powComplex, imaginary part

Percentage Accurate: 40.3% → 78.3%

Time: 1.2min

Alternatives: 21

Speedup: 2.6×

• 35.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 40.3% 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: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_2 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_3 := e^{y.re \cdot t_2 - t_0}\\ \mathbf{if}\;y.re \leq -200000000000:\\ \;\;\;\;t_3 \cdot \sqrt{{\sin t_1}^{2}}\\ \mathbf{elif}\;y.re \leq 2.85 \cdot 10^{+69}:\\ \;\;\;\;t_3 \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(t_2, y.im, t_1\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - t_0} \cdot \sin \left(t_1 + \left|y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (* y.re (atan2 x.im x.re)))
        (t_2 (log (hypot x.re x.im)))
        (t_3 (exp (- (* y.re t_2) t_0))))
   (if (<= y.re -200000000000.0)
     (* t_3 (sqrt (pow (sin t_1) 2.0)))
     (if (<= y.re 2.85e+69)
       (* t_3 (sin (pow (cbrt (fma t_2 y.im t_1)) 3.0)))
       (*
        (exp (- (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im))))) t_0))
        (sin (+ t_1 (fabs (* 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 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = y_46_re * atan2(x_46_im, x_46_re);
	double t_2 = log(hypot(x_46_re, x_46_im));
	double t_3 = exp(((y_46_re * t_2) - t_0));
	double tmp;
	if (y_46_re <= -200000000000.0) {
		tmp = t_3 * sqrt(pow(sin(t_1), 2.0));
	} else if (y_46_re <= 2.85e+69) {
		tmp = t_3 * sin(pow(cbrt(fma(t_2, y_46_im, t_1)), 3.0));
	} else {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - t_0)) * sin((t_1 + fabs((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 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_2 = log(hypot(x_46_re, x_46_im))
	t_3 = exp(Float64(Float64(y_46_re * t_2) - t_0))
	tmp = 0.0
	if (y_46_re <= -200000000000.0)
		tmp = Float64(t_3 * sqrt((sin(t_1) ^ 2.0)));
	elseif (y_46_re <= 2.85e+69)
		tmp = Float64(t_3 * sin((cbrt(fma(t_2, y_46_im, t_1)) ^ 3.0)));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))))) - t_0)) * sin(Float64(t_1 + abs(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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Exp[N[(N[(y$46$re * t$95$2), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -200000000000.0], N[(t$95$3 * N[Sqrt[N[Power[N[Sin[t$95$1], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.85e+69], N[(t$95$3 * N[Sin[N[Power[N[Power[N[(t$95$2 * y$46$im + t$95$1), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$1 + N[Abs[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2e11

   1. Initial program 40.6%

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

      1. hypot-def 40.6%

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

      2. fma-def 40.6%

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

      3. hypot-def 78.3%

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

      4. *-commutative 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in y.im around 0 82.6%

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

      1. add-sqr-sqrt 44.9%

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

      2. sqrt-unprod 91.3%

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

      3. pow2 91.3%

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

      4. *-commutative 91.3%

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

   6. Applied egg-rr 91.3%

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

   if -2e11 < y.re < 2.85e69

   1. Initial program 34.2%

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

      1. hypot-def 34.2%

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

      2. fma-def 34.2%

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

      3. hypot-def 82.3%

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

      4. *-commutative 82.3%

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

   3. Simplified 82.3%

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

      1. fma-udef 82.3%

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

      2. hypot-udef 34.2%

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

      3. *-commutative 34.2%

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

      4. add-cube-cbrt 33.3%

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

      5. pow3 34.0%

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

      6. hypot-udef 84.3%

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

      7. *-commutative 84.3%

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

      8. fma-udef 84.3%

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

      9. *-commutative 84.3%

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

   5. Applied egg-rr 84.3%

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

   if 2.85e69 < y.re

   1. Initial program 46.3%

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

      1. add-sqr-sqrt 19.5%

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

      2. sqrt-unprod 34.1%

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

      3. pow2 34.1%

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

      4. hypot-udef 56.1%

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

   3. Applied egg-rr 56.1%

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

      1. unpow2 56.1%

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

      2. rem-sqrt-square 75.6%

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

      3. hypot-def 46.3%

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

      4. unpow2 46.3%

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

      5. unpow2 46.3%

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

      6. +-commutative 46.3%

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

      7. *-commutative 46.3%

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

      8. unpow2 46.3%

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

      9. unpow2 46.3%

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

      10. hypot-def 75.6%

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

   5. Simplified 75.6%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -200000000000:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \mathbf{elif}\;y.re \leq 2.85 \cdot 10^{+69}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left({\left(\sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + \left|y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right|\right)\\ \end{array} \]

Alternative 2: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -45000000000000:\\ \;\;\;\;t_0 \cdot \sqrt{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {\left(\sqrt[3]{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, {\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot \sqrt[3]{y.im}\right)}^{3}\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (- (* y.re (log (hypot x.re x.im))) (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -45000000000000.0)
     (* t_0 (sqrt (pow (sin (* y.re (atan2 x.im x.re))) 2.0)))
     (*
      t_0
      (pow
       (cbrt
        (sin
         (fma
          (atan2 x.im x.re)
          y.re
          (pow (* (cbrt (log (hypot x.im x.re))) (cbrt y.im)) 3.0))))
       3.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -45000000000000.0) {
		tmp = t_0 * sqrt(pow(sin((y_46_re * atan2(x_46_im, x_46_re))), 2.0));
	} else {
		tmp = t_0 * pow(cbrt(sin(fma(atan2(x_46_im, x_46_re), y_46_re, pow((cbrt(log(hypot(x_46_im, x_46_re))) * cbrt(y_46_im)), 3.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -45000000000000.0)
		tmp = Float64(t_0 * sqrt((sin(Float64(y_46_re * atan(x_46_im, x_46_re))) ^ 2.0)));
	else
		tmp = Float64(t_0 * (cbrt(sin(fma(atan(x_46_im, x_46_re), y_46_re, (Float64(cbrt(log(hypot(x_46_im, x_46_re))) * cbrt(y_46_im)) ^ 3.0)))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -45000000000000.0], N[(t$95$0 * N[Sqrt[N[Power[N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Power[N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[Power[N[(N[Power[N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[y$46$im, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.5e13

