powComplex, real part

Percentage Accurate: 41.1% → 81.0%

Time: 25.5s

Alternatives: 8

Speedup: 4.0×

• 35.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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)))
    (cos (+ (* 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))) * cos(((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))) * cos(((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.cos(((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.cos(((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))) * cos(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))) * cos(((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[Cos[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: 41.1% accurate, 1.0× speedup

Math

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

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)))
    (cos (+ (* 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))) * cos(((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))) * cos(((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.cos(((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.cos(((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))) * cos(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))) * cos(((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[Cos[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: 81.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;x.im \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{fma}\left(t\_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.im}{y.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= x.im -1.45e-198)
     (*
      (exp (fma t_0 y.re (* (atan2 x.im x.re) (- y.im))))
      (cos (pow (cbrt (fma t_0 y.im (* y.re (atan2 x.im x.re)))) 3.0)))
     (exp
      (*
       y.re
       (- (log (hypot x.im x.re)) (* (atan2 x.im x.re) (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (x_46_im <= -1.45e-198) {
		tmp = exp(fma(t_0, y_46_re, (atan2(x_46_im, x_46_re) * -y_46_im))) * cos(pow(cbrt(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re)))), 3.0));
	} else {
		tmp = exp((y_46_re * (log(hypot(x_46_im, x_46_re)) - (atan2(x_46_im, x_46_re) * (y_46_im / y_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (x_46_im <= -1.45e-198)
		tmp = Float64(exp(fma(t_0, y_46_re, Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))) * cos((cbrt(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))) ^ 3.0)));
	else
		tmp = exp(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) - Float64(atan(x_46_im, x_46_re) * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -1.45e-198], N[(N[Exp[N[(t$95$0 * y$46$re + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Power[N[Power[N[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -1.45e-198

   1. Initial program 44.4%

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

      1. fma-neg 44.4%

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

      2. hypot-define 44.4%

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

      3. distribute-rgt-neg-out 44.4%

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

      4. fma-define 44.4%

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

      5. hypot-define 82.5%

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

      6. *-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. fma-undefine 82.5%

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

      2. hypot-define 44.4%

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

      3. *-commutative 44.4%

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

      4. add-cube-cbrt 47.2%

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

      5. pow3 48.2%

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

      6. fma-define 48.2%

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

      7. hypot-define 90.1%

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

   6. Applied egg-rr 90.1%

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

   if -1.45e-198 < x.im

   1. Initial program 42.8%

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

   3. Taylor expanded in y.im around 0 68.9%

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

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

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

   5. Taylor expanded in y.re around inf 70.4%

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

      1. +-commutative 70.4%

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

      2. unpow2 70.4%

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

      3. unpow2 70.4%

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

      4. hypot-undefine 86.1%

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

      5. mul-1-neg 86.1%

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

      6. unsub-neg 86.1%

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

      7. hypot-undefine 70.4%

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

      8. unpow2 70.4%

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

      9. unpow2 70.4%

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

      10. +-commutative 70.4%

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

      11. unpow2 70.4%

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

      12. unpow2 70.4%

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

      13. hypot-define 86.1%

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

      14. *-commutative 86.1%

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

      15. associate-/l* 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -1.45 \cdot 10^{-198}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left({\left(\sqrt[3]{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.im}{y.re}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;x.im \leq -3 \cdot 10^{-199}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left({\left(\sqrt[3]{y.im}\right)}^{2} \cdot \left(t\_0 \cdot \sqrt[3]{y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.im}{y.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= x.im -3e-199)
     (*
      (exp (fma t_0 y.re (* (atan2 x.im x.re) (- y.im))))
      (cos (* (pow (cbrt y.im) 2.0) (* t_0 (cbrt y.im)))))
     (exp
      (*
       y.re
       (- (log (hypot x.im x.re)) (* (atan2 x.im x.re) (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (x_46_im <= -3e-199) {
		tmp = exp(fma(t_0, y_46_re, (atan2(x_46_im, x_46_re) * -y_46_im))) * cos((pow(cbrt(y_46_im), 2.0) * (t_0 * cbrt(y_46_im))));
	} else {
		tmp = exp((y_46_re * (log(hypot(x_46_im, x_46_re)) - (atan2(x_46_im, x_46_re) * (y_46_im / y_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (x_46_im <= -3e-199)
		tmp = Float64(exp(fma(t_0, y_46_re, Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))) * cos(Float64((cbrt(y_46_im) ^ 2.0) * Float64(t_0 * cbrt(y_46_im)))));
	else
		tmp = exp(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) - Float64(atan(x_46_im, x_46_re) * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -3e-199], N[(N[Exp[N[(t$95$0 * y$46$re + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(N[Power[N[Power[y$46$im, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[(t$95$0 * N[Power[y$46$im, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -2.99999999999999983e-199

