powComplex, imaginary part

Percentage Accurate: 42.1% → 80.3%

Time: 30.5s

Alternatives: 15

Speedup: 2.0×

• 35.5% 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: 42.1% 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: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := e^{\mathsf{fma}\left(t\_1, y.re, -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;t\_2 \cdot \sin \left(\left|\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), t\_0\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2 \cdot \sin \left(\mathsf{fma}\left(t\_1, y.im, t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (hypot x.re x.im)))
        (t_2 (exp (fma t_1 y.re (- (* y.im (atan2 x.im x.re)))))))
   (if (<= y.im -1.15e+89)
     (* t_2 (sin (fabs (fma y.im (log (hypot x.im x.re)) t_0))))
     (* t_2 (sin (fma t_1 y.im t_0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = exp(fma(t_1, y_46_re, -(y_46_im * atan2(x_46_im, x_46_re))));
	double tmp;
	if (y_46_im <= -1.15e+89) {
		tmp = t_2 * sin(fabs(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0)));
	} else {
		tmp = t_2 * sin(fma(t_1, y_46_im, 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(hypot(x_46_re, x_46_im))
	t_2 = exp(fma(t_1, y_46_re, Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))))
	tmp = 0.0
	if (y_46_im <= -1.15e+89)
		tmp = Float64(t_2 * sin(abs(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0))));
	else
		tmp = Float64(t_2 * sin(fma(t_1, y_46_im, t_0)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Exp[N[(t$95$1 * y$46$re + (-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -1.15e+89], N[(t$95$2 * N[Sin[N[Abs[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Sin[N[(t$95$1 * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.1499999999999999e89

   1. Initial program 24.1%

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

      1. fmm-def 24.1%

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

      2. hypot-define 24.1%

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

      3. distribute-rgt-neg-out 24.1%

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

      4. fma-define 24.1%

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

      5. hypot-define 65.8%

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

      6. *-commutative 65.8%

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

   3. Simplified 65.8%

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

      1. fma-undefine 65.8%

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

      2. hypot-define 24.1%

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

      3. *-commutative 24.1%

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

      4. add-sqr-sqrt 12.1%

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

      5. sqrt-unprod 5.2%

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

      6. pow2 5.2%

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

      7. fma-define 5.2%

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

      8. hypot-define 14.3%

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

   6. Applied egg-rr 14.3%

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

      1. unpow2 14.3%

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

      2. rem-sqrt-square 74.6%

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

      3. fma-undefine 74.6%

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

      4. hypot-undefine 32.8%

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

      5. unpow2 32.8%

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

      6. unpow2 32.8%

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

      7. +-commutative 32.8%

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

      8. *-commutative 32.8%

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

      9. *-commutative 32.8%

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

      10. fma-define 32.8%

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

      11. unpow2 32.8%

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

      12. unpow2 32.8%

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

      13. hypot-undefine 74.6%

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

   8. Simplified 74.6%

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

   if -1.1499999999999999e89 < y.im

   1. Initial program 36.7%

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

      1. fmm-def 36.7%

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

      2. hypot-define 36.7%

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

      3. distribute-rgt-neg-out 36.7%

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

      4. fma-define 36.7%

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

      5. hypot-define 82.1%

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

      6. *-commutative 82.1%

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

   3. Simplified 82.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+89}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\left|\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right|\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

   1. fmm-def 33.8%

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

   2. hypot-define 33.8%

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

   3. distribute-rgt-neg-out 33.8%

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

   4. fma-define 33.8%

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

   5. hypot-define 78.4%

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

   6. *-commutative 78.4%

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

3. Simplified 78.4%

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

5. Final simplification 78.4%

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

Alternative 3: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-100} \lor \neg \left(y.im \leq 3.4 \cdot 10^{-119}\right):\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, y.re, -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t\_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (or (<= y.im -2.5e-100) (not (<= y.im 3.4e-119)))
     (*
      (exp (fma t_0 y.re (- (* y.im (atan2 x.im x.re)))))
      (sin (* y.im (log (hypot x.im x.re)))))
     (*
      (sin (fma t_0 y.im (* y.re (atan2 x.im x.re))))
      (pow (hypot x.re x.im) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if ((y_46_im <= -2.5e-100) || !(y_46_im <= 3.4e-119)) {
		tmp = exp(fma(t_0, y_46_re, -(y_46_im * atan2(x_46_im, x_46_re)))) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = sin(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re)))) * pow(hypot(x_46_re, x_46_im), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if ((y_46_im <= -2.5e-100) || !(y_46_im <= 3.4e-119))
		tmp = Float64(exp(fma(t_0, y_46_re, Float64(-Float64(y_46_im * atan(x_46_im, x_46_re))))) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = Float64(sin(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[y$46$im, -2.5e-100], N[Not[LessEqual[y$46$im, 3.4e-119]], $MachinePrecision]], N[(N[Exp[N[(t$95$0 * y$46$re + (-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])), $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], N[(N[Sin[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.5e-100 or 3.40000000000000024e-119 < y.im

