powComplex, real part

Percentage Accurate: 40.5% → 81.3%

Time: 24.5s

Alternatives: 13

Speedup: 4.0×

• Could not converge on a ground truth

  1. reg0 = (bf "-1.513844885970746595657489142075087232927e292")
  2. reg1 = (bf "-9.228843904086586886445332970384535420164e-164")
  3. reg2 = (bf #e-5515.824637141881794377695769071578979492)
  4. reg3 = (bf "2.603359574786005327133572614937277872292e165")

• 35.1% 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: 40.5% accurate, 1.0× speedup

Math

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

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: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\\ t_1 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_1, y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(t\_1, y.im, t\_0\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left|\cos \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), t\_0\right)\right)\right|\\ \end{array} \end{array} \]

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))))
   (if (<= y.re 1.95e+15)
     (*
      (exp (fma t_1 y.re (* (atan2 x.im x.re) (- y.im))))
      (cos (fma t_1 y.im t_0)))
     (*
      (exp
       (-
        (* y.re (log (sqrt (+ (* x.re x.re) (* x.im x.im)))))
        (* (atan2 x.im x.re) y.im)))
      (fabs (cos (fma y.im (log (hypot x.im x.re)) t_0)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = y_46_re * atan2(x_46_im, x_46_re);
	double t_1 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= 1.95e+15) {
		tmp = exp(fma(t_1, y_46_re, (atan2(x_46_im, x_46_re) * -y_46_im))) * cos(fma(t_1, y_46_im, t_0));
	} else {
		tmp = exp(((y_46_re * log(sqrt(((x_46_re * x_46_re) + (x_46_im * x_46_im))))) - (atan2(x_46_im, x_46_re) * y_46_im))) * fabs(cos(fma(y_46_im, log(hypot(x_46_im, x_46_re)), t_0)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * atan(x_46_im, x_46_re))
	t_1 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= 1.95e+15)
		tmp = Float64(exp(fma(t_1, y_46_re, Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))) * cos(fma(t_1, y_46_im, t_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))))) - Float64(atan(x_46_im, x_46_re) * y_46_im))) * abs(cos(fma(y_46_im, log(hypot(x_46_im, x_46_re)), 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]}, If[LessEqual[y$46$re, 1.95e+15], N[(N[Exp[N[(t$95$1 * y$46$re + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(t$95$1 * y$46$im + t$95$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] - N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Abs[N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if y.re < 1.95e15