   1. Initial program 40.9%

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

      1. hypot-def 40.9%

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

      2. fma-def 40.9%

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

      3. hypot-def 78.8%

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

      4. *-commutative 78.8%

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

   3. Simplified 78.8%

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

   4. Taylor expanded in y.im around 0 83.3%

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

      1. add-sqr-sqrt 45.5%

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

      2. sqrt-unprod 90.9%

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

      3. pow2 90.9%

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

      4. *-commutative 90.9%

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

   6. Applied egg-rr 90.9%

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

   if -4.5e13 < y.re

   1. Initial program 36.8%

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

      1. hypot-def 36.8%

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

      2. fma-def 36.8%

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

      3. hypot-def 79.6%

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

      4. *-commutative 79.6%

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

   3. Simplified 79.6%

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

      1. fma-udef 79.6%

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

      2. hypot-udef 36.8%

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

      3. *-commutative 36.8%

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

      4. add-sqr-sqrt 20.1%

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

      5. sqrt-unprod 15.8%

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

      6. pow2 15.8%

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

      7. fma-def 15.8%

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

      8. hypot-udef 32.1%

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

   5. Applied egg-rr 32.1%

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

      1. unpow2 32.1%

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

      2. rem-sqrt-square 59.2%

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

      3. hypot-def 28.7%

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

      4. unpow2 28.7%

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

      5. unpow2 28.7%

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

      6. +-commutative 28.7%

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

      7. *-commutative 28.7%

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

      8. fma-def 28.7%

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

      9. *-commutative 28.7%

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

      10. +-commutative 28.7%

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

      11. fma-def 28.7%

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

      12. unpow2 28.7%

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

      13. unpow2 28.7%

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

      14. hypot-def 59.2%

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

   7. Simplified 59.2%

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

      1. add-cube-cbrt 58.9%

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

      2. pow3 58.9%

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

      3. add-sqr-sqrt 42.4%

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

      4. fabs-sqr 42.4%

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

      5. add-sqr-sqrt 79.0%

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

      6. fma-udef 79.0%

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

      7. *-commutative 79.0%

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

      8. fma-def 79.0%

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

   9. Applied egg-rr 79.0%

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

       1. add-cube-cbrt 80.5%

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

       2. pow3 81.5%

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

   11. Applied egg-rr 81.5%

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

       1. *-commutative 81.5%

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

       2. cbrt-prod 82.5%

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

   13. Applied egg-rr 82.5%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -45000000000000:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot {\left(\sqrt[3]{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, {\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \cdot \sqrt[3]{y.im}\right)}^{3}\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 3: 79.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -5000000000:\\ \;\;\;\;t_0 \cdot \sqrt{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot {\left(\sqrt[3]{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, {\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{3}\right)\right)}\right)}^{3}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (exp
          (- (* y.re (log (hypot x.re x.im))) (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -5000000000.0)
     (* t_0 (sqrt (pow (sin (* y.re (atan2 x.im x.re))) 2.0)))
     (*
      t_0
      (pow
       (cbrt
        (sin
         (fma
          (atan2 x.im x.re)
          y.re
          (pow (cbrt (* y.im (log (hypot x.im x.re)))) 3.0))))
       3.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -5000000000.0) {
		tmp = t_0 * sqrt(pow(sin((y_46_re * atan2(x_46_im, x_46_re))), 2.0));
	} else {
		tmp = t_0 * pow(cbrt(sin(fma(atan2(x_46_im, x_46_re), y_46_re, pow(cbrt((y_46_im * log(hypot(x_46_im, x_46_re)))), 3.0)))), 3.0);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -5000000000.0)
		tmp = Float64(t_0 * sqrt((sin(Float64(y_46_re * atan(x_46_im, x_46_re))) ^ 2.0)));
	else
		tmp = Float64(t_0 * (cbrt(sin(fma(atan(x_46_im, x_46_re), y_46_re, (cbrt(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) ^ 3.0)))) ^ 3.0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -5000000000.0], N[(t$95$0 * N[Sqrt[N[Power[N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Power[N[Power[N[Sin[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$re + N[Power[N[Power[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5e9

   1. Initial program 40.6%

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

      1. hypot-def 40.6%

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

      2. fma-def 40.6%

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

      3. hypot-def 78.3%

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

      4. *-commutative 78.3%

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

   3. Simplified 78.3%

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

   4. Taylor expanded in y.im around 0 82.6%

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

      1. add-sqr-sqrt 44.9%

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

      2. sqrt-unprod 91.3%

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

      3. pow2 91.3%

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

      4. *-commutative 91.3%

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

   6. Applied egg-rr 91.3%

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

   if -5e9 < y.re

   1. Initial program 36.9%

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

      1. hypot-def 36.9%

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

      2. fma-def 36.9%

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

      3. hypot-def 79.8%

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

      4. *-commutative 79.8%

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

   3. Simplified 79.8%

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

      1. fma-udef 79.8%

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

      2. hypot-udef 36.9%

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

      3. *-commutative 36.9%

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

      4. add-sqr-sqrt 20.4%

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

      5. sqrt-unprod 15.5%

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

      6. pow2 15.5%

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

      7. fma-def 15.5%

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

      8. hypot-udef 31.5%

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

   5. Applied egg-rr 31.5%

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

      1. unpow2 31.5%

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

      2. rem-sqrt-square 59.1%

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

      3. hypot-def 28.6%

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

      4. unpow2 28.6%

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

      5. unpow2 28.6%

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

      6. +-commutative 28.6%

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

      7. *-commutative 28.6%

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

      8. fma-def 28.6%

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

      9. *-commutative 28.6%

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

      10. +-commutative 28.6%

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

      11. fma-def 28.6%

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

      12. unpow2 28.6%

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

      13. unpow2 28.6%

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

      14. hypot-def 59.1%

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

   7. Simplified 59.1%

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

      1. add-cube-cbrt 58.8%

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

      2. pow3 58.8%

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

      3. add-sqr-sqrt 42.6%

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

      4. fabs-sqr 42.6%

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

      5. add-sqr-sqrt 79.2%

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

      6. fma-udef 79.2%

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

      7. *-commutative 79.2%

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

      8. fma-def 79.2%

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

   9. Applied egg-rr 79.2%

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

       1. add-cube-cbrt 80.7%

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

       2. pow3 81.8%

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

   11. Applied egg-rr 81.8%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5000000000:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot {\left(\sqrt[3]{\sin \left(\mathsf{fma}\left(\tan^{-1}_* \frac{x.im}{x.re}, y.re, {\left(\sqrt[3]{y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}\right)}^{3}\right)\right)}\right)}^{3}\\ \end{array} \]

Alternative 4: 79.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := e^{y.re \cdot t_1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{+106}:\\ \;\;\;\;t_2 \cdot \sqrt{{\sin t_0}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \sin \left(\mathsf{fma}\left(t_1, y.im, t_0\right)\right)\\ \end{array} \end{array} \]

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 (- (* y.re t_1) (* (atan2 x.im x.re) y.im)))))
   (if (<= y.re -4e+106)
     (* t_2 (sqrt (pow (sin t_0) 2.0)))
     (* 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(((y_46_re * t_1) - (atan2(x_46_im, x_46_re) * y_46_im)));
	double tmp;
	if (y_46_re <= -4e+106) {
		tmp = t_2 * sqrt(pow(sin(t_0), 2.0));
	} 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(Float64(Float64(y_46_re * t_1) - Float64(atan(x_46_im, x_46_re) * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -4e+106)
		tmp = Float64(t_2 * sqrt((sin(t_0) ^ 2.0)));
	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[(N[(y$46$re * t$95$1), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -4e+106], N[(t$95$2 * N[Sqrt[N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $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.re < -4.00000000000000036e106

   1. Initial program 35.6%

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

      1. hypot-def 35.6%

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

      2. fma-def 35.6%

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

      3. hypot-def 73.3%

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

      4. *-commutative 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in y.im around 0 82.2%

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

      1. add-sqr-sqrt 46.7%

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

      2. sqrt-unprod 93.3%

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

      3. pow2 93.3%

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

      4. *-commutative 93.3%

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

   6. Applied egg-rr 93.3%

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

   if -4.00000000000000036e106 < y.re

   1. Initial program 38.4%

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

      1. hypot-def 38.4%

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

      2. fma-def 38.4%

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

      3. hypot-def 80.7%

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

      4. *-commutative 80.7%

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

   3. Simplified 80.7%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{+106}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternative 5: 76.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(t_1, y.im, t_0\right)\right)\\ \mathbf{if}\;y.re \leq -0.00022:\\ \;\;\;\;e^{y.re \cdot t_1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt{{\sin t_0}^{2}}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;t_2 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (hypot x.re x.im)))
        (t_2 (sin (fma t_1 y.im t_0))))
   (if (<= y.re -0.00022)
     (*
      (exp (- (* y.re t_1) (* (atan2 x.im x.re) y.im)))
      (sqrt (pow (sin t_0) 2.0)))
     (if (<= y.re 4.5e-7)
       (* t_2 (exp (* (atan2 x.im x.re) (- y.im))))
       (* t_2 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = sin(fma(t_1, y_46_im, t_0));
	double tmp;
	if (y_46_re <= -0.00022) {
		tmp = exp(((y_46_re * t_1) - (atan2(x_46_im, x_46_re) * y_46_im))) * sqrt(pow(sin(t_0), 2.0));
	} else if (y_46_re <= 4.5e-7) {
		tmp = t_2 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_2 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(hypot(x_46_re, x_46_im))
	t_2 = sin(fma(t_1, y_46_im, t_0))
	tmp = 0.0
	if (y_46_re <= -0.00022)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_1) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sqrt((sin(t_0) ^ 2.0)));
	elseif (y_46_re <= 4.5e-7)
		tmp = Float64(t_2 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64(t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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[Sin[N[(t$95$1 * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -0.00022], N[(N[Exp[N[(N[(y$46$re * t$95$1), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.5e-7], N[(t$95$2 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.20000000000000008e-4