   1. Initial program 31.0%

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

      1. fmm-def 31.0%

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

      2. hypot-define 31.0%

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

      3. distribute-rgt-neg-out 31.0%

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

      4. fma-define 31.0%

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

      5. hypot-define 74.2%

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

      6. *-commutative 74.2%

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

   3. Simplified 74.2%

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

   5. Taylor expanded in y.im around inf 30.0%

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

      1. unpow2 30.0%

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

      2. unpow2 30.0%

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

      3. hypot-undefine 75.0%

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

   7. Simplified 75.0%

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

   if -2.5e-100 < y.im < 3.40000000000000024e-119

   1. Initial program 41.0%

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

      1. exp-diff 41.0%

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

      2. exp-to-pow 41.0%

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

      3. hypot-define 41.0%

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

      4. *-commutative 41.0%

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

      5. exp-prod 41.0%

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

      6. fma-define 41.0%

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

      7. hypot-define 88.9%

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

      8. *-commutative 88.9%

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

   3. Simplified 88.9%

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

   5. Taylor expanded in y.im around 0 88.9%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{-100} \lor \neg \left(y.im \leq 3.4 \cdot 10^{-119}\right):\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_1 := -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+36}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{t\_1}\right)\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+57}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t\_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, y.re, t\_1\right)} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))) (t_1 (- (* y.im (atan2 x.im x.re)))))
   (if (<= y.im -8e+36)
     (* y.re (* (atan2 x.im x.re) (exp t_1)))
     (if (<= y.im 9.2e+57)
       (*
        (sin (fma t_0 y.im (* y.re (atan2 x.im x.re))))
        (pow (hypot x.re x.im) y.re))
       (* (exp (fma t_0 y.re t_1)) (* y.im (log (hypot x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double t_1 = -(y_46_im * atan2(x_46_im, x_46_re));
	double tmp;
	if (y_46_im <= -8e+36) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(t_1));
	} else if (y_46_im <= 9.2e+57) {
		tmp = sin(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re)))) * pow(hypot(x_46_re, x_46_im), y_46_re);
	} else {
		tmp = exp(fma(t_0, y_46_re, t_1)) * (y_46_im * log(hypot(x_46_im, x_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	t_1 = Float64(-Float64(y_46_im * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (y_46_im <= -8e+36)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(t_1)));
	elseif (y_46_im <= 9.2e+57)
		tmp = Float64(sin(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))) * (hypot(x_46_re, x_46_im) ^ y_46_re));
	else
		tmp = Float64(exp(fma(t_0, y_46_re, t_1)) * Float64(y_46_im * log(hypot(x_46_im, x_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = (-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[y$46$im, -8e+36], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.2e+57], N[(N[Sin[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(t$95$0 * y$46$re + t$95$1), $MachinePrecision]], $MachinePrecision] * N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -8.00000000000000034e36

   1. Initial program 21.2%

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

   3. Taylor expanded in y.im around 0 44.5%

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

   4. Taylor expanded in y.re around 0 62.4%

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

      1. *-commutative 62.4%

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

      2. *-commutative 62.4%

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

      3. distribute-rgt-neg-in 62.4%

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

   6. Simplified 62.4%

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

   if -8.00000000000000034e36 < y.im < 9.1999999999999995e57

   1. Initial program 40.3%

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

      1. exp-diff 38.8%

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

      2. exp-to-pow 38.8%

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

      3. hypot-define 38.8%

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

      4. *-commutative 38.8%

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

      5. exp-prod 37.9%

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

      6. fma-define 37.9%

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

      7. hypot-define 80.8%

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

      8. *-commutative 80.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in y.im around 0 82.1%

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

   if 9.1999999999999995e57 < y.im

   1. Initial program 32.5%

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

      1. fmm-def 32.5%

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

      2. hypot-define 32.5%

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

      3. distribute-rgt-neg-out 32.5%

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

      4. fma-define 32.5%

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

      5. hypot-define 76.9%

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

      6. *-commutative 76.9%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in y.im around inf 30.5%

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

      1. unpow2 30.5%

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

      2. unpow2 30.5%

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

      3. hypot-undefine 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in y.im around 0 44.2%

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

      1. unpow2 44.2%

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

      2. unpow2 44.2%

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

      3. hypot-undefine 70.2%

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

   10. Simplified 70.2%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8 \cdot 10^{+36}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+57}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;x.im \leq 1.1 \cdot 10^{+160}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, -t\_0\right)} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re))))
   (if (<= x.im 1.1e+160)
     (*
      (exp (fma (log (hypot x.re x.im)) y.re (- t_0)))
      (* y.im (log (hypot x.im x.re))))
     (*
      (exp (- (* y.re (log x.im)) t_0))
      (sin (+ (* y.re (atan2 x.im x.re)) (* y.im (log x.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_im * atan2(x_46_im, x_46_re);
	double tmp;
	if (x_46_im <= 1.1e+160) {
		tmp = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, -t_0)) * (y_46_im * log(hypot(x_46_im, x_46_re)));
	} else {
		tmp = exp(((y_46_re * log(x_46_im)) - t_0)) * sin(((y_46_re * atan2(x_46_im, x_46_re)) + (y_46_im * log(x_46_im))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (x_46_im <= 1.1e+160)
		tmp = Float64(exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, Float64(-t_0))) * 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_0)) * sin(Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) + Float64(y_46_im * log(x_46_im)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, 1.1e+160], N[(N[Exp[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re + (-t$95$0)), $MachinePrecision]], $MachinePrecision] * N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] + N[(y$46$im * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < 1.09999999999999996e160