   1. Initial program 40.1%

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

      1. cancel-sign-sub-inv 40.1%

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

      2. fma-define 40.1%

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

      3. hypot-define 40.1%

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

      4. distribute-lft-neg-in 40.1%

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

      5. distribute-rgt-neg-out 40.1%

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

      6. fma-define 40.1%

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

      7. hypot-define 85.1%

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

      8. *-commutative 85.1%

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

   3. Simplified 85.1%

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

   if 1.95e15 < y.re

   1. Initial program 30.3%

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

      1. fma-define 30.3%

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

      2. hypot-define 53.1%

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

      3. *-commutative 53.1%

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

      4. add-sqr-sqrt 37.9%

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

      5. sqrt-unprod 77.3%

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

      6. pow2 77.3%

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

   4. Applied egg-rr 77.3%

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

      1. unpow2 77.3%

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

      2. rem-sqrt-square 77.3%

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

      3. fma-define 77.3%

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

      4. hypot-undefine 40.9%

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

      5. unpow2 40.9%

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

      6. unpow2 40.9%

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

      7. +-commutative 40.9%

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

      8. +-commutative 40.9%

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

      9. fma-define 40.9%

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

   6. Simplified 77.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq 1.95 \cdot 10^{+15}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{y.re \cdot \log \left(\sqrt{x.re \cdot x.re + x.im \cdot x.im}\right) - \tan^{-1}_* \frac{x.im}{x.re} \cdot y.im} \cdot \left|\cos \left(\mathsf{fma}\left(y.im, \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right), y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.re, x.im\right)\right)\\ \mathbf{if}\;y.re \leq 7.5 \cdot 10^{+70}:\\ \;\;\;\;e^{\mathsf{fma}\left(t\_0, y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(\mathsf{fma}\left(t\_0, y.im, y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.re x.im))))
   (if (<= y.re 7.5e+70)
     (*
      (exp (fma t_0 y.re (* (atan2 x.im x.re) (- y.im))))
      (cos (fma t_0 y.im (* y.re (atan2 x.im x.re)))))
     (/ (pow (hypot x.im x.re) y.re) (pow (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 = log(hypot(x_46_re, x_46_im));
	double tmp;
	if (y_46_re <= 7.5e+70) {
		tmp = exp(fma(t_0, y_46_re, (atan2(x_46_im, x_46_re) * -y_46_im))) * cos(fma(t_0, y_46_im, (y_46_re * atan2(x_46_im, x_46_re))));
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) / pow(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 = log(hypot(x_46_re, x_46_im))
	tmp = 0.0
	if (y_46_re <= 7.5e+70)
		tmp = Float64(exp(fma(t_0, y_46_re, Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))) * cos(fma(t_0, y_46_im, Float64(y_46_re * atan(x_46_im, x_46_re)))));
	else
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) / (exp(atan(x_46_im, x_46_re)) ^ 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[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, 7.5e+70], 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[(t$95$0 * y$46$im + N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Power[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < 7.50000000000000031e70

   1. Initial program 39.4%

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

      1. cancel-sign-sub-inv 39.4%

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

      2. fma-define 39.4%

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

      3. hypot-define 39.4%

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

      4. distribute-lft-neg-in 39.4%

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

      5. distribute-rgt-neg-out 39.4%

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

      6. fma-define 39.4%

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

      7. hypot-define 83.4%

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

      8. *-commutative 83.4%

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

   3. Simplified 83.4%

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

   if 7.50000000000000031e70 < y.re

   1. Initial program 30.0%

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

      1. exp-diff 28.0%

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

      2. exp-to-pow 28.0%

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

      3. hypot-define 28.0%

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

      4. *-commutative 28.0%

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

      5. exp-prod 24.0%

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

      6. fma-define 24.0%

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

      7. hypot-define 44.0%

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

      8. *-commutative 44.0%

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

   3. Simplified 44.0%

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

      1. add-cube-cbrt 44.0%

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

      2. exp-prod 44.0%

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

      3. pow2 44.0%

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

   6. Applied egg-rr 44.0%

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

   7. Taylor expanded in y.re around 0 32.0%

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

      1. unpow2 32.0%

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

      2. unpow2 32.0%

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

      3. hypot-undefine 60.0%

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

   9. Simplified 60.0%

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

   10. Taylor expanded in y.im around 0 68.0%

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

       1. +-commutative 68.0%

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

       2. unpow2 68.0%

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

       3. unpow2 68.0%

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

       4. hypot-undefine 68.0%

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

       5. hypot-undefine 68.0%

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

       6. unpow2 68.0%

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

       7. unpow2 68.0%

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

       8. +-commutative 68.0%

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

       9. unpow2 68.0%

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

       10. unpow2 68.0%

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

       11. hypot-undefine 68.0%

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

       12. *-commutative 68.0%

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

       13. exp-prod 70.0%

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

   12. Simplified 70.0%

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

Alternative 3: 78.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-16}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \frac{t\_0}{y.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot t\_0\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (log (hypot x.im x.re))))
   (if (<= y.re -2.1e-16)
     (*
      (exp (fma (log (hypot x.re x.im)) y.re (* (atan2 x.im x.re) (- y.im))))
      (cos (* y.re (+ (atan2 x.im x.re) (* y.im (/ t_0 y.re))))))
     (if (<= y.re 1.8e-51)
       (/ (cos (* y.im t_0)) (exp (* (atan2 x.im x.re) y.im)))
       (/ (pow (hypot x.im x.re) y.re) (pow (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 = log(hypot(x_46_im, x_46_re));
	double tmp;
	if (y_46_re <= -2.1e-16) {
		tmp = exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, (atan2(x_46_im, x_46_re) * -y_46_im))) * cos((y_46_re * (atan2(x_46_im, x_46_re) + (y_46_im * (t_0 / y_46_re)))));
	} else if (y_46_re <= 1.8e-51) {
		tmp = cos((y_46_im * t_0)) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) / pow(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 = log(hypot(x_46_im, x_46_re))
	tmp = 0.0
	if (y_46_re <= -2.1e-16)
		tmp = Float64(exp(fma(log(hypot(x_46_re, x_46_im)), y_46_re, Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)))) * cos(Float64(y_46_re * Float64(atan(x_46_im, x_46_re) + Float64(y_46_im * Float64(t_0 / y_46_re))))));
	elseif (y_46_re <= 1.8e-51)
		tmp = Float64(cos(Float64(y_46_im * t_0)) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) / (exp(atan(x_46_im, x_46_re)) ^ 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[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e-16], N[(N[Exp[N[(N[Log[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision]], $MachinePrecision] * y$46$re + N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(y$46$re * N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] + N[(y$46$im * N[(t$95$0 / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.8e-51], N[(N[Cos[N[(y$46$im * t$95$0), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[Power[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.1000000000000001e-16