   1. Initial program 44.4%

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

      1. fma-neg 44.4%

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

      2. hypot-define 44.4%

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

      3. distribute-rgt-neg-out 44.4%

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

      4. fma-define 44.4%

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

      5. hypot-define 82.5%

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

      6. *-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. fma-undefine 82.5%

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

      2. hypot-define 44.4%

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

      3. *-commutative 44.4%

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

      4. add-cube-cbrt 47.2%

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

      5. pow3 48.2%

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

      6. fma-define 48.2%

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

      7. hypot-define 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in y.im around inf 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. unpow2 45.3%

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

      4. hypot-undefine 88.2%

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

   9. Simplified 88.2%

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

       1. rem-cube-cbrt 85.3%

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

       2. add-cube-cbrt 89.1%

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

       3. hypot-define 46.3%

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

       4. associate-*l* 45.3%

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

       5. pow2 45.3%

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

       6. hypot-define 89.1%

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

   11. Applied egg-rr 89.1%

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

   if -2.99999999999999983e-199 < x.im

   1. Initial program 42.8%

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

   3. Taylor expanded in y.im around 0 68.9%

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

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

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

   5. Taylor expanded in y.re around inf 70.4%

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

      1. +-commutative 70.4%

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

      2. unpow2 70.4%

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

      3. unpow2 70.4%

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

      4. hypot-undefine 86.1%

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

      5. mul-1-neg 86.1%

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

      6. unsub-neg 86.1%

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

      7. hypot-undefine 70.4%

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

      8. unpow2 70.4%

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

      9. unpow2 70.4%

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

      10. +-commutative 70.4%

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

      11. unpow2 70.4%

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

      12. unpow2 70.4%

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

      13. hypot-define 86.1%

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

      14. *-commutative 86.1%

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

      15. associate-/l* 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -3 \cdot 10^{-199}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left({\left(\sqrt[3]{y.im}\right)}^{2} \cdot \left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot \sqrt[3]{y.im}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.im}{y.re}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;x.im \leq -4 \cdot 10^{-198}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left({\left(\sqrt[3]{t\_0 \cdot y.im}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.im}{y.re}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= x.im -4e-198)
     (*
      (exp (fma t_0 y.re (* (atan2 x.im x.re) (- y.im))))
      (cos (pow (cbrt (* t_0 y.im)) 3.0)))
     (exp
      (*
       y.re
       (- (log (hypot x.im x.re)) (* (atan2 x.im x.re) (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (x_46_im <= -4e-198) {
		tmp = exp(fma(t_0, y_46_re, (atan2(x_46_im, x_46_re) * -y_46_im))) * cos(pow(cbrt((t_0 * y_46_im)), 3.0));
	} else {
		tmp = exp((y_46_re * (log(hypot(x_46_im, x_46_re)) - (atan2(x_46_im, x_46_re) * (y_46_im / y_46_re)))));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (x_46_im <= -4e-198)
		tmp = Float64(exp(fma(t_0, y_46_re, Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))) * cos((cbrt(Float64(t_0 * y_46_im)) ^ 3.0)));
	else
		tmp = exp(Float64(y_46_re * Float64(log(hypot(x_46_im, x_46_re)) - Float64(atan(x_46_im, x_46_re) * Float64(y_46_im / y_46_re)))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x$46$im, -4e-198], N[(N[Exp[N[(t$95$0 * y$46$re + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[Power[N[Power[N[(t$95$0 * y$46$im), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(y$46$re * N[(N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x.im < -3.9999999999999996e-198