   1. Initial program 41.4%

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

      1. hypot-def 41.4%

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

      2. fma-def 41.4%

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

      3. hypot-def 78.5%

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

      4. *-commutative 78.5%

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

   3. Simplified 78.5%

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

   4. Taylor expanded in y.im around 0 82.8%

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

      1. add-sqr-sqrt 45.7%

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

      2. sqrt-unprod 90.6%

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

      3. pow2 90.6%

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

      4. *-commutative 90.6%

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

   6. Applied egg-rr 90.6%

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

   if -2.20000000000000008e-4 < y.re < 4.4999999999999998e-7

   1. Initial program 33.1%

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

      1. exp-diff 33.1%

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

      2. exp-to-pow 33.1%

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

      3. hypot-def 33.1%

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

      4. *-commutative 33.1%

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

      5. exp-prod 33.0%

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

      6. fma-def 33.0%

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

      7. hypot-def 81.9%

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

      8. *-commutative 81.9%

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

   3. Simplified 81.9%

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

   4. Taylor expanded in y.re around 0 81.6%

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

      1. rec-exp 81.6%

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

      2. distribute-rgt-neg-in 81.6%

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

   6. Simplified 81.6%

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

   if 4.4999999999999998e-7 < y.re

   1. Initial program 45.3%

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

      1. exp-diff 39.6%

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

      2. exp-to-pow 39.6%

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

      3. hypot-def 39.6%

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

      4. *-commutative 39.6%

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

      5. exp-prod 39.6%

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

      6. fma-def 39.6%

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

      7. hypot-def 64.2%

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

      8. *-commutative 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in y.im around 0 71.8%

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

      1. unpow2 71.8%

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

      2. unpow2 71.8%

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

      3. hypot-def 71.8%

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

   6. Simplified 71.8%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.00022:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sqrt{{\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{2}}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 6: 76.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := \sin \left(\mathsf{fma}\left(t_1, y.im, t_0\right)\right)\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-9}:\\ \;\;\;\;e^{y.re \cdot t_1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin t_0\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;t_2 \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (hypot x.re x.im)))
        (t_2 (sin (fma t_1 y.im t_0))))
   (if (<= y.re -2.15e-9)
     (* (exp (- (* y.re t_1) (* (atan2 x.im x.re) y.im))) (sin t_0))
     (if (<= y.re 3.6e-7)
       (* t_2 (exp (* (atan2 x.im x.re) (- y.im))))
       (* t_2 (pow (hypot x.im x.re) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = sin(fma(t_1, y_46_im, t_0));
	double tmp;
	if (y_46_re <= -2.15e-9) {
		tmp = exp(((y_46_re * t_1) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(t_0);
	} else if (y_46_re <= 3.6e-7) {
		tmp = t_2 * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_2 * pow(hypot(x_46_im, x_46_re), y_46_re);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(hypot(x_46_re, x_46_im))
	t_2 = sin(fma(t_1, y_46_im, t_0))
	tmp = 0.0
	if (y_46_re <= -2.15e-9)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_1) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(t_0));
	elseif (y_46_re <= 3.6e-7)
		tmp = Float64(t_2 * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	else
		tmp = Float64(t_2 * (hypot(x_46_im, x_46_re) ^ y_46_re));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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[Sin[N[(t$95$1 * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -2.15e-9], N[(N[Exp[N[(N[(y$46$re * t$95$1), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.6e-7], N[(t$95$2 * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.14999999999999981e-9

   1. Initial program 40.2%

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

      1. hypot-def 40.2%

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

      2. fma-def 40.2%

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

      3. hypot-def 76.6%

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

      4. *-commutative 76.6%

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

   3. Simplified 76.6%

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

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

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

   if -2.14999999999999981e-9 < y.re < 3.59999999999999994e-7

   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. Step-by-step derivation

      1. exp-diff 33.6%

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

      2. exp-to-pow 33.6%

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

      3. hypot-def 33.6%

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

      4. *-commutative 33.6%

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

      5. exp-prod 33.5%

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

      6. fma-def 33.5%

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

      7. hypot-def 83.1%

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

      8. *-commutative 83.1%

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

   3. Simplified 83.1%

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

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

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

      1. rec-exp 82.7%

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

      2. distribute-rgt-neg-in 82.7%

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

   6. Simplified 82.7%

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

   if 3.59999999999999994e-7 < y.re

   1. Initial program 45.3%

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

      1. exp-diff 39.6%

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

      2. exp-to-pow 39.6%

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

      3. hypot-def 39.6%

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

      4. *-commutative 39.6%

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

      5. exp-prod 39.6%

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

      6. fma-def 39.6%

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

      7. hypot-def 64.2%

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

      8. *-commutative 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in y.im around 0 71.8%

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

      1. unpow2 71.8%

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

      2. unpow2 71.8%

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

      3. hypot-def 71.8%

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

   6. Simplified 71.8%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-9}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-7}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \end{array} \]

Alternative 7: 77.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ t_2 := e^{y.re \cdot t_1 - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin t_0\\ \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2300000000:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(t_1, y.im, t_0\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+226}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\left|t_0\right| \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

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 (- (* y.re t_1) (* (atan2 x.im x.re) y.im))) (sin t_0))))
   (if (<= y.im -6.5e+31)
     t_2
     (if (<= y.im 2300000000.0)
       (* (sin (fma t_1 y.im t_0)) (pow (hypot x.im x.re) y.re))
       (if (<= y.im 8e+226)
         t_2
         (* (fabs t_0) (exp (* (atan2 x.im x.re) (- y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(hypot(x_46_re, x_46_im));
	double t_2 = exp(((y_46_re * t_1) - (atan2(x_46_im, x_46_re) * y_46_im))) * sin(t_0);
	double tmp;
	if (y_46_im <= -6.5e+31) {
		tmp = t_2;
	} else if (y_46_im <= 2300000000.0) {
		tmp = sin(fma(t_1, y_46_im, t_0)) * pow(hypot(x_46_im, x_46_re), y_46_re);
	} else if (y_46_im <= 8e+226) {
		tmp = t_2;
	} else {
		tmp = fabs(t_0) * exp((atan2(x_46_im, x_46_re) * -y_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(hypot(x_46_re, x_46_im))
	t_2 = Float64(exp(Float64(Float64(y_46_re * t_1) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * sin(t_0))
	tmp = 0.0
	if (y_46_im <= -6.5e+31)
		tmp = t_2;
	elseif (y_46_im <= 2300000000.0)
		tmp = Float64(sin(fma(t_1, y_46_im, t_0)) * (hypot(x_46_im, x_46_re) ^ y_46_re));
	elseif (y_46_im <= 8e+226)
		tmp = t_2;
	else
		tmp = Float64(abs(t_0) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(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[(N[Exp[N[(N[(y$46$re * t$95$1), $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.5e+31], t$95$2, If[LessEqual[y$46$im, 2300000000.0], N[(N[Sin[N[(t$95$1 * y$46$im + t$95$0), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8e+226], t$95$2, N[(N[Abs[t$95$0], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.5000000000000004e31 or 2.3e9 < y.im < 7.99999999999999969e226

   1. Initial program 33.3%

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

      1. hypot-def 33.3%

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

      2. fma-def 33.3%

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

      3. hypot-def 65.1%

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

      4. *-commutative 65.1%

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

   3. Simplified 65.1%

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

   4. Taylor expanded in y.im around 0 70.4%

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

   if -6.5000000000000004e31 < y.im < 2.3e9

   1. Initial program 43.0%

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

      1. exp-diff 42.2%

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

      2. exp-to-pow 42.3%

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

      3. hypot-def 42.3%

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

      4. *-commutative 42.3%

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

      5. exp-prod 41.4%

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

      6. fma-def 41.4%

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

      7. hypot-def 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. Taylor expanded in y.im around 0 61.4%