   1. Initial program 33.3%

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

      1. cancel-sign-sub-inv 33.3%

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

      2. fma-define 33.3%

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

      3. hypot-define 33.3%

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

      4. distribute-lft-neg-in 33.3%

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

      5. distribute-rgt-neg-out 33.3%

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

      6. fma-define 33.3%

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

      7. hypot-define 79.6%

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

      8. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   5. Taylor expanded in y.re around inf 33.3%

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

      1. +-commutative 33.3%

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

      2. associate-/l* 33.3%

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

      3. unpow2 33.3%

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

      4. unpow2 33.3%

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

      5. hypot-undefine 81.1%

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

   7. Simplified 81.1%

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

   if -2.1000000000000001e-16 < y.re < 1.8e-51

   1. Initial program 45.3%

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

      1. exp-diff 45.3%

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

      2. exp-to-pow 45.3%

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

      3. hypot-define 45.3%

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

      4. *-commutative 45.3%

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

      5. exp-prod 45.1%

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

      6. fma-define 45.1%

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

      7. hypot-define 88.8%

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

      8. *-commutative 88.8%

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

   3. Simplified 88.8%

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

      1. add-cube-cbrt 88.8%

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

      2. exp-prod 88.8%

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

      3. pow2 88.8%

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

   6. Applied egg-rr 88.8%

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

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

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

      1. unpow2 45.1%

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

      2. unpow2 45.1%

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

      3. hypot-undefine 88.8%

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

   9. Simplified 88.8%

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

   10. Taylor expanded in y.re around 0 45.3%

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

       1. unpow2 45.3%

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

       2. unpow2 45.3%

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

       3. hypot-undefine 89.0%

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

   12. Simplified 89.0%

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

   if 1.8e-51 < y.re

   1. Initial program 30.3%

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

      1. exp-diff 25.0%

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

      2. exp-to-pow 25.0%

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

      3. hypot-define 25.0%

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

      4. *-commutative 25.0%

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

      5. exp-prod 22.4%

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

      6. fma-define 22.4%

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

      7. hypot-define 48.7%

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

      8. *-commutative 48.7%

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

   3. Simplified 48.7%

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

      1. add-cube-cbrt 48.7%

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

      2. exp-prod 48.7%

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

      3. pow2 48.7%

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

   6. Applied egg-rr 48.7%

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

   7. Taylor expanded in y.re around 0 27.6%

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

      1. unpow2 27.6%

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

      2. unpow2 27.6%

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

      3. hypot-undefine 56.6%

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

   9. Simplified 56.6%

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

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

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

       1. +-commutative 63.3%

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

       2. unpow2 63.3%

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

       3. unpow2 63.3%

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

       4. hypot-undefine 67.1%

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

       5. hypot-undefine 63.3%

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

       6. unpow2 63.3%

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

       7. unpow2 63.3%

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

       8. +-commutative 63.3%

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

       9. unpow2 63.3%

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

       10. unpow2 63.3%

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

       11. hypot-undefine 67.1%

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

       12. *-commutative 67.1%

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

       13. exp-prod 68.4%

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

   12. Simplified 68.4%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-16}:\\ \;\;\;\;e^{\mathsf{fma}\left(\log \left(\mathsf{hypot}\left(x.re, x.im\right)\right), y.re, \tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)\right)} \cdot \cos \left(y.re \cdot \left(\tan^{-1}_* \frac{x.im}{x.re} + y.im \cdot \frac{\log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)}{y.re}\right)\right)\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \cos \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re -2.4e-7)
     (* t_0 (cos (pow (cbrt (* y.re (atan2 x.im x.re))) 3.0)))
     (if (<= y.re 3.4e-51)
       (/
        (cos (* y.im (log (hypot x.im x.re))))
        (exp (* (atan2 x.im x.re) y.im)))
       (/ t_0 (pow (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 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.4e-7) {
		tmp = t_0 * cos(pow(cbrt((y_46_re * atan2(x_46_im, x_46_re))), 3.0));
	} else if (y_46_re <= 3.4e-51) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / pow(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 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.4e-7) {
		tmp = t_0 * Math.cos(Math.pow(Math.cbrt((y_46_re * Math.atan2(x_46_im, x_46_re))), 3.0));
	} else if (y_46_re <= 3.4e-51) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / Math.pow(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 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -2.4e-7)
		tmp = Float64(t_0 * cos((cbrt(Float64(y_46_re * atan(x_46_im, x_46_re))) ^ 3.0)));
	elseif (y_46_re <= 3.4e-51)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = Float64(t_0 / (exp(atan(x_46_im, x_46_re)) ^ 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -2.4e-7], N[(t$95$0 * N[Cos[N[Power[N[Power[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.4e-51], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.39999999999999979e-7

   1. Initial program 32.4%

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

      1. cancel-sign-sub-inv 32.4%

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

      2. fma-define 32.4%

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

      3. hypot-define 32.4%

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

      4. distribute-lft-neg-in 32.4%

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

      5. distribute-rgt-neg-out 32.4%

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

      6. fma-define 32.4%

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

      7. hypot-define 79.3%

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

      8. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in y.im around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

      3. hypot-undefine 77.7%

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

   7. Simplified 77.7%

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

      1. add-cube-cbrt 82.1%

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

      2. pow3 80.6%

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

   9. Applied egg-rr 80.6%

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

   if -2.39999999999999979e-7 < y.re < 3.40000000000000003e-51

   1. Initial program 45.8%

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

      1. exp-diff 45.7%

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

      2. exp-to-pow 45.7%

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

      3. hypot-define 45.7%

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

      4. *-commutative 45.7%

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

      5. exp-prod 45.6%

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

      6. fma-define 45.6%

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

      7. hypot-define 88.9%

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

      8. *-commutative 88.9%

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

   3. Simplified 88.9%

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

      1. add-cube-cbrt 88.9%

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

      2. exp-prod 88.9%

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

      3. pow2 88.9%

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

   6. Applied egg-rr 88.9%

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

   7. Taylor expanded in y.re around 0 45.4%

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

      1. unpow2 45.4%

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

      2. unpow2 45.4%

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

      3. hypot-undefine 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in y.re around 0 45.3%