   1. Initial program 44.4%

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

      1. fma-neg 44.4%

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

      2. hypot-define 44.4%

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

      3. distribute-rgt-neg-out 44.4%

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

      4. fma-define 44.4%

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

      5. hypot-define 82.5%

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

      6. *-commutative 82.5%

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

   3. Simplified 82.5%

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

      1. fma-undefine 82.5%

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

      2. hypot-define 44.4%

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

      3. *-commutative 44.4%

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

      4. add-cube-cbrt 47.2%

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

      5. pow3 48.2%

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

      6. fma-define 48.2%

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

      7. hypot-define 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in y.im around inf 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. unpow2 45.3%

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

      4. hypot-undefine 88.2%

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

   9. Simplified 88.2%

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

   if -3.9999999999999996e-198 < x.im

   1. Initial program 42.8%

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

   3. Taylor expanded in y.im around 0 68.9%

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

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

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

   5. Taylor expanded in y.re around inf 70.4%

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

      1. +-commutative 70.4%

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

      2. unpow2 70.4%

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

      3. unpow2 70.4%

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

      4. hypot-undefine 86.1%

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

      5. mul-1-neg 86.1%

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

      6. unsub-neg 86.1%

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

      7. hypot-undefine 70.4%

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

      8. unpow2 70.4%

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

      9. unpow2 70.4%

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

      10. +-commutative 70.4%

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

      11. unpow2 70.4%

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

      12. unpow2 70.4%

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

      13. hypot-define 86.1%

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

      14. *-commutative 86.1%

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

      15. associate-/l* 86.1%

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

   7. Simplified 86.1%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4 \cdot 10^{-198}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left({\left(\sqrt[3]{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.im}\right)}^{3}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \left(\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot \frac{y.im}{y.re}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.5%

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

3. Taylor expanded in y.im around 0 68.2%

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

4. Taylor expanded in y.re around 0 69.1%

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

5. Taylor expanded in y.re around inf 69.1%

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

   1. +-commutative 69.1%

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

   2. unpow2 69.1%

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

   3. unpow2 69.1%

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

   4. hypot-undefine 84.6%

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

   5. mul-1-neg 84.6%

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

   6. unsub-neg 84.6%

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

   7. hypot-undefine 69.1%

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

   8. unpow2 69.1%

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

   9. unpow2 69.1%

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

   10. +-commutative 69.1%

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

   11. unpow2 69.1%

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

   12. unpow2 69.1%

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

   13. hypot-define 84.6%

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

   14. *-commutative 84.6%

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

   15. associate-/l* 84.6%

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

7. Simplified 84.6%

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

8. Final simplification 84.6%

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

Alternative 5: 76.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+19} \lor \neg \left(y.re \leq 2.2\right):\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.8e+19) (not (<= y.re 2.2)))
   (exp (* (log (hypot x.re x.im)) y.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 <= -5.8e+19) || !(y_46_re <= 2.2)) {
		tmp = exp((log(hypot(x_46_re, x_46_im)) * y_46_re));
	} else {
		tmp = 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 tmp;
	if ((y_46_re <= -5.8e+19) || !(y_46_re <= 2.2)) {
		tmp = Math.exp((Math.log(Math.hypot(x_46_re, x_46_im)) * y_46_re));
	} else {
		tmp = 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 <= -5.8e+19) or not (y_46_re <= 2.2):
		tmp = math.exp((math.log(math.hypot(x_46_re, x_46_im)) * y_46_re))
	else:
		tmp = 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 <= -5.8e+19) || !(y_46_re <= 2.2))
		tmp = exp(Float64(log(hypot(x_46_re, x_46_im)) * y_46_re));
	else
		tmp = 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 <= -5.8e+19) || ~((y_46_re <= 2.2)))
		tmp = exp((log(hypot(x_46_re, x_46_im)) * y_46_re));
	else
		tmp = 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, -5.8e+19], N[Not[LessEqual[y$46$re, 2.2]], $MachinePrecision]], N[Exp[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re), $MachinePrecision]], $MachinePrecision], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.8e19 or 2.2000000000000002 < y.re