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

      1. unpow2 61.4%

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

      2. unpow2 61.4%

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

      3. hypot-def 90.1%

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

   6. Simplified 90.1%

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

   if 7.99999999999999969e226 < y.im

   1. Initial program 26.8%

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

      1. hypot-def 26.8%

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

      2. fma-def 26.8%

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

      3. hypot-def 74.7%

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

      4. *-commutative 74.7%

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

   3. Simplified 74.7%

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

   4. Taylor expanded in y.im around 0 40.4%

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

   5. Taylor expanded in y.re around 0 47.1%

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

      1. *-commutative 47.1%

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

      2. distribute-rgt-neg-in 47.1%

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

   7. Simplified 47.1%

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

      1. *-commutative 47.1%

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

      2. add-sqr-sqrt 33.5%

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

      3. sqrt-unprod 53.8%

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

      4. pow2 53.8%

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

   9. Applied egg-rr 53.8%

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

       1. *-commutative 53.8%

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

       2. unpow2 53.8%

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

       3. rem-sqrt-square 73.8%

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

   11. Simplified 73.8%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.im \leq 2300000000:\\ \;\;\;\;\sin \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right) \cdot {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+226}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]

Alternative 8: 64.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(-x.im\right)\\ t_2 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1.25 \cdot 10^{-116}:\\ \;\;\;\;e^{y.re \cdot t_1 - t_2} \cdot \sin \left(t_0 + y.im \cdot t_1\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - t_2} \cdot \sin t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (log (- x.im)))
        (t_2 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -1.25e-116)
     (* (exp (- (* y.re t_1) t_2)) (sin (+ t_0 (* y.im t_1))))
     (* (exp (- (* y.re (log (hypot x.re x.im))) t_2)) (sin t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(-x_46_im);
	double t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -1.25e-116) {
		tmp = exp(((y_46_re * t_1) - t_2)) * sin((t_0 + (y_46_im * t_1)));
	} else {
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - t_2)) * sin(t_0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.log(-x_46_im);
	double t_2 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -1.25e-116) {
		tmp = Math.exp(((y_46_re * t_1) - t_2)) * Math.sin((t_0 + (y_46_im * t_1)));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))) - t_2)) * Math.sin(t_0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.log(-x_46_im)
	t_2 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -1.25e-116:
		tmp = math.exp(((y_46_re * t_1) - t_2)) * math.sin((t_0 + (y_46_im * t_1)))
	else:
		tmp = math.exp(((y_46_re * math.log(math.hypot(x_46_re, x_46_im))) - t_2)) * math.sin(t_0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(Float64(-x_46_im))
	t_2 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.25e-116)
		tmp = Float64(exp(Float64(Float64(y_46_re * t_1) - t_2)) * sin(Float64(t_0 + Float64(y_46_im * t_1))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - t_2)) * sin(t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = y_46_re * atan2(x_46_im, x_46_re);
	t_1 = log(-x_46_im);
	t_2 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -1.25e-116)
		tmp = exp(((y_46_re * t_1) - t_2)) * sin((t_0 + (y_46_im * t_1)));
	else
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - t_2)) * sin(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Log[(-x$46$im)], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -1.25e-116], N[(N[Exp[N[(N[(y$46$re * t$95$1), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(t$95$0 + N[(y$46$im * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sin[t$95$0], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.2500000000000001e-116

   1. Initial program 37.6%

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

   2. Taylor expanded in x.im around -inf 37.1%

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

      1. mul-1-neg 37.1%

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

   4. Simplified 37.1%

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

   5. Taylor expanded in x.im around -inf 75.0%

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

      1. mul-1-neg 37.1%

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

   7. Simplified 75.0%

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

   if -1.2500000000000001e-116 < x.im

   1. Initial program 38.0%

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

      1. hypot-def 38.0%

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

      2. fma-def 38.0%

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

      3. hypot-def 76.8%

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

      4. *-commutative 76.8%

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

   3. Simplified 76.8%

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

   4. Taylor expanded in y.im around 0 64.4%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.25 \cdot 10^{-116}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \log \left(-x.im\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 9: 58.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ t_1 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -7.5 \cdot 10^{-229}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - t_1} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 1.95 \cdot 10^{+30}:\\ \;\;\;\;t_0 \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{t_1 + 1}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot e^{y.re \cdot \log x.im - t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (sin (* y.re (atan2 x.im x.re))))
        (t_1 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -7.5e-229)
     (*
      (exp (- (* y.re (log (- x.im))) t_1))
      (sin (* y.im (log (hypot x.im x.re)))))
     (if (<= x.im 1.95e+30)
       (* t_0 (/ (pow (hypot x.re x.im) y.re) (+ t_1 1.0)))
       (* t_0 (exp (- (* y.re (log x.im)) t_1)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -7.5e-229) {
		tmp = exp(((y_46_re * log(-x_46_im)) - t_1)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else if (x_46_im <= 1.95e+30) {
		tmp = t_0 * (pow(hypot(x_46_re, x_46_im), y_46_re) / (t_1 + 1.0));
	} else {
		tmp = t_0 * exp(((y_46_re * log(x_46_im)) - t_1));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double t_1 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -7.5e-229) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_im)) - t_1)) * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else if (x_46_im <= 1.95e+30) {
		tmp = t_0 * (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / (t_1 + 1.0));
	} else {
		tmp = t_0 * Math.exp(((y_46_re * Math.log(x_46_im)) - t_1));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	t_1 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -7.5e-229:
		tmp = math.exp(((y_46_re * math.log(-x_46_im)) - t_1)) * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	elif x_46_im <= 1.95e+30:
		tmp = t_0 * (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / (t_1 + 1.0))
	else:
		tmp = t_0 * math.exp(((y_46_re * math.log(x_46_im)) - t_1))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	t_1 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -7.5e-229)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_1)) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	elseif (x_46_im <= 1.95e+30)
		tmp = Float64(t_0 * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(t_1 + 1.0)));
	else
		tmp = Float64(t_0 * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	t_1 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -7.5e-229)
		tmp = exp(((y_46_re * log(-x_46_im)) - t_1)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	elseif (x_46_im <= 1.95e+30)
		tmp = t_0 * ((hypot(x_46_re, x_46_im) ^ y_46_re) / (t_1 + 1.0));
	else
		tmp = t_0 * exp(((y_46_re * log(x_46_im)) - t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -7.5e-229], N[(N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.95e+30], N[(t$95$0 * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(t$95$1 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -7.4999999999999999e-229