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

       1. unpow2 45.3%

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

       2. unpow2 45.3%

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

       3. hypot-undefine 88.7%

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

   12. Simplified 88.7%

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

   if 3.40000000000000003e-51 < y.re

   1. Initial program 30.3%

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

      1. exp-diff 25.0%

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

      2. exp-to-pow 25.0%

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

      3. hypot-define 25.0%

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

      4. *-commutative 25.0%

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

      5. exp-prod 22.4%

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

      6. fma-define 22.4%

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

      7. hypot-define 48.7%

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

      8. *-commutative 48.7%

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

   3. Simplified 48.7%

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

      1. add-cube-cbrt 48.7%

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

      2. exp-prod 48.7%

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

      3. pow2 48.7%

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

   6. Applied egg-rr 48.7%

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

   7. Taylor expanded in y.re around 0 27.6%

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

      1. unpow2 27.6%

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

      2. unpow2 27.6%

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

      3. hypot-undefine 56.6%

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

   9. Simplified 56.6%

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

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

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

       1. +-commutative 63.3%

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

       2. unpow2 63.3%

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

       3. unpow2 63.3%

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

       4. hypot-undefine 67.1%

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

       5. hypot-undefine 63.3%

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

       6. unpow2 63.3%

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

       7. unpow2 63.3%

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

       8. +-commutative 63.3%

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

       9. unpow2 63.3%

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

       10. unpow2 63.3%

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

       11. hypot-undefine 67.1%

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

       12. *-commutative 67.1%

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

       13. exp-prod 68.4%

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

   12. Simplified 68.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{-7}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left({\left(\sqrt[3]{y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}}\right)}^{3}\right)\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.re \cdot \sqrt[3]{{\tan^{-1}_* \frac{x.im}{x.re}}^{3}}\right)\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re -7.5e-6)
     (* t_0 (cos (* y.re (cbrt (pow (atan2 x.im x.re) 3.0)))))
     (if (<= y.re 4.5e-52)
       (/
        (cos (* y.im (log (hypot x.im x.re))))
        (exp (* (atan2 x.im x.re) y.im)))
       (/ t_0 (pow (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 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -7.5e-6) {
		tmp = t_0 * cos((y_46_re * cbrt(pow(atan2(x_46_im, x_46_re), 3.0))));
	} else if (y_46_re <= 4.5e-52) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / pow(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 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -7.5e-6) {
		tmp = t_0 * Math.cos((y_46_re * Math.cbrt(Math.pow(Math.atan2(x_46_im, x_46_re), 3.0))));
	} else if (y_46_re <= 4.5e-52) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / Math.pow(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 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -7.5e-6)
		tmp = Float64(t_0 * cos(Float64(y_46_re * cbrt((atan(x_46_im, x_46_re) ^ 3.0)))));
	elseif (y_46_re <= 4.5e-52)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = Float64(t_0 / (exp(atan(x_46_im, x_46_re)) ^ 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[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -7.5e-6], N[(t$95$0 * N[Cos[N[(y$46$re * N[Power[N[Power[N[ArcTan[x$46$im / x$46$re], $MachinePrecision], 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 4.5e-52], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -7.50000000000000019e-6

   1. Initial program 32.4%

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

      1. cancel-sign-sub-inv 32.4%

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

      2. fma-define 32.4%

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

      3. hypot-define 32.4%

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

      4. distribute-lft-neg-in 32.4%

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

      5. distribute-rgt-neg-out 32.4%

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

      6. fma-define 32.4%

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

      7. hypot-define 79.3%

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

      8. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in y.im around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

      3. hypot-undefine 77.7%

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

   7. Simplified 77.7%

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

      1. add-cbrt-cube 79.1%

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

      2. pow3 79.1%

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

   9. Applied egg-rr 79.1%

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

   if -7.50000000000000019e-6 < y.re < 4.5e-52

   1. Initial program 45.8%

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

      1. exp-diff 45.7%

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

      2. exp-to-pow 45.7%

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

      3. hypot-define 45.7%

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

      4. *-commutative 45.7%

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

      5. exp-prod 45.6%

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

      6. fma-define 45.6%

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

      7. hypot-define 88.9%

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

      8. *-commutative 88.9%

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

   3. Simplified 88.9%

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

      1. add-cube-cbrt 88.9%

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

      2. exp-prod 88.9%

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

      3. pow2 88.9%

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

   6. Applied egg-rr 88.9%

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

   7. Taylor expanded in y.re around 0 45.4%

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

      1. unpow2 45.4%

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

      2. unpow2 45.4%

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

      3. hypot-undefine 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in y.re around 0 45.3%