   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. Step-by-step derivation

      1. fmm-def 38.7%

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

      2. hypot-define 38.7%

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

      3. distribute-rgt-neg-out 38.7%

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

      4. fma-define 38.7%

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

      5. hypot-define 78.4%

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

      6. *-commutative 78.4%

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

   3. Simplified 78.4%

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

   5. Taylor expanded in y.im around inf 34.9%

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

      1. unpow2 34.9%

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

      2. unpow2 34.9%

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

      3. hypot-undefine 75.2%

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

   7. Simplified 75.2%

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

   8. Taylor expanded in y.im around 0 43.1%

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

      1. unpow2 43.1%

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

      2. unpow2 43.1%

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

      3. hypot-undefine 71.5%

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

   10. Simplified 71.5%

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

   if 1.09999999999999996e160 < x.im

   1. Initial program 0.0%

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

      1. fmm-def 0.0%

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

      2. hypot-define 0.0%

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

      3. distribute-rgt-neg-out 0.0%

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

      4. fma-define 0.0%

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

      5. hypot-define 78.0%

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

      6. *-commutative 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in x.re around 0 74.9%

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

   6. Taylor expanded in x.re around 0 74.9%

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

      1. +-commutative 74.9%

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

      2. neg-mul-1 74.9%

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

      3. unsub-neg 74.9%

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

      4. *-commutative 74.9%

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

   8. Simplified 74.9%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq 1.1 \cdot 10^{+160}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, -y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)} \cdot \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log x.im - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log x.im\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot e^{\left(-t\_0\right) - y.re \cdot \log \left(\frac{-1}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq 1.65 \cdot 10^{-12} \lor \neg \left(x.re \leq 8.8 \cdot 10^{+24}\right):\\ \;\;\;\;t\_1 \cdot e^{y.re \cdot \log x.re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(x.re + {x.im}^{2} \cdot \frac{0.5}{x.re}\right) - t\_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (sin (* y.im (log (hypot x.im x.re))))))
   (if (<= x.re -2e-310)
     (* t_1 (exp (- (- t_0) (* y.re (log (/ -1.0 x.re))))))
     (if (or (<= x.re 1.65e-12) (not (<= x.re 8.8e+24)))
       (* t_1 (exp (- (* y.re (log x.re)) t_0)))
       (*
        (exp (- (* y.re (log (+ x.re (* (pow x.im 2.0) (/ 0.5 x.re))))) t_0))
        (sin (* y.re (atan2 x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (x_46_re <= -2e-310) {
		tmp = t_1 * exp((-t_0 - (y_46_re * log((-1.0 / x_46_re)))));
	} else if ((x_46_re <= 1.65e-12) || !(x_46_re <= 8.8e+24)) {
		tmp = t_1 * exp(((y_46_re * log(x_46_re)) - t_0));
	} else {
		tmp = exp(((y_46_re * log((x_46_re + (pow(x_46_im, 2.0) * (0.5 / x_46_re))))) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (x_46_re <= -2e-310) {
		tmp = t_1 * Math.exp((-t_0 - (y_46_re * Math.log((-1.0 / x_46_re)))));
	} else if ((x_46_re <= 1.65e-12) || !(x_46_re <= 8.8e+24)) {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(x_46_re)) - t_0));
	} else {
		tmp = Math.exp(((y_46_re * Math.log((x_46_re + (Math.pow(x_46_im, 2.0) * (0.5 / x_46_re))))) - t_0)) * Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if x_46_re <= -2e-310:
		tmp = t_1 * math.exp((-t_0 - (y_46_re * math.log((-1.0 / x_46_re)))))
	elif (x_46_re <= 1.65e-12) or not (x_46_re <= 8.8e+24):
		tmp = t_1 * math.exp(((y_46_re * math.log(x_46_re)) - t_0))
	else:
		tmp = math.exp(((y_46_re * math.log((x_46_re + (math.pow(x_46_im, 2.0) * (0.5 / x_46_re))))) - t_0)) * math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	tmp = 0.0
	if (x_46_re <= -2e-310)
		tmp = Float64(t_1 * exp(Float64(Float64(-t_0) - Float64(y_46_re * log(Float64(-1.0 / x_46_re))))));
	elseif ((x_46_re <= 1.65e-12) || !(x_46_re <= 8.8e+24))
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(Float64(x_46_re + Float64((x_46_im ^ 2.0) * Float64(0.5 / x_46_re))))) - t_0)) * sin(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (x_46_re <= -2e-310)
		tmp = t_1 * exp((-t_0 - (y_46_re * log((-1.0 / x_46_re)))));
	elseif ((x_46_re <= 1.65e-12) || ~((x_46_re <= 8.8e+24)))
		tmp = t_1 * exp(((y_46_re * log(x_46_re)) - t_0));
	else
		tmp = exp(((y_46_re * log((x_46_re + ((x_46_im ^ 2.0) * (0.5 / x_46_re))))) - t_0)) * sin((y_46_re * atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -2e-310], N[(t$95$1 * N[Exp[N[((-t$95$0) - N[(y$46$re * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$re, 1.65e-12], N[Not[LessEqual[x$46$re, 8.8e+24]], $MachinePrecision]], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[(x$46$re + N[(N[Power[x$46$im, 2.0], $MachinePrecision] * N[(0.5 / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.999999999999994e-310

   1. Initial program 32.3%

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

      1. fmm-def 32.3%

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

      2. hypot-define 32.3%

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

      3. distribute-rgt-neg-out 32.3%

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

      4. fma-define 32.3%

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

      5. hypot-define 79.9%

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

      6. *-commutative 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in y.im around inf 30.2%