   1. Initial program 37.9%

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

   2. Taylor expanded in x.im around -inf 36.6%

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

      1. mul-1-neg 36.6%

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

   4. Simplified 36.6%

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

   5. Taylor expanded in y.re around 0 31.1%

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

      1. unpow2 31.1%

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

      2. unpow2 31.1%

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

      3. hypot-def 70.4%

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

   7. Simplified 70.4%

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

   if -7.4999999999999999e-229 < x.im < 1.95000000000000005e30

   1. Initial program 48.8%

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

      1. hypot-def 48.8%

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

      2. fma-def 48.8%

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

      3. hypot-def 75.5%

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

      4. *-commutative 75.5%

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

   3. Simplified 75.5%

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

   4. Taylor expanded in y.im around 0 63.3%

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

      1. expm1-log1p-u 55.8%

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

      2. expm1-udef 46.3%

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

      3. *-commutative 46.3%

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

      4. *-commutative 46.3%

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

      5. exp-diff 42.5%

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

      6. exp-to-pow 42.5%

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

      7. exp-prod 41.4%

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

   6. Applied egg-rr 41.4%

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

      1. expm1-def 50.9%

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

      2. expm1-log1p 58.4%

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

      3. *-commutative 58.4%

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

   8. Simplified 58.4%

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

   9. Taylor expanded in y.im around 0 54.0%

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

   if 1.95000000000000005e30 < x.im

   1. Initial program 25.0%

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

      1. hypot-def 25.0%

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

      2. fma-def 25.0%

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

      3. hypot-def 77.8%

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

      4. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in y.im around 0 68.3%

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

   5. Taylor expanded in x.re around 0 68.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -7.5 \cdot 10^{-229}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{elif}\;x.im \leq 1.95 \cdot 10^{+30}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 10: 62.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ \mathbf{if}\;x.im \leq -1.05 \cdot 10^{-268}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - t_0} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - t_0} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im)))
   (if (<= x.im -1.05e-268)
     (*
      (exp (- (* y.re (log (- x.im))) t_0))
      (sin (* y.im (log (hypot x.im x.re)))))
     (*
      (exp (- (* y.re (log (hypot x.re x.im))) 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 = atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -1.05e-268) {
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	} else {
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - 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 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double tmp;
	if (x_46_im <= -1.05e-268) {
		tmp = Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0)) * Math.sin((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re))));
	} else {
		tmp = Math.exp(((y_46_re * Math.log(Math.hypot(x_46_re, x_46_im))) - 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 = math.atan2(x_46_im, x_46_re) * y_46_im
	tmp = 0
	if x_46_im <= -1.05e-268:
		tmp = math.exp(((y_46_re * math.log(-x_46_im)) - t_0)) * math.sin((y_46_im * math.log(math.hypot(x_46_im, x_46_re))))
	else:
		tmp = math.exp(((y_46_re * math.log(math.hypot(x_46_re, x_46_im))) - 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(atan(x_46_im, x_46_re) * y_46_im)
	tmp = 0.0
	if (x_46_im <= -1.05e-268)
		tmp = Float64(exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0)) * sin(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))));
	else
		tmp = Float64(exp(Float64(Float64(y_46_re * log(hypot(x_46_re, x_46_im))) - 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 = atan2(x_46_im, x_46_re) * y_46_im;
	tmp = 0.0;
	if (x_46_im <= -1.05e-268)
		tmp = exp(((y_46_re * log(-x_46_im)) - t_0)) * sin((y_46_im * log(hypot(x_46_im, x_46_re))));
	else
		tmp = exp(((y_46_re * log(hypot(x_46_re, x_46_im))) - 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[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, If[LessEqual[x$46$im, -1.05e-268], N[(N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(N[(y$46$re * N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $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 2 regimes

2. if x.im < -1.04999999999999999e-268

   1. Initial program 38.7%

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

   2. Taylor expanded in x.im around -inf 37.5%

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

      1. mul-1-neg 37.5%

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

   4. Simplified 37.5%

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

   5. Taylor expanded in y.re around 0 30.2%

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

      1. unpow2 30.2%

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

      2. unpow2 30.2%

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

      3. hypot-def 68.5%

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

   7. Simplified 68.5%

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

   if -1.04999999999999999e-268 < x.im

   1. Initial program 37.2%

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

      1. hypot-def 37.2%

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

      2. fma-def 37.2%

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

      3. hypot-def 76.7%

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

      4. *-commutative 76.7%

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

   3. Simplified 76.7%

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

   4. Taylor expanded in y.im around 0 67.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.05 \cdot 10^{-268}:\\ \;\;\;\;e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \end{array} \]

Alternative 11: 57.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im\\ t_1 := \sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;x.im \leq -8 \cdot 10^{-105}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log \left(-x.im\right) - t_0}\\ \mathbf{elif}\;x.im \leq 1.95 \cdot 10^{+30}:\\ \;\;\;\;t_1 \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{t_0 + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot e^{y.re \cdot \log x.im - t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (atan2 x.im x.re) y.im))
        (t_1 (sin (* y.re (atan2 x.im x.re)))))
   (if (<= x.im -8e-105)
     (* t_1 (exp (- (* y.re (log (- x.im))) t_0)))
     (if (<= x.im 1.95e+30)
       (* t_1 (/ (pow (hypot x.re x.im) y.re) (+ t_0 1.0)))
       (* t_1 (exp (- (* y.re (log x.im)) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -8e-105) {
		tmp = t_1 * exp(((y_46_re * log(-x_46_im)) - t_0));
	} else if (x_46_im <= 1.95e+30) {
		tmp = t_1 * (pow(hypot(x_46_re, x_46_im), y_46_re) / (t_0 + 1.0));
	} else {
		tmp = t_1 * exp(((y_46_re * log(x_46_im)) - t_0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = Math.atan2(x_46_im, x_46_re) * y_46_im;
	double t_1 = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re)));
	double tmp;
	if (x_46_im <= -8e-105) {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(-x_46_im)) - t_0));
	} else if (x_46_im <= 1.95e+30) {
		tmp = t_1 * (Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re) / (t_0 + 1.0));
	} else {
		tmp = t_1 * Math.exp(((y_46_re * Math.log(x_46_im)) - t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.atan2(x_46_im, x_46_re) * y_46_im
	t_1 = math.sin((y_46_re * math.atan2(x_46_im, x_46_re)))
	tmp = 0
	if x_46_im <= -8e-105:
		tmp = t_1 * math.exp(((y_46_re * math.log(-x_46_im)) - t_0))
	elif x_46_im <= 1.95e+30:
		tmp = t_1 * (math.pow(math.hypot(x_46_re, x_46_im), y_46_re) / (t_0 + 1.0))
	else:
		tmp = t_1 * math.exp(((y_46_re * math.log(x_46_im)) - t_0))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(atan(x_46_im, x_46_re) * y_46_im)
	t_1 = sin(Float64(y_46_re * atan(x_46_im, x_46_re)))
	tmp = 0.0
	if (x_46_im <= -8e-105)
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(Float64(-x_46_im))) - t_0)));
	elseif (x_46_im <= 1.95e+30)
		tmp = Float64(t_1 * Float64((hypot(x_46_re, x_46_im) ^ y_46_re) / Float64(t_0 + 1.0)));
	else
		tmp = Float64(t_1 * exp(Float64(Float64(y_46_re * log(x_46_im)) - t_0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = atan2(x_46_im, x_46_re) * y_46_im;
	t_1 = sin((y_46_re * atan2(x_46_im, x_46_re)));
	tmp = 0.0;
	if (x_46_im <= -8e-105)
		tmp = t_1 * exp(((y_46_re * log(-x_46_im)) - t_0));
	elseif (x_46_im <= 1.95e+30)
		tmp = t_1 * ((hypot(x_46_re, x_46_im) ^ y_46_re) / (t_0 + 1.0));
	else
		tmp = t_1 * exp(((y_46_re * log(x_46_im)) - t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -8e-105], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[(-x$46$im)], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.95e+30], N[(t$95$1 * N[(N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(t$95$0 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[Exp[N[(N[(y$46$re * N[Log[x$46$im], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x.im < -7.99999999999999972e-105

   1. Initial program 36.5%

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

   2. Taylor expanded in x.im around -inf 36.0%

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

      1. mul-1-neg 36.0%

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

   4. Simplified 36.0%

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

   5. Taylor expanded in y.im around 0 64.9%

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

   if -7.99999999999999972e-105 < x.im < 1.95000000000000005e30

   1. Initial program 47.2%

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

      1. hypot-def 47.2%

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

      2. fma-def 47.2%

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

      3. hypot-def 75.8%

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

      4. *-commutative 75.8%

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

   3. Simplified 75.8%

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

   4. Taylor expanded in y.im around 0 61.1%

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

      1. expm1-log1p-u 55.4%

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

      2. expm1-udef 44.7%

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

      3. *-commutative 44.7%

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

      4. *-commutative 44.7%

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

      5. exp-diff 40.0%

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

      6. exp-to-pow 40.0%

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

      7. exp-prod 39.1%

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

   6. Applied egg-rr 39.1%

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

      1. expm1-def 49.9%

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

      2. expm1-log1p 55.5%

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

      3. *-commutative 55.5%

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

   8. Simplified 55.5%

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

   9. Taylor expanded in y.im around 0 54.1%

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

   if 1.95000000000000005e30 < x.im

   1. Initial program 25.0%

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

      1. hypot-def 25.0%

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

      2. fma-def 25.0%

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

      3. hypot-def 77.8%

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

      4. *-commutative 77.8%

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

   3. Simplified 77.8%

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

   4. Taylor expanded in y.im around 0 68.3%

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

   5. Taylor expanded in x.re around 0 68.3%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -8 \cdot 10^{-105}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log \left(-x.im\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \mathbf{elif}\;x.im \leq 1.95 \cdot 10^{+30}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot \frac{{\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot e^{y.re \cdot \log x.im - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}\\ \end{array} \]