   1. Initial program 41.6%

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

   3. Taylor expanded in y.im around 0 78.9%

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

   4. Taylor expanded in y.re around 0 81.1%

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

   5. Taylor expanded in y.re around inf 78.2%

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

      1. +-commutative 78.2%

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

      2. unpow2 78.2%

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

      3. unpow2 78.2%

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

      4. hypot-undefine 78.2%

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

   7. Simplified 78.2%

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

   if -5.8e19 < y.re < 2.2000000000000002

   1. Initial program 45.6%

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

   3. Taylor expanded in y.im around 0 56.0%

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

   4. Taylor expanded in y.re around 0 55.4%

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

   5. Taylor expanded in y.re around 0 83.4%

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

      1. mul-1-neg 83.4%

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

      2. *-commutative 83.4%

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

      3. distribute-rgt-neg-in 83.4%

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

   7. Simplified 83.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+19} \lor \neg \left(y.re \leq 2.2\right):\\ \;\;\;\;e^{\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right) \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.6% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.5%

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

3. Taylor expanded in y.im around 0 68.2%

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

4. Taylor expanded in y.re around 0 69.1%

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

5. Taylor expanded in y.re around 0 56.0%

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

   1. mul-1-neg 56.0%

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

   2. *-commutative 56.0%

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

   3. distribute-rgt-neg-in 56.0%

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

7. Simplified 56.0%

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

8. Final simplification 56.0%

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

Alternative 7: 29.1% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 43.5%

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

3. Taylor expanded in y.im around 0 68.2%

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

4. Taylor expanded in y.re around 0 69.1%

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

5. Taylor expanded in y.re around 0 56.0%

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

   1. mul-1-neg 56.0%

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

   2. *-commutative 56.0%

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

   3. distribute-rgt-neg-in 56.0%

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

7. Simplified 56.0%

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

   1. add-sqr-sqrt 27.4%

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

   2. sqrt-unprod 41.3%

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

   3. sqr-neg 41.3%

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

   4. sqrt-unprod 14.0%

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

   5. add-sqr-sqrt 26.1%

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

   6. add-log-exp 26.2%

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

   7. log-pow 26.3%

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

   8. *-un-lft-identity 26.3%

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

   9. log-prod 26.3%

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

   10. metadata-eval 26.3%

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

   11. pow-exp 26.1%

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

   12. rem-log-exp 26.1%

       \[\leadsto e^{0 + \color{blue}{y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]

9. Applied egg-rr 26.1%

   \[\leadsto e^{\color{blue}{0 + y.im \cdot \tan^{-1}_* \frac{x.im}{x.re}}} \cdot 1 \]
10. Step-by-step derivation

    1. +-lft-identity 26.1%

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

11. Simplified 26.1%

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

12. Final simplification 26.1%

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

Alternative 8: 25.7% accurate, 7.8× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (- 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 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 = 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 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 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(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 = 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[(1.0 - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 43.5%

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

3. Taylor expanded in y.im around 0 68.2%

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

4. Taylor expanded in y.re around 0 69.1%

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

5. Taylor expanded in y.re around 0 56.0%

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

   1. mul-1-neg 56.0%

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

   2. *-commutative 56.0%

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

   3. distribute-rgt-neg-in 56.0%

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

7. Simplified 56.0%

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

8. Taylor expanded in y.im around 0 21.2%

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

   1. mul-1-neg 21.2%

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

10. Simplified 21.2%

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

11. Final simplification 21.2%

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

Reproduce

herbie shell --seed 2024085 
(FPCore (x.re x.im y.re y.im)
  :name "powComplex, real part"
  :precision binary64
  (* (exp (- (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.re) (* (atan2 x.im x.re) y.im))) (cos (+ (* (log (sqrt (+ (* x.re x.re) (* x.im x.im)))) y.im) (* (atan2 x.im x.re) y.re)))))