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

      1. unpow2 30.2%

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

      2. unpow2 30.2%

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

      3. hypot-undefine 77.3%

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

   7. Simplified 77.3%

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

   8. Taylor expanded in x.re around -inf 72.7%

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

      1. +-commutative 72.7%

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

      2. neg-mul-1 72.7%

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

      3. unsub-neg 72.7%

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

      4. mul-1-neg 72.7%

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

      5. distribute-rgt-neg-in 72.7%

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

   10. Simplified 72.7%

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

   if -1.999999999999994e-310 < x.re < 1.65e-12 or 8.80000000000000007e24 < x.re

   1. Initial program 32.5%

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

      1. fmm-def 32.5%

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

      2. hypot-define 32.5%

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

      3. distribute-rgt-neg-out 32.5%

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

      4. fma-define 32.5%

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

      5. hypot-define 77.4%

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

      6. *-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in y.im around inf 29.3%

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

      1. unpow2 29.3%

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

      2. unpow2 29.3%

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

      3. hypot-undefine 71.8%

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

   7. Simplified 71.8%

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

   8. Taylor expanded in x.im around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. neg-mul-1 69.3%

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

      3. unsub-neg 69.3%

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

   10. Simplified 69.3%

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

   if 1.65e-12 < x.re < 8.80000000000000007e24

   1. Initial program 69.8%

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

   3. Taylor expanded in y.im around 0 90.0%

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

   4. Taylor expanded in x.im around 0 90.2%

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

      1. *-commutative 90.2%

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

      2. associate-*l/ 90.2%

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

      3. associate-*r/ 90.2%

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

   6. Simplified 90.2%

      \[\leadsto e^{\log \color{blue}{\left(x.re + {x.im}^{2} \cdot \frac{0.5}{x.re}\right)} \cdot y.re - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\left(-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - y.re \cdot \log \left(\frac{-1}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq 1.65 \cdot 10^{-12} \lor \neg \left(x.re \leq 8.8 \cdot 10^{+24}\right):\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(x.re + {x.im}^{2} \cdot \frac{0.5}{x.re}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0}\\ \mathbf{if}\;x.re \leq -5.5 \cdot 10^{-208}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x.re \leq 9 \cdot 10^{-302}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-t\_0}\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{-12} \lor \neg \left(x.re \leq 8.8 \cdot 10^{+24}\right):\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1
         (*
          (* y.re (atan2 x.im x.re))
          (exp
           (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0)))))
   (if (<= x.re -5.5e-208)
     t_1
     (if (<= x.re 9e-302)
       (* y.re (* (atan2 x.im x.re) (exp (- t_0))))
       (if (or (<= x.re 1.35e-12) (not (<= x.re 8.8e+24)))
         (*
          (sin (* y.im (log (hypot x.im x.re))))
          (exp (- (* y.re (log x.re)) t_0)))
         t_1)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = (y_46_re * atan2(x_46_im, x_46_re)) * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (x_46_re <= -5.5e-208) {
		tmp = t_1;
	} else if (x_46_re <= 9e-302) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-t_0));
	} else if ((x_46_re <= 1.35e-12) || !(x_46_re <= 8.8e+24)) {
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_re)) - t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	double tmp;
	if (x_46_re <= -5.5e-208) {
		tmp = t_1;
	} else if (x_46_re <= 9e-302) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp(-t_0));
	} else if ((x_46_re <= 1.35e-12) || !(x_46_re <= 8.8e+24)) {
		tmp = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) * Math.exp(((y_46_re * Math.log(x_46_re)) - t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	tmp = 0
	if x_46_re <= -5.5e-208:
		tmp = t_1
	elif x_46_re <= 9e-302:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp(-t_0))
	elif (x_46_re <= 1.35e-12) or not (x_46_re <= 8.8e+24):
		tmp = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) * math.exp(((y_46_re * math.log(x_46_re)) - t_0))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)))
	tmp = 0.0
	if (x_46_re <= -5.5e-208)
		tmp = t_1;
	elseif (x_46_re <= 9e-302)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(-t_0))));
	elseif ((x_46_re <= 1.35e-12) || !(x_46_re <= 8.8e+24))
		tmp = Float64(sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = (y_46_re * atan2(x_46_im, x_46_re)) * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	tmp = 0.0;
	if (x_46_re <= -5.5e-208)
		tmp = t_1;
	elseif (x_46_re <= 9e-302)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-t_0));
	elseif ((x_46_re <= 1.35e-12) || ~((x_46_re <= 8.8e+24)))
		tmp = sin((y_46_im * log(hypot(x_46_im, x_46_re)))) * exp(((y_46_re * log(x_46_re)) - t_0));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$re, -5.5e-208], t$95$1, If[LessEqual[x$46$re, 9e-302], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[(-t$95$0)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$re, 1.35e-12], N[Not[LessEqual[x$46$re, 8.8e+24]], $MachinePrecision]], N[(N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -5.4999999999999997e-208 or 1.3499999999999999e-12 < x.re < 8.80000000000000007e24

   1. Initial program 33.6%

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

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

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

   4. Taylor expanded in y.re around 0 57.3%

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

   if -5.4999999999999997e-208 < x.re < 9.00000000000000018e-302

   1. Initial program 41.3%

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

   3. Taylor expanded in y.im around 0 29.1%

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

   4. Taylor expanded in y.re around 0 51.4%

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

      1. *-commutative 51.4%

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

      2. *-commutative 51.4%

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

      3. distribute-rgt-neg-in 51.4%

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

   6. Simplified 51.4%

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

   if 9.00000000000000018e-302 < x.re < 1.3499999999999999e-12 or 8.80000000000000007e24 < x.re

