-x.im\right) \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 16: 50.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.re \leq -4.9 \cdot 10^{+104} \lor \neg \left(y.re \leq 1.42 \cdot 10^{+38}\right):\\ \;\;\;\;\sin t_0 \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.re -4.9e+104) (not (<= y.re 1.42e+38)))
     (* (sin t_0) (pow x.im y.re))
     (* t_0 (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) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -4.9e+104) || !(y_46_re <= 1.42e+38)) {
		tmp = sin(t_0) * pow(x_46_im, y_46_re);
	} else {
		tmp = t_0 * exp(-(atan2(x_46_im, x_46_re) * y_46_im));
	}
	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) :: tmp
    t_0 = y_46re * atan2(x_46im, x_46re)
    if ((y_46re <= (-4.9d+104)) .or. (.not. (y_46re <= 1.42d+38))) then
        tmp = sin(t_0) * (x_46im ** y_46re)
    else
        tmp = t_0 * exp(-(atan2(x_46im, x_46re) * y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_re <= -4.9e+104) || !(y_46_re <= 1.42e+38)) {
		tmp = Math.sin(t_0) * Math.pow(x_46_im, y_46_re);
	} else {
		tmp = t_0 * Math.exp(-(Math.atan2(x_46_im, x_46_re) * y_46_im));
	}
	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 <= -4.9e+104) or not (y_46_re <= 1.42e+38):
		tmp = math.sin(t_0) * math.pow(x_46_im, y_46_re)
	else:
		tmp = t_0 * math.exp(-(math.atan2(x_46_im, x_46_re) * y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_re <= -4.9e+104) || !(y_46_re <= 1.42e+38))
		tmp = Float64(sin(t_0) * (x_46_im ^ y_46_re));
	else
		tmp = Float64(t_0 * exp(Float64(-Float64(atan(x_46_im, x_46_re) * y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	tmp = 0.0;
	if ((y_46_re <= -4.9e+104) || ~((y_46_re <= 1.42e+38)))
		tmp = sin(t_0) * (x_46_im ^ y_46_re);
	else
		tmp = t_0 * exp(-(atan2(x_46_im, x_46_re) * y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$re, -4.9e+104], N[Not[LessEqual[y$46$re, 1.42e+38]], $MachinePrecision]], N[(N[Sin[t$95$0], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[(-N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.89999999999999985e104 or 1.4200000000000001e38 < y.re

   1. Initial program 40.9%

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

   2. Taylor expanded in y.im around 0 81.9%

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

   3. Taylor expanded in x.re around 0 33.2%

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

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

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

   if -4.89999999999999985e104 < y.re < 1.4200000000000001e38

   1. Initial program 41.2%

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

   2. Taylor expanded in y.im around 0 39.0%

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

   3. Taylor expanded in x.re around 0 17.8%

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

   4. Taylor expanded in y.re around 0 45.1%

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

      1. *-commutative 45.1%

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

      2. distribute-rgt-neg-in 45.1%

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

   6. Simplified 45.1%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.9 \cdot 10^{+104} \lor \neg \left(y.re \leq 1.42 \cdot 10^{+38}\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{-\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 17: 31.4% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 41.1%

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

2. Taylor expanded in y.im around 0 53.7%

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

3. Taylor expanded in x.re around 0 23.1%

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

4. Taylor expanded in y.im around 0 24.7%

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

5. Final simplification 24.7%

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

Reproduce

herbie shell --seed 2023187 
(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)))))