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

       1. unpow2 45.3%

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

       2. unpow2 45.3%

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

       3. hypot-undefine 88.7%

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

   12. Simplified 88.7%

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

   if 4.5e-52 < y.re

   1. Initial program 30.3%

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

      1. exp-diff 25.0%

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

      2. exp-to-pow 25.0%

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

      3. hypot-define 25.0%

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

      4. *-commutative 25.0%

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

      5. exp-prod 22.4%

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

      6. fma-define 22.4%

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

      7. hypot-define 48.7%

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

      8. *-commutative 48.7%

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

   3. Simplified 48.7%

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

      1. add-cube-cbrt 48.7%

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

      2. exp-prod 48.7%

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

      3. pow2 48.7%

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

   6. Applied egg-rr 48.7%

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

   7. Taylor expanded in y.re around 0 27.6%

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

      1. unpow2 27.6%

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

      2. unpow2 27.6%

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

      3. hypot-undefine 56.6%

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

   9. Simplified 56.6%

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

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

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

       1. +-commutative 63.3%

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

       2. unpow2 63.3%

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

       3. unpow2 63.3%

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

       4. hypot-undefine 67.1%

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

       5. hypot-undefine 63.3%

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

       6. unpow2 63.3%

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

       7. unpow2 63.3%

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

       8. +-commutative 63.3%

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

       9. unpow2 63.3%

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

       10. unpow2 63.3%

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

       11. hypot-undefine 67.1%

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

       12. *-commutative 67.1%

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

       13. exp-prod 68.4%

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

   12. Simplified 68.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(y.re \cdot \sqrt[3]{{\tan^{-1}_* \frac{x.im}{x.re}}^{3}}\right)\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-51}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re -4.8e-7)
     (* t_0 (cos (* y.re (atan2 x.im x.re))))
     (if (<= y.re 1.16e-51)
       (/
        (cos (* y.im (log (hypot x.im x.re))))
        (exp (* (atan2 x.im x.re) y.im)))
       (/ t_0 (pow (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 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.8e-7) {
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1.16e-51) {
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / pow(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 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -4.8e-7) {
		tmp = t_0 * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 1.16e-51) {
		tmp = Math.cos((y_46_im * Math.log(Math.hypot(x_46_im, x_46_re)))) / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	} else {
		tmp = t_0 / Math.pow(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 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -4.8e-7:
		tmp = t_0 * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 1.16e-51:
		tmp = math.cos((y_46_im * math.log(math.hypot(x_46_im, x_46_re)))) / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	else:
		tmp = t_0 / math.pow(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 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -4.8e-7)
		tmp = Float64(t_0 * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 1.16e-51)
		tmp = Float64(cos(Float64(y_46_im * log(hypot(x_46_im, x_46_re)))) / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	else
		tmp = Float64(t_0 / (exp(atan(x_46_im, x_46_re)) ^ y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -4.8e-7)
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 1.16e-51)
		tmp = cos((y_46_im * log(hypot(x_46_im, x_46_re)))) / exp((atan2(x_46_im, x_46_re) * y_46_im));
	else
		tmp = t_0 / (exp(atan2(x_46_im, x_46_re)) ^ y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -4.8e-7], N[(t$95$0 * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.16e-51], N[(N[Cos[N[(y$46$im * N[Log[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Power[N[Exp[N[ArcTan[x$46$im / x$46$re], $MachinePrecision]], $MachinePrecision], y$46$im], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.79999999999999957e-7

   1. Initial program 32.4%

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

      1. cancel-sign-sub-inv 32.4%

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

      2. fma-define 32.4%

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

      3. hypot-define 32.4%

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

      4. distribute-lft-neg-in 32.4%

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

      5. distribute-rgt-neg-out 32.4%

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

      6. fma-define 32.4%

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

      7. hypot-define 79.3%

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

      8. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in y.im around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