   1. Initial program 32.2%

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

      1. fmm-def 32.2%

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

      2. hypot-define 32.2%

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

      3. distribute-rgt-neg-out 32.2%

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

      4. fma-define 32.2%

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

      5. hypot-define 77.1%

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

      6. *-commutative 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in y.im around inf 29.8%

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

      1. unpow2 29.8%

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

      2. unpow2 29.8%

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

      3. hypot-undefine 73.0%

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

   7. Simplified 73.0%

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

   8. Taylor expanded in x.im around 0 70.5%

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

      1. +-commutative 70.5%

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

      2. neg-mul-1 70.5%

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

      3. unsub-neg 70.5%

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

   10. Simplified 70.5%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -5.5 \cdot 10^{-208}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{elif}\;x.re \leq 9 \cdot 10^{-302}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{elif}\;x.re \leq 1.35 \cdot 10^{-12} \lor \neg \left(x.re \leq 8.8 \cdot 10^{+24}\right):\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{if}\;x.re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;t\_1 \cdot e^{\left(-t\_0\right) - y.re \cdot \log \left(\frac{-1}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq 2.7 \cdot 10^{-12} \lor \neg \left(x.re \leq 1.05 \cdot 10^{+25}\right):\\ \;\;\;\;t\_1 \cdot e^{y.re \cdot \log x.re - t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.im (atan2 x.im x.re)))
        (t_1 (sin (* y.im (log (hypot x.im x.re))))))
   (if (<= x.re -2e-310)
     (* t_1 (exp (- (- t_0) (* y.re (log (/ -1.0 x.re))))))
     (if (or (<= x.re 2.7e-12) (not (<= x.re 1.05e+25)))
       (* t_1 (exp (- (* y.re (log x.re)) t_0)))
       (*
        (* y.re (atan2 x.im x.re))
        (exp
         (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * atan2(x_46_im, x_46_re);
	double t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	double tmp;
	if (x_46_re <= -2e-310) {
		tmp = t_1 * exp((-t_0 - (y_46_re * log((-1.0 / x_46_re)))));
	} else if ((x_46_re <= 2.7e-12) || !(x_46_re <= 1.05e+25)) {
		tmp = t_1 * exp(((y_46_re * log(x_46_re)) - t_0));
	} else {
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_im * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	double tmp;
	if (x_46_re <= -2e-310) {
		tmp = t_1 * Math.exp((-t_0 - (y_46_re * Math.log((-1.0 / x_46_re)))));
	} else if ((x_46_re <= 2.7e-12) || !(x_46_re <= 1.05e+25)) {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(x_46_re)) - t_0));
	} else {
		tmp = (y_46_re * Math.atan2(x_46_im, x_46_re)) * Math.exp(((y_46_re * Math.log(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_im * math.atan2(x_46_im, x_46_re)
	t_1 = math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	tmp = 0
	if x_46_re <= -2e-310:
		tmp = t_1 * math.exp((-t_0 - (y_46_re * math.log((-1.0 / x_46_re)))))
	elif (x_46_re <= 2.7e-12) or not (x_46_re <= 1.05e+25):
		tmp = t_1 * math.exp(((y_46_re * math.log(x_46_re)) - t_0))
	else:
		tmp = (y_46_re * math.atan2(x_46_im, x_46_re)) * math.exp(((y_46_re * math.log(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_im * atan(x_46_im, x_46_re))
	t_1 = sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re))))
	tmp = 0.0
	if (x_46_re <= -2e-310)
		tmp = Float64(t_1 * exp(Float64(Float64(-t_0) - Float64(y_46_re * log(Float64(-1.0 / x_46_re))))));
	elseif ((x_46_re <= 2.7e-12) || !(x_46_re <= 1.05e+25))
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(x_46_re)) - t_0)));
	else
		tmp = Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_im * atan2(x_46_im, x_46_re);
	t_1 = sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	tmp = 0.0;
	if (x_46_re <= -2e-310)
		tmp = t_1 * exp((-t_0 - (y_46_re * log((-1.0 / x_46_re)))));
	elseif ((x_46_re <= 2.7e-12) || ~((x_46_re <= 1.05e+25)))
		tmp = t_1 * exp(((y_46_re * log(x_46_re)) - t_0));
	else
		tmp = (y_46_re * atan2(x_46_im, x_46_re)) * exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$re, -2e-310], N[(t$95$1 * N[Exp[N[((-t$95$0) - N[(y$46$re * N[Log[N[(-1.0 / x$46$re), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[x$46$re, 2.7e-12], N[Not[LessEqual[x$46$re, 1.05e+25]], $MachinePrecision]], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[x$46$re], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.re < -1.999999999999994e-310