Alternative 12: 58.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+39} \lor \neg \left(y.re \leq 0.0037\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.95e+39) (not (<= y.re 0.0037)))
   (*
    (sin (* y.re (atan2 x.im x.re)))
    (pow (sqrt (+ (* x.re x.re) (* x.im x.im))) y.re))
   (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.95e+39) || !(y_46_re <= 0.0037)) {
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * pow(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.95d+39)) .or. (.not. (y_46re <= 0.0037d0))) then
        tmp = sin((y_46re * atan2(x_46im, x_46re))) * (sqrt(((x_46re * x_46re) + (x_46im * x_46im))) ** y_46re)
    else
        tmp = y_46re * (atan2(x_46im, x_46re) * exp((atan2(x_46im, x_46re) * -y_46im)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.95e+39) || !(y_46_re <= 0.0037)) {
		tmp = Math.sin((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(Math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re);
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.95e+39) or not (y_46_re <= 0.0037):
		tmp = math.sin((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(math.sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))), y_46_re)
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.95e+39) || !(y_46_re <= 0.0037))
		tmp = Float64(sin(Float64(y_46_re * atan(x_46_im, x_46_re))) * (sqrt(Float64(Float64(x_46_re * x_46_re) + Float64(x_46_im * x_46_im))) ^ y_46_re));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	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 <= -1.95e+39) || ~((y_46_re <= 0.0037)))
		tmp = sin((y_46_re * atan2(x_46_im, x_46_re))) * (sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))) ^ y_46_re);
	else
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.95e+39], N[Not[LessEqual[y$46$re, 0.0037]], $MachinePrecision]], N[(N[Sin[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[N[Sqrt[N[(N[(x$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], y$46$re], $MachinePrecision]), $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.95e39 or 0.0037000000000000002 < y.re

   1. Initial program 43.3%

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

      1. hypot-def 43.3%

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

      2. fma-def 43.3%

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

      3. hypot-def 76.1%

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

      4. *-commutative 76.1%

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

   3. Simplified 76.1%

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

   4. Taylor expanded in y.im around 0 77.9%

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

   5. Taylor expanded in y.im around 0 70.1%

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

      1. *-commutative 70.1%

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

      2. unpow2 70.1%

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

      3. unpow2 70.1%

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

   7. Simplified 70.1%

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

   if -1.95e39 < y.re < 0.0037000000000000002

   1. Initial program 33.5%

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

      1. hypot-def 33.5%

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

      2. fma-def 33.6%

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

      3. hypot-def 82.0%

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

      4. *-commutative 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in y.im around 0 53.4%

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

   5. Taylor expanded in y.re around 0 52.8%

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

      1. *-commutative 52.8%

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

      2. distribute-rgt-neg-in 52.8%

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

   7. Simplified 52.8%

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

      1. *-commutative 52.8%

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

      2. add-cbrt-cube 37.4%

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

      3. pow1/3 21.8%

         \[\leadsto \color{blue}{{\left(\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      4. pow3 21.8%

         \[\leadsto {\color{blue}{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}}^{0.3333333333333333} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Applied egg-rr 21.8%

      \[\leadsto \color{blue}{{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   10. Taylor expanded in x.im around 0 53.5%

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

       1. mul-1-neg 53.5%

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

   12. Simplified 53.5%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+39} \lor \neg \left(y.re \leq 0.0037\right):\\ \;\;\;\;\sin \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {\left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right)}^{y.re}\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \end{array} \]

Alternative 13: 43.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\left|t_0\right| \cdot t_1\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot t_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re)))
        (t_1 (exp (* (atan2 x.im x.re) (- y.im)))))
   (if (<= y.im -1e+51)
     (* (fabs t_0) t_1)
     (if (<= y.im 1.4e-58)
       (log1p (expm1 t_0))
       (* y.re (* (atan2 x.im x.re) t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	double tmp;
	if (y_46_im <= -1e+51) {
		tmp = fabs(t_0) * t_1;
	} else if (y_46_im <= 1.4e-58) {
		tmp = log1p(expm1(t_0));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * t_1);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double t_1 = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	double tmp;
	if (y_46_im <= -1e+51) {
		tmp = Math.abs(t_0) * t_1;
	} else if (y_46_im <= 1.4e-58) {
		tmp = Math.log1p(Math.expm1(t_0));
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	t_1 = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	tmp = 0
	if y_46_im <= -1e+51:
		tmp = math.fabs(t_0) * t_1
	elif y_46_im <= 1.4e-58:
		tmp = math.log1p(math.expm1(t_0))
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * t_1)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))
	tmp = 0.0
	if (y_46_im <= -1e+51)
		tmp = Float64(abs(t_0) * t_1);
	elseif (y_46_im <= 1.4e-58)
		tmp = log1p(expm1(t_0));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * t_1));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$im, -1e+51], N[(N[Abs[t$95$0], $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$im, 1.4e-58], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1e51

   1. Initial program 28.1%

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

      1. hypot-def 28.1%

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

      2. fma-def 28.1%

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

      3. hypot-def 59.7%

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

      4. *-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y.im around 0 67.3%

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

   5. Taylor expanded in y.re around 0 52.1%

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

      1. *-commutative 52.1%

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

      2. distribute-rgt-neg-in 52.1%

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

   7. Simplified 52.1%

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

      1. *-commutative 52.1%

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

      2. add-sqr-sqrt 34.8%

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

      3. sqrt-unprod 39.5%

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

      4. pow2 39.5%

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

   9. Applied egg-rr 39.5%

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

       1. *-commutative 39.5%

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

       2. unpow2 39.5%

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

       3. rem-sqrt-square 55.2%

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

   11. Simplified 55.2%

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

   if -1e51 < y.im < 1.4e-58

   1. Initial program 41.7%

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

      1. hypot-def 41.7%

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

      2. fma-def 41.7%

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

      3. hypot-def 90.7%

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

      4. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in y.im around 0 65.6%

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

   5. Taylor expanded in y.re around 0 26.2%

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

      1. *-commutative 26.2%

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

      2. distribute-rgt-neg-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in y.im around 0 25.4%

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

      1. *-commutative 25.4%

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

      2. log1p-expm1-u 34.7%

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

   10. Applied egg-rr 34.7%

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

   if 1.4e-58 < y.im

   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. Step-by-step derivation

      1. hypot-def 40.1%

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

      2. fma-def 40.1%

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

      3. hypot-def 77.9%

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

      4. *-commutative 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in y.im around 0 59.1%

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

   5. Taylor expanded in y.re around 0 52.3%

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

      1. *-commutative 52.3%

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

      2. distribute-rgt-neg-in 52.3%

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

   7. Simplified 52.3%

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

      1. *-commutative 52.3%

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

      2. add-cbrt-cube 39.8%

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

      3. pow1/3 30.1%

         \[\leadsto \color{blue}{{\left(\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      4. pow3 30.1%

         \[\leadsto {\color{blue}{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}}^{0.3333333333333333} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   10. Taylor expanded in x.im around 0 53.7%

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

       1. mul-1-neg 53.7%

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

   12. Simplified 53.7%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\left|y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right| \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \end{array} \]

Alternative 14: 43.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\frac{t_0}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.im -1e+51)
     (/ t_0 (pow (exp y.im) (atan2 x.im x.re)))
     (if (<= y.im 9.2e-60)
       (log1p (expm1 t_0))
       (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= -1e+51) {
		tmp = t_0 / pow(exp(y_46_im), atan2(x_46_im, x_46_re));
	} else if (y_46_im <= 9.2e-60) {
		tmp = log1p(expm1(t_0));
	} else {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	}
	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 <= -1e+51) {
		tmp = t_0 / Math.pow(Math.exp(y_46_im), Math.atan2(x_46_im, x_46_re));
	} else if (y_46_im <= 9.2e-60) {
		tmp = Math.log1p(Math.expm1(t_0));
	} else {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if y_46_im <= -1e+51:
		tmp = t_0 / math.pow(math.exp(y_46_im), math.atan2(x_46_im, x_46_re))
	elif y_46_im <= 9.2e-60:
		tmp = math.log1p(math.expm1(t_0))
	else:
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= -1e+51)
		tmp = Float64(t_0 / (exp(y_46_im) ^ atan(x_46_im, x_46_re)));
	elseif (y_46_im <= 9.2e-60)
		tmp = log1p(expm1(t_0));
	else
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1e+51], N[(t$95$0 / N[Power[N[Exp[y$46$im], $MachinePrecision], N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.2e-60], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1e51