      3. hypot-undefine 77.7%

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

   7. Simplified 77.7%

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

   if -4.79999999999999957e-7 < y.re < 1.1600000000000001e-51

   1. Initial program 45.8%

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

      1. exp-diff 45.7%

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

      2. exp-to-pow 45.7%

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

      3. hypot-define 45.7%

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

      4. *-commutative 45.7%

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

      5. exp-prod 45.6%

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

      6. fma-define 45.6%

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

      7. hypot-define 88.9%

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

      8. *-commutative 88.9%

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

   3. Simplified 88.9%

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

      1. add-cube-cbrt 88.9%

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

      2. exp-prod 88.9%

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

      3. pow2 88.9%

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

   6. Applied egg-rr 88.9%

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

   7. Taylor expanded in y.re around 0 45.4%

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

      1. unpow2 45.4%

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

      2. unpow2 45.4%

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

      3. hypot-undefine 88.7%

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

   9. Simplified 88.7%

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

   10. Taylor expanded in y.re around 0 45.3%

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

       1. unpow2 45.3%

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

       2. unpow2 45.3%

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

       3. hypot-undefine 88.7%

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

   12. Simplified 88.7%

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

   if 1.1600000000000001e-51 < y.re

   1. Initial program 30.3%

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

      1. exp-diff 25.0%

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

      2. exp-to-pow 25.0%

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

      3. hypot-define 25.0%

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

      4. *-commutative 25.0%

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

      5. exp-prod 22.4%

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

      6. fma-define 22.4%

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

      7. hypot-define 48.7%

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

      8. *-commutative 48.7%

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

   3. Simplified 48.7%

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

      1. add-cube-cbrt 48.7%

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

      2. exp-prod 48.7%

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

      3. pow2 48.7%

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

   6. Applied egg-rr 48.7%

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

   7. Taylor expanded in y.re around 0 27.6%

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

      1. unpow2 27.6%

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

      2. unpow2 27.6%

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

      3. hypot-undefine 56.6%

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

   9. Simplified 56.6%

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

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

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

       1. +-commutative 63.3%

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

       2. unpow2 63.3%

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

       3. unpow2 63.3%

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

       4. hypot-undefine 67.1%

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

       5. hypot-undefine 63.3%

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

       6. unpow2 63.3%

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

       7. unpow2 63.3%

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

       8. +-commutative 63.3%

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

       9. unpow2 63.3%

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

       10. unpow2 63.3%

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

       11. hypot-undefine 67.1%

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

       12. *-commutative 67.1%

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

       13. exp-prod 68.4%

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

   12. Simplified 68.4%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.8 \cdot 10^{-7}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 1.16 \cdot 10^{-51}:\\ \;\;\;\;\frac{\cos \left(y.im \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)\right)}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{{\left(e^{\tan^{-1}_* \frac{x.im}{x.re}}\right)}^{y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+80}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re -1.55e+80)
     (* t_0 (cos (* y.re (atan2 x.im x.re))))
     (/ t_0 (exp (* (atan2 x.im x.re) y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.55e+80) {
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0 / 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 = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -1.55e+80) {
		tmp = t_0 * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else {
		tmp = t_0 / Math.exp((Math.atan2(x_46_im, x_46_re) * y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -1.55e+80:
		tmp = t_0 * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	else:
		tmp = t_0 / math.exp((math.atan2(x_46_im, x_46_re) * y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -1.55e+80)