   1. Initial program 32.3%

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

      1. fmm-def 32.3%

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

      2. hypot-define 32.3%

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

      3. distribute-rgt-neg-out 32.3%

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

      4. fma-define 32.3%

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

      5. hypot-define 79.9%

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

      6. *-commutative 79.9%

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

   3. Simplified 79.9%

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

   5. Taylor expanded in y.im around inf 30.2%

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

      1. unpow2 30.2%

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

      2. unpow2 30.2%

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

      3. hypot-undefine 77.3%

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

   7. Simplified 77.3%

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

   8. Taylor expanded in x.re around -inf 72.7%

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

      1. +-commutative 72.7%

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

      2. neg-mul-1 72.7%

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

      3. unsub-neg 72.7%

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

      4. mul-1-neg 72.7%

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

      5. distribute-rgt-neg-in 72.7%

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

   10. Simplified 72.7%

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

   if -1.999999999999994e-310 < x.re < 2.6999999999999998e-12 or 1.05e25 < x.re

   1. Initial program 32.5%

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

      1. fmm-def 32.5%

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

      2. hypot-define 32.5%

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

      3. distribute-rgt-neg-out 32.5%

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

      4. fma-define 32.5%

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

      5. hypot-define 77.4%

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

      6. *-commutative 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in y.im around inf 29.3%

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

      1. unpow2 29.3%

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

      2. unpow2 29.3%

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

      3. hypot-undefine 71.8%

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

   7. Simplified 71.8%

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

   8. Taylor expanded in x.im around 0 69.3%

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

      1. +-commutative 69.3%

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

      2. neg-mul-1 69.3%

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

      3. unsub-neg 69.3%

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

   10. Simplified 69.3%

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

   if 2.6999999999999998e-12 < x.re < 1.05e25

   1. Initial program 69.8%

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

   3. Taylor expanded in y.im around 0 90.0%

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

   4. Taylor expanded in y.re around 0 90.0%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.re \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{\left(-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) - y.re \cdot \log \left(\frac{-1}{x.re}\right)}\\ \mathbf{elif}\;x.re \leq 2.7 \cdot 10^{-12} \lor \neg \left(x.re \leq 1.05 \cdot 10^{+25}\right):\\ \;\;\;\;\sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right) \cdot e^{y.re \cdot \log x.re - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{y.im}\right)}^{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.2e+39)
   (* y.re (* (atan2 x.im x.re) (exp (- (* y.im (atan2 x.im x.re))))))
   (if (<= y.im 2.2e+58)
     (* (sin (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))
     (* y.re (* (atan2 x.im x.re) (pow (exp y.im) (- (atan2 x.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+39) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-(y_46_im * atan2(x_46_im, x_46_re))));
	} else if (y_46_im <= 2.2e+58) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * pow(exp(y_46_im), -atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.2e+39) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else if (y_46_im <= 2.2e+58) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.pow(Math.exp(y_46_im), -Math.atan2(x_46_im, x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.2e+39:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))))
	elif y_46_im <= 2.2e+58:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.pow(math.exp(y_46_im), -math.atan2(x_46_im, x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+39)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re))))));
	elseif (y_46_im <= 2.2e+58)
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * (exp(y_46_im) ^ Float64(-atan(x_46_im, x_46_re)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.2e+39)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-(y_46_im * atan2(x_46_im, x_46_re))));
	elseif (y_46_im <= 2.2e+58)
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * (exp(y_46_im) ^ -atan2(x_46_im, x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.2e+39], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.2e+58], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Power[N[Exp[y$46$im], $MachinePrecision], (-N[ArcTan[x$46$im / x$46$re], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.2e39

   1. Initial program 21.2%

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

   3. Taylor expanded in y.im around 0 44.5%

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

   4. Taylor expanded in y.re around 0 62.4%

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

      1. *-commutative 62.4%

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

      2. *-commutative 62.4%

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

      3. distribute-rgt-neg-in 62.4%

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

   6. Simplified 62.4%

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

   if -1.2e39 < y.im < 2.2000000000000001e58

   1. Initial program 40.3%

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

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

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

   4. Taylor expanded in y.im around 0 43.6%

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

      1. unpow2 43.6%

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

      2. unpow2 43.6%

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

      3. hypot-undefine 48.7%

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

   6. Simplified 48.7%

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

   if 2.2000000000000001e58 < y.im

   1. Initial program 32.5%

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

   3. Taylor expanded in y.im around 0 50.5%

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

   4. Taylor expanded in y.re around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. *-commutative 60.5%

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

      3. distribute-rgt-neg-in 60.5%

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

   6. Simplified 60.5%

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

   7. Taylor expanded in y.re around 0 60.5%

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

      1. *-commutative 60.5%

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

      2. neg-mul-1 60.5%

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

      3. distribute-rgt-neg-in 60.5%

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

      4. mul-1-neg 60.5%

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

      5. exp-prod 62.4%

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

      6. mul-1-neg 62.4%

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

   9. Simplified 62.4%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+58}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot {\left(e^{y.im}\right)}^{\left(-\tan^{-1}_* \frac{x.im}{x.re}\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+37} \lor \neg \left(y.im \leq 8 \cdot 10^{+57}\right):\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -9e+37) (not (<= y.im 8e+57)))
   (* y.re (* (atan2 x.im x.re) (exp (- (* y.im (atan2 x.im x.re))))))
   (* (sin (* y.re (atan2 x.im x.re))) (pow (hypot x.im x.re) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9e+37) || !(y_46_im <= 8e+57)) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-(y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -9e+37) || !(y_46_im <= 8e+57)) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -9e+37) or not (y_46_im <= 8e+57):
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))))
	else:
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -9e+37) || !(y_46_im <= 8e+57))
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re))))));
	else
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -9e+37) || ~((y_46_im <= 8e+57)))
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-(y_46_im * atan2(x_46_im, x_46_re))));
	else
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (hypot(x_46_im, x_46_re) ^ y_46_re);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -9e+37], N[Not[LessEqual[y$46$im, 8e+57]], $MachinePrecision]], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.99999999999999923e37 or 8.00000000000000039e57 < y.im