   1. Initial program 28.1%

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

      1. hypot-def 28.1%

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

      2. fma-def 28.1%

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

      3. hypot-def 59.7%

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

      4. *-commutative 59.7%

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

   3. Simplified 59.7%

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

   4. Taylor expanded in y.im around 0 67.3%

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

      1. expm1-log1p-u 56.4%

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

      2. expm1-udef 56.4%

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

      3. *-commutative 56.4%

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

      4. *-commutative 56.4%

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

      5. exp-diff 42.3%

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

      6. exp-to-pow 42.3%

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

      7. exp-prod 42.5%

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

   6. Applied egg-rr 42.5%

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

      1. expm1-def 42.5%

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

      2. expm1-log1p 51.9%

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

      3. *-commutative 51.9%

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

   8. Simplified 51.9%

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

   9. Taylor expanded in y.re around 0 52.1%

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

       1. exp-prod 55.1%

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

   11. Simplified 55.1%

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

   if -1e51 < y.im < 9.2000000000000005e-60

   1. Initial program 41.7%

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

      1. hypot-def 41.7%

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

      2. fma-def 41.7%

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

      3. hypot-def 90.7%

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

      4. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in y.im around 0 65.6%

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

   5. Taylor expanded in y.re around 0 26.2%

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

      1. *-commutative 26.2%

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

      2. distribute-rgt-neg-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in y.im around 0 25.4%

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

      1. *-commutative 25.4%

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

      2. log1p-expm1-u 34.7%

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

   10. Applied egg-rr 34.7%

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

   if 9.2000000000000005e-60 < y.im

   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. Step-by-step derivation

      1. hypot-def 40.1%

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

      2. fma-def 40.1%

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

      3. hypot-def 77.9%

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

      4. *-commutative 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in y.im around 0 59.1%

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

   5. Taylor expanded in y.re around 0 52.3%

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

      1. *-commutative 52.3%

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

      2. distribute-rgt-neg-in 52.3%

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

   7. Simplified 52.3%

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

      1. *-commutative 52.3%

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

      2. add-cbrt-cube 39.8%

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

      3. pow1/3 30.1%

         \[\leadsto \color{blue}{{\left(\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      4. pow3 30.1%

         \[\leadsto {\color{blue}{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}}^{0.3333333333333333} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   10. Taylor expanded in x.im around 0 53.7%

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

       1. mul-1-neg 53.7%

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

   12. Simplified 53.7%

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

4. Final simplification 45.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+51}:\\ \;\;\;\;\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{{\left(e^{y.im}\right)}^{\tan^{-1}_* \frac{x.im}{x.re}}}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-60}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \end{array} \]

Alternative 15: 43.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+51} \lor \neg \left(y.im \leq 9.5 \cdot 10^{-63}\right):\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1e+51) (not (<= y.im 9.5e-63)))
   (* y.re (* (atan2 x.im x.re) (exp (* (atan2 x.im x.re) (- y.im)))))
   (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) {
	double tmp;
	if ((y_46_im <= -1e+51) || !(y_46_im <= 9.5e-63)) {
		tmp = y_46_re * (atan2(x_46_im, x_46_re) * exp((atan2(x_46_im, x_46_re) * -y_46_im)));
	} else {
		tmp = log1p(expm1((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 tmp;
	if ((y_46_im <= -1e+51) || !(y_46_im <= 9.5e-63)) {
		tmp = y_46_re * (Math.atan2(x_46_im, x_46_re) * Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im)));
	} else {
		tmp = Math.log1p(Math.expm1((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):
	tmp = 0
	if (y_46_im <= -1e+51) or not (y_46_im <= 9.5e-63):
		tmp = y_46_re * (math.atan2(x_46_im, x_46_re) * math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im)))
	else:
		tmp = math.log1p(math.expm1((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)
	tmp = 0.0
	if ((y_46_im <= -1e+51) || !(y_46_im <= 9.5e-63))
		tmp = Float64(y_46_re * Float64(atan(x_46_im, x_46_re) * exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))));
	else
		tmp = log1p(expm1(Float64(y_46_re * atan(x_46_im, x_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1e+51], N[Not[LessEqual[y$46$im, 9.5e-63]], $MachinePrecision]], N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1e51 or 9.50000000000000016e-63 < y.im

   1. Initial program 34.5%

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

      1. hypot-def 34.5%

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

      2. fma-def 34.5%

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

      3. hypot-def 69.3%

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

      4. *-commutative 69.3%

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

   3. Simplified 69.3%

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

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

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

   5. Taylor expanded in y.re around 0 52.2%

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

      1. *-commutative 52.2%

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

      2. distribute-rgt-neg-in 52.2%

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

   7. Simplified 52.2%

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

      1. *-commutative 52.2%

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

      2. add-cbrt-cube 37.4%

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

      3. pow1/3 28.6%

         \[\leadsto \color{blue}{{\left(\left(\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right) \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

      4. pow3 28.6%

         \[\leadsto {\color{blue}{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}}^{0.3333333333333333} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   9. Applied egg-rr 28.6%

      \[\leadsto \color{blue}{{\left({\left(\tan^{-1}_* \frac{x.im}{x.re} \cdot y.re\right)}^{3}\right)}^{0.3333333333333333}} \cdot e^{y.im \cdot \left(-\tan^{-1}_* \frac{x.im}{x.re}\right)} \]

   10. Taylor expanded in x.im around 0 52.9%

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

       1. mul-1-neg 52.9%

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

   12. Simplified 52.9%

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

   if -1e51 < y.im < 9.50000000000000016e-63

   1. Initial program 41.7%

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

      1. hypot-def 41.7%

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

      2. fma-def 41.7%

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

      3. hypot-def 90.7%

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

      4. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in y.im around 0 65.6%

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

   5. Taylor expanded in y.re around 0 26.2%

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

      1. *-commutative 26.2%

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

      2. distribute-rgt-neg-in 26.2%

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

   7. Simplified 26.2%

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

   8. Taylor expanded in y.im around 0 25.4%

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

      1. *-commutative 25.4%

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

      2. log1p-expm1-u 34.7%

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

   10. Applied egg-rr 34.7%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+51} \lor \neg \left(y.im \leq 9.5 \cdot 10^{-63}\right):\\ \;\;\;\;y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} \cdot e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternative 16: 43.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq -1.36 \cdot 10^{+51} \lor \neg \left(y.im \leq 4.1 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{t_0}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \end{array} \end{array} \]

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 -1.36e+51) (not (<= y.im 4.1e-32)))
     (/ t_0 (exp (* (atan2 x.im x.re) y.im)))
     (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 <= -1.36e+51) || !(y_46_im <= 4.1e-32)) {
		tmp = t_0 / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} 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 <= -1.36e+51) || !(y_46_im <= 4.1e-32)) {
		tmp = t_0 / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = Math.log1p(Math.expm1(t_0));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = y_46_re * math.atan2(x_46_im, x_46_re)
	tmp = 0
	if (y_46_im <= -1.36e+51) or not (y_46_im <= 4.1e-32):
		tmp = t_0 / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	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 <= -1.36e+51) || !(y_46_im <= 4.1e-32))
		tmp = Float64(t_0 / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	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, -1.36e+51], N[Not[LessEqual[y$46$im, 4.1e-32]], $MachinePrecision]], N[(t$95$0 / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.3599999999999999e51 or 4.09999999999999975e-32 < y.im

   1. Initial program 33.5%

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

      1. hypot-def 33.5%

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

      2. fma-def 33.5%

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

      3. hypot-def 68.2%

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

      4. *-commutative 68.2%

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

   3. Simplified 68.2%

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

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

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

      1. expm1-log1p-u 53.0%

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

      2. expm1-udef 53.1%

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

      3. *-commutative 53.1%

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

      4. *-commutative 53.1%

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

      5. exp-diff 42.4%

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

      6. exp-to-pow 42.4%

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

      7. exp-prod 43.2%

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

   6. Applied egg-rr 43.2%

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

      1. expm1-def 43.2%

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

      2. expm1-log1p 52.4%

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

      3. *-commutative 52.4%

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

   8. Simplified 52.4%

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

   9. Taylor expanded in y.re around 0 54.1%

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

   if -1.3599999999999999e51 < y.im < 4.09999999999999975e-32

   1. Initial program 42.4%

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

      1. hypot-def 42.4%

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

      2. fma-def 42.4%

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

      3. hypot-def 91.1%

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

      4. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in y.im around 0 63.9%

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

   5. Taylor expanded in y.re around 0 25.3%

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

      1. *-commutative 25.3%

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

      2. distribute-rgt-neg-in 25.3%

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

   7. Simplified 25.3%

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

   8. Taylor expanded in y.im around 0 24.5%

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

      1. *-commutative 24.5%

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

      2. log1p-expm1-u 33.5%

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

   10. Applied egg-rr 33.5%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.36 \cdot 10^{+51} \lor \neg \left(y.im \leq 4.1 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternative 17: 34.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{if}\;y.im \leq -2.3 \cdot 10^{-69} \lor \neg \left(y.im \leq 9.5 \cdot 10^{-24}\right):\\ \;\;\;\;\log \left(1 + t_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(t_0\right)\\ \end{array} \end{array} \]