		tmp = Float64(t_0 * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	else
		tmp = Float64(t_0 / exp(Float64(atan(x_46_im, x_46_re) * y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.55e+80)
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	else
		tmp = t_0 / exp((atan2(x_46_im, x_46_re) * y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -1.55e+80], N[(t$95$0 * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$0 / N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.54999999999999994e80

   1. Initial program 35.8%

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

      1. cancel-sign-sub-inv 35.8%

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

      2. fma-define 35.8%

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

      3. hypot-define 35.8%

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

      4. distribute-lft-neg-in 35.8%

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

      5. distribute-rgt-neg-out 35.8%

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

      6. fma-define 35.8%

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

      7. hypot-define 81.1%

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

      8. *-commutative 81.1%

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

   3. Simplified 81.1%

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

   5. Taylor expanded in y.im around 0 79.4%

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

      1. unpow2 79.4%

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

      2. unpow2 79.4%

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

      3. hypot-undefine 79.4%

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

   7. Simplified 79.4%

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

   if -1.54999999999999994e80 < y.re

   1. Initial program 38.1%

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

      1. exp-diff 36.1%

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

      2. exp-to-pow 36.1%

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

      3. hypot-define 36.1%

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

      4. *-commutative 36.1%

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

      5. exp-prod 35.0%

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

      6. fma-define 35.0%

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

      7. hypot-define 72.7%

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

      8. *-commutative 72.7%

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

   3. Simplified 72.7%

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

      1. add-cube-cbrt 72.7%

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

      2. exp-prod 72.7%

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

      3. pow2 72.7%

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

   6. Applied egg-rr 72.7%

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

   7. Taylor expanded in y.re around 0 37.3%

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

      1. unpow2 37.3%

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

      2. unpow2 37.3%

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

      3. hypot-undefine 75.6%

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

   9. Simplified 75.6%

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

   10. Taylor expanded in y.im around 0 58.6%

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

       1. unpow2 58.6%

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

       2. unpow2 58.6%

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

       3. hypot-undefine 76.8%

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

   12. Simplified 76.8%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+80}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0 \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 9000000:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (pow (hypot x.im x.re) y.re)))
   (if (<= y.re -2.8e-7)
     (* t_0 (cos (* y.re (atan2 x.im x.re))))
     (if (<= y.re 9000000.0)
       (exp (* (atan2 x.im x.re) (- y.im)))
       (/ t_0 (+ (* (atan2 x.im x.re) y.im) 1.0))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = pow(hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.8e-7) {
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9000000.0) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0 / ((atan2(x_46_im, x_46_re) * y_46_im) + 1.0);
	}
	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.pow(Math.hypot(x_46_im, x_46_re), y_46_re);
	double tmp;
	if (y_46_re <= -2.8e-7) {
		tmp = t_0 * Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re)));
	} else if (y_46_re <= 9000000.0) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = t_0 / ((Math.atan2(x_46_im, x_46_re) * y_46_im) + 1.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = math.pow(math.hypot(x_46_im, x_46_re), y_46_re)
	tmp = 0
	if y_46_re <= -2.8e-7:
		tmp = t_0 * math.cos((y_46_re * math.atan2(x_46_im, x_46_re)))
	elif y_46_re <= 9000000.0:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = t_0 / ((math.atan2(x_46_im, x_46_re) * y_46_im) + 1.0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re
	tmp = 0.0
	if (y_46_re <= -2.8e-7)
		tmp = Float64(t_0 * cos(Float64(y_46_re * atan(x_46_im, x_46_re))));
	elseif (y_46_re <= 9000000.0)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = Float64(t_0 / Float64(Float64(atan(x_46_im, x_46_re) * y_46_im) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = hypot(x_46_im, x_46_re) ^ y_46_re;
	tmp = 0.0;
	if (y_46_re <= -2.8e-7)
		tmp = t_0 * cos((y_46_re * atan2(x_46_im, x_46_re)));
	elseif (y_46_re <= 9000000.0)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = t_0 / ((atan2(x_46_im, x_46_re) * y_46_im) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]}, If[LessEqual[y$46$re, -2.8e-7], N[(t$95$0 * N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9000000.0], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[(t$95$0 / N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.80000000000000019e-7