   1. Initial program 26.1%

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

   3. Taylor expanded in y.im around 0 47.1%

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

   4. Taylor expanded in y.re around 0 61.6%

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

      1. *-commutative 61.6%

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

      2. *-commutative 61.6%

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

      3. distribute-rgt-neg-in 61.6%

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

   6. Simplified 61.6%

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

   if -8.99999999999999923e37 < y.im < 8.00000000000000039e57

   1. Initial program 40.3%

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

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

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

   4. Taylor expanded in y.im around 0 43.6%

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

      1. unpow2 43.6%

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

      2. unpow2 43.6%

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

      3. hypot-undefine 48.7%

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

   6. Simplified 48.7%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+37} \lor \neg \left(y.im \leq 8 \cdot 10^{+57}\right):\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -720000000000 \lor \neg \left(y.im \leq 25000000000\right):\\ \;\;\;\;\log \left(1 + t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (expm1 (* y.re (atan2 x.im x.re)))))
   (if (or (<= y.im -720000000000.0) (not (<= y.im 25000000000.0)))
     (log (+ 1.0 t_0))
     (log1p t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = expm1((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if ((y_46_im <= -720000000000.0) || !(y_46_im <= 25000000000.0)) {
		tmp = log((1.0 + t_0));
	} else {
		tmp = log1p(t_0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.expm1((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if ((y_46_im <= -720000000000.0) || !(y_46_im <= 25000000000.0)) {
		tmp = Math.log((1.0 + t_0));
	} else {
		tmp = Math.log1p(t_0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.expm1((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if (y_46_im <= -720000000000.0) or not (y_46_im <= 25000000000.0):
		tmp = math.log((1.0 + t_0))
	else:
		tmp = math.log1p(t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = expm1(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if ((y_46_im <= -720000000000.0) || !(y_46_im <= 25000000000.0))
		tmp = log(Float64(1.0 + t_0));
	else
		tmp = log1p(t_0);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(Exp[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -720000000000.0], N[Not[LessEqual[y$46$im, 25000000000.0]], $MachinePrecision]], N[Log[N[(1.0 + t$95$0), $MachinePrecision]], $MachinePrecision], N[Log[1 + t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.2e11 or 2.5e10 < y.im

   1. Initial program 27.6%

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

   3. Taylor expanded in y.im around 0 47.0%

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

   4. Taylor expanded in y.re around 0 56.3%

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

      1. *-commutative 56.3%

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

      2. *-commutative 56.3%

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

      3. distribute-rgt-neg-in 56.3%

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

   6. Simplified 56.3%

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

   7. Taylor expanded in y.im around 0 4.9%

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

      1. log1p-expm1-u 14.3%

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

      2. log1p-undefine 30.5%

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

   9. Applied egg-rr 30.5%

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

   if -7.2e11 < y.im < 2.5e10

   1. Initial program 40.1%

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

   3. Taylor expanded in y.im around 0 44.0%

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

   4. Taylor expanded in y.re around 0 15.9%

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

      1. *-commutative 15.9%

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

      2. *-commutative 15.9%

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

      3. distribute-rgt-neg-in 15.9%

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

   6. Simplified 15.9%

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

   7. Taylor expanded in y.im around 0 15.2%

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

      1. log1p-expm1-u 21.0%

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

   9. Applied egg-rr 21.0%

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

4. Final simplification 25.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -720000000000 \lor \neg \left(y.im \leq 25000000000\right):\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+46} \lor \neg \left(y.im \leq 26000000000\right):\\ \;\;\;\;\sqrt[3]{{t\_0}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (or (<= y.im -4.5e+46) (not (<= y.im 26000000000.0)))
     (cbrt (pow t_0 3.0))
     (log1p (expm1 t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_im <= -4.5e+46) || !(y_46_im <= 26000000000.0)) {
		tmp = cbrt(pow(t_0, 3.0));
	} else {
		tmp = log1p(expm1(t_0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if ((y_46_im <= -4.5e+46) || !(y_46_im <= 26000000000.0)) {
		tmp = Math.cbrt(Math.pow(t_0, 3.0));
	} else {
		tmp = Math.log1p(Math.expm1(t_0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if ((y_46_im <= -4.5e+46) || !(y_46_im <= 26000000000.0))
		tmp = cbrt((t_0 ^ 3.0));
	else
		tmp = log1p(expm1(t_0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[y$46$im, -4.5e+46], N[Not[LessEqual[y$46$im, 26000000000.0]], $MachinePrecision]], N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.5000000000000001e46 or 2.6e10 < y.im