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 -2.3e-69) (not (<= y.im 9.5e-24)))
     (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 <= -2.3e-69) || !(y_46_im <= 9.5e-24)) {
		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 <= -2.3e-69) || !(y_46_im <= 9.5e-24)) {
		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 <= -2.3e-69) or not (y_46_im <= 9.5e-24):
		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 <= -2.3e-69) || !(y_46_im <= 9.5e-24))
		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, -2.3e-69], N[Not[LessEqual[y$46$im, 9.5e-24]], $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 < -2.3000000000000001e-69 or 9.50000000000000029e-24 < 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. hypot-def 36.7%

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

      2. fma-def 36.7%

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

      3. hypot-def 70.9%

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

      4. *-commutative 70.9%

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

   3. Simplified 70.9%

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

   4. Taylor expanded in y.im around 0 63.9%

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

   5. Taylor expanded in y.re around 0 47.3%

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

      1. *-commutative 47.3%

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

      2. distribute-rgt-neg-in 47.3%

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

   7. Simplified 47.3%

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

   8. Taylor expanded in y.im around 0 4.1%

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

      1. *-commutative 4.1%

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

      2. log1p-expm1-u 14.6%

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

      3. log1p-udef 33.2%

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

   10. Applied egg-rr 33.2%

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

   if -2.3000000000000001e-69 < y.im < 9.50000000000000029e-24

   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. Step-by-step derivation

      1. hypot-def 39.7%

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

      2. fma-def 39.7%

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

      3. hypot-def 91.9%

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

      4. *-commutative 91.9%

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

   3. Simplified 91.9%

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

   4. Taylor expanded in y.im around 0 64.7%

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

   5. Taylor expanded in y.re around 0 29.2%

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

      1. *-commutative 29.2%

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

      2. distribute-rgt-neg-in 29.2%

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

   7. Simplified 29.2%

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

   8. Taylor expanded in y.im around 0 29.2%

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

      1. *-commutative 29.2%

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

      2. log1p-expm1-u 36.3%

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

   10. Applied egg-rr 36.3%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.3 \cdot 10^{-69} \lor \neg \left(y.im \leq 9.5 \cdot 10^{-24}\right):\\ \;\;\;\;\log \left(1 + \mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \end{array} \]

Alternative 18: 25.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ \mathbf{if}\;y.im \leq 1.2 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(t_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{t_0}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* y.re (atan2 x.im x.re))))
   (if (<= y.im 1.2e-143) (log1p (expm1 t_0)) (cbrt (pow t_0 3.0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= 1.2e-143) {
		tmp = log1p(expm1(t_0));
	} else {
		tmp = cbrt(pow(t_0, 3.0));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * Math.atan2(x_46_im, x_46_re);
	double tmp;
	if (y_46_im <= 1.2e-143) {
		tmp = Math.log1p(Math.expm1(t_0));
	} else {
		tmp = Math.cbrt(Math.pow(t_0, 3.0));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_im <= 1.2e-143)
		tmp = log1p(expm1(t_0));
	else
		tmp = cbrt((t_0 ^ 3.0));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, 1.2e-143], N[Log[1 + N[(Exp[t$95$0] - 1), $MachinePrecision]], $MachinePrecision], N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < 1.1999999999999999e-143

   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. Step-by-step derivation

      1. hypot-def 35.2%

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

      2. fma-def 35.2%

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

      3. hypot-def 78.9%

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

      4. *-commutative 78.9%

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

   3. Simplified 78.9%

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

   4. Taylor expanded in y.im around 0 66.5%

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

   5. Taylor expanded in y.re around 0 37.8%

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

      1. *-commutative 37.8%

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

      2. distribute-rgt-neg-in 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in y.im around 0 18.6%

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

      1. *-commutative 18.6%

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

      2. log1p-expm1-u 27.6%

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

   10. Applied egg-rr 27.6%

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

   if 1.1999999999999999e-143 < y.im

   1. Initial program 42.8%

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

      1. hypot-def 42.8%

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

      2. fma-def 42.8%

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

      3. hypot-def 80.3%

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

      4. *-commutative 80.3%

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

   3. Simplified 80.3%

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

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

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

   5. Taylor expanded in y.re around 0 44.0%

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

      1. *-commutative 44.0%

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

      2. distribute-rgt-neg-in 44.0%

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

   7. Simplified 44.0%

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

   8. Taylor expanded in y.im around 0 6.1%

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

      1. *-commutative 6.1%

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

      2. add-cbrt-cube 21.6%

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

      3. pow3 21.6%

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

   10. Applied egg-rr 21.6%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq 1.2 \cdot 10^{-143}:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{expm1}\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}^{3}}\\ \end{array} \]

Alternative 19: 23.8% 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 37.9%

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

   1. hypot-def 37.9%

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

   2. fma-def 37.9%

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

   3. hypot-def 79.4%

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

   4. *-commutative 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in y.im around 0 64.2%

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

5. Taylor expanded in y.re around 0 40.0%

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

   1. *-commutative 40.0%

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

   2. distribute-rgt-neg-in 40.0%

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

7. Simplified 40.0%

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

8. Taylor expanded in y.im around 0 14.2%

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

   1. *-commutative 14.2%

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

   2. log1p-expm1-u 23.3%

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

10. Applied egg-rr 23.3%

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

11. Final simplification 23.3%

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

Alternative 20: 15.9% accurate, 3.9× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (* (* y.re (atan2 x.im x.re)) (- 1.0 (* (atan2 x.im x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * atan2(x_46_im, x_46_re)) * (1.0 - (atan2(x_46_im, x_46_re) * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = (y_46re * atan2(x_46im, x_46re)) * (1.0d0 - (atan2(x_46im, x_46re) * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (y_46_re * Math.atan2(x_46_im, x_46_re)) * (1.0 - (Math.atan2(x_46_im, x_46_re) * y_46_im));
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (y_46_re * math.atan2(x_46_im, x_46_re)) * (1.0 - (math.atan2(x_46_im, x_46_re) * y_46_im))

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(y_46_re * atan(x_46_im, x_46_re)) * Float64(1.0 - Float64(atan(x_46_im, x_46_re) * y_46_im)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (y_46_re * atan2(x_46_im, x_46_re)) * (1.0 - (atan2(x_46_im, x_46_re) * y_46_im));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 37.9%

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

   1. hypot-def 37.9%

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

   2. fma-def 37.9%

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

   3. hypot-def 79.4%

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

   4. *-commutative 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in y.im around 0 64.2%

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

5. Taylor expanded in y.re around 0 40.0%

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

   1. *-commutative 40.0%

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

   2. distribute-rgt-neg-in 40.0%

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

7. Simplified 40.0%

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

8. Taylor expanded in y.im around 0 16.3%

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

9. Final simplification 16.3%

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

Alternative 21: 13.9% 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 37.9%

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

   1. hypot-def 37.9%

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

   2. fma-def 37.9%

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

   3. hypot-def 79.4%

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

   4. *-commutative 79.4%

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

3. Simplified 79.4%

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

4. Taylor expanded in y.im around 0 64.2%

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

5. Taylor expanded in y.re around 0 40.0%

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

   1. *-commutative 40.0%

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

   2. distribute-rgt-neg-in 40.0%

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

7. Simplified 40.0%

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

8. Taylor expanded in y.im around 0 14.2%

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

9. Final simplification 14.2%

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

Reproduce

herbie shell --seed 2023274 
(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)))))