   1. Initial program 27.4%

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

   3. Taylor expanded in y.im around 0 48.1%

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

   4. Taylor expanded in y.re around 0 57.8%

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

      1. *-commutative 57.8%

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

      2. *-commutative 57.8%

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

      3. distribute-rgt-neg-in 57.8%

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

   6. Simplified 57.8%

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

   7. Taylor expanded in y.im around 0 5.1%

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

      1. add-cbrt-cube 23.5%

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

      2. pow3 23.5%

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

   9. Applied egg-rr 23.5%

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

   if -4.5000000000000001e46 < y.im < 2.6e10

   1. Initial program 39.7%

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

   3. Taylor expanded in y.im around 0 43.2%

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

   4. Taylor expanded in y.re around 0 16.6%

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

      1. *-commutative 16.6%

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

      2. *-commutative 16.6%

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

      3. distribute-rgt-neg-in 16.6%

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

   6. Simplified 16.6%

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

   7. Taylor expanded in y.im around 0 14.5%

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

      1. log1p-expm1-u 20.7%

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

   9. Applied egg-rr 20.7%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+46} \lor \neg \left(y.im \leq 26000000000\right):\\ \;\;\;\;\sqrt[3]{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq 1.8 \cdot 10^{+203}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re 1.8e+203)
   (* y.re (* (atan2 x.im x.re) (exp (- (* y.im (atan2 x.im x.re))))))
   (sqrt (pow (* y.re (atan2 x.im x.re)) 2.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= 1.8e+203) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-(y_46_im * atan2(x_46_im, x_46_re))));
	} else {
		tmp = sqrt(pow((y_46_re * atan2(x_46_im, x_46_re)), 2.0));
	}
	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) :: tmp
    if (y_46re <= 1.8d+203) then
        tmp = y_46re * (atan2(x_46im, x_46re) * exp(-(y_46im * atan2(x_46im, x_46re))))
    else
        tmp = sqrt(((y_46re * atan2(x_46im, x_46re)) ** 2.0d0))
    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 tmp;
	if (y_46_re <= 1.8e+203) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp(-(y_46_im * Math.atan2(x_46_im, x_46_re))));
	} else {
		tmp = Math.sqrt(Math.pow((y_46_re * Math.atan2(x_46_im, x_46_re)), 2.0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= 1.8e+203:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp(-(y_46_im * math.atan2(x_46_im, x_46_re))))
	else:
		tmp = math.sqrt(math.pow((y_46_re * math.atan2(x_46_im, x_46_re)), 2.0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= 1.8e+203)
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(-Float64(y_46_im * atan(x_46_im, x_46_re))))));
	else
		tmp = sqrt((Float64(y_46_re * atan(x_46_im, x_46_re)) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= 1.8e+203)
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp(-(y_46_im * atan2(x_46_im, x_46_re))));
	else
		tmp = sqrt(((y_46_re * atan2(x_46_im, x_46_re)) ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, 1.8e+203], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[(-N[(y$46$im * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < 1.79999999999999991e203

   1. Initial program 35.2%

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

   3. Taylor expanded in y.im around 0 45.5%

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

   4. Taylor expanded in y.re around 0 38.6%

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

      1. *-commutative 38.6%

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

      2. *-commutative 38.6%

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

      3. distribute-rgt-neg-in 38.6%

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

   6. Simplified 38.6%

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

   if 1.79999999999999991e203 < y.re

   1. Initial program 20.8%

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

   3. Taylor expanded in y.im around 0 45.9%

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

   4. Taylor expanded in y.re around 0 13.7%

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

      1. *-commutative 13.7%

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

      2. *-commutative 13.7%

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

      3. distribute-rgt-neg-in 13.7%

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

   6. Simplified 13.7%

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

      1. add-sqr-sqrt 9.2%

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

      2. sqrt-unprod 25.8%

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

      3. pow2 25.8%

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

      4. associate-*r* 25.8%

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

      5. *-commutative 25.8%

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

      6. associate-*l* 25.8%

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

      7. exp-prod 25.8%

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

      8. add-sqr-sqrt 17.3%

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

      9. sqrt-unprod 21.7%

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

      10. sqr-neg 21.7%

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

      11. sqrt-unprod 9.0%

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

      12. add-sqr-sqrt 34.1%

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

   8. Applied egg-rr 34.1%

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

   9. Taylor expanded in y.im around 0 33.6%

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

       1. *-commutative 33.6%

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

       2. unpow2 33.6%

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

       3. unpow2 33.6%

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

       4. swap-sqr 33.9%

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

       5. unpow2 33.9%

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

       6. *-commutative 33.9%

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

   11. Simplified 33.9%

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

4. Final simplification 38.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 1.8 \cdot 10^{+203}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{-y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 22.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (log1p (expm1 (* y.re (atan2 x.im x.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return log1p(expm1((y_46_re * atan2(x_46_im, x_46_re))));
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return Math.log1p(Math.expm1((y_46_re * Math.atan2(x_46_im, x_46_re))));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return math.log1p(math.expm1((y_46_re * math.atan2(x_46_im, x_46_re))))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return log1p(expm1(Float64(y_46_re * atan(x_46_im, x_46_re))))
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[Log[1 + N[(Exp[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in y.im around 0 45.5%

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

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

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

   1. *-commutative 36.3%

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

   2. *-commutative 36.3%

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

   3. distribute-rgt-neg-in 36.3%

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

6. Simplified 36.3%

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

7. Taylor expanded in y.im around 0 10.0%

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

   1. log1p-expm1-u 17.6%

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

9. Applied egg-rr 17.6%

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

10. Final simplification 17.6%

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

Alternative 15: 13.3% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 33.8%

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

3. Taylor expanded in y.im around 0 45.5%

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

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

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

   1. *-commutative 36.3%

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

   2. *-commutative 36.3%

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

   3. distribute-rgt-neg-in 36.3%

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

6. Simplified 36.3%

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

7. Taylor expanded in y.im around 0 10.0%

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

8. Final simplification 10.0%

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

Reproduce

herbie shell --seed 2024112 
(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)))))