   1. Initial program 32.4%

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

      1. cancel-sign-sub-inv 32.4%

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

      2. fma-define 32.4%

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

      3. hypot-define 32.4%

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

      4. distribute-lft-neg-in 32.4%

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

      5. distribute-rgt-neg-out 32.4%

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

      6. fma-define 32.4%

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

      7. hypot-define 79.3%

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

      8. *-commutative 79.3%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in y.im around 0 76.6%

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

      1. unpow2 76.6%

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

      2. unpow2 76.6%

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

      3. hypot-undefine 77.7%

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

   7. Simplified 77.7%

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

   if -2.80000000000000019e-7 < y.re < 9e6

   1. Initial program 45.2%

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

      1. exp-diff 45.2%

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

      2. exp-to-pow 45.2%

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

      3. hypot-define 45.2%

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

      4. *-commutative 45.2%

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

      5. exp-prod 45.0%

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

      6. fma-define 45.0%

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

      7. hypot-define 88.0%

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

      8. *-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. add-cube-cbrt 88.0%

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

      2. exp-prod 88.0%

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

      3. pow2 88.0%

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

   6. Applied egg-rr 88.0%

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

   7. Taylor expanded in y.re around 0 44.8%

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

      1. unpow2 44.8%

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

      2. unpow2 44.8%

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

      3. hypot-undefine 87.8%

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

   9. Simplified 87.8%

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

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

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

       1. +-commutative 53.3%

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

       2. unpow2 53.3%

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

       3. unpow2 53.3%

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

       4. hypot-undefine 83.9%

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

       5. hypot-undefine 53.3%

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

       6. unpow2 53.3%

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

       7. unpow2 53.3%

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

       8. +-commutative 53.3%

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

       9. unpow2 53.3%

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

       10. unpow2 53.3%

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

       11. hypot-undefine 83.9%

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

       12. *-commutative 83.9%

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

       13. exp-prod 80.7%

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

   12. Simplified 80.7%

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

   13. Taylor expanded in y.re around 0 83.9%

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

       1. rec-exp 83.9%

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

       2. distribute-rgt-neg-in 83.9%

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

   15. Simplified 83.9%

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

   if 9e6 < y.re

   1. Initial program 29.4%

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

      1. exp-diff 23.5%

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

      2. exp-to-pow 23.5%

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

      3. hypot-define 23.5%

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

      4. *-commutative 23.5%

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

      5. exp-prod 20.6%

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

      6. fma-define 20.6%

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

      7. hypot-define 45.6%

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

      8. *-commutative 45.6%

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

   3. Simplified 45.6%

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

      1. add-cube-cbrt 45.6%

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

      2. exp-prod 45.6%

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

      3. pow2 45.6%

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

   6. Applied egg-rr 45.6%

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

   7. Taylor expanded in y.re around 0 26.5%

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

      1. unpow2 26.5%

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

      2. unpow2 26.5%

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

      3. hypot-undefine 54.4%

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

   9. Simplified 54.4%

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

   10. Taylor expanded in y.im around 0 64.7%

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

       1. +-commutative 64.7%

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

       2. unpow2 64.7%

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

       3. unpow2 64.7%

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

       4. hypot-undefine 64.7%

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

       5. hypot-undefine 64.7%

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

       6. unpow2 64.7%

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

       7. unpow2 64.7%

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

       8. +-commutative 64.7%

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

       9. unpow2 64.7%

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

       10. unpow2 64.7%

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

       11. hypot-undefine 64.7%

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

       12. *-commutative 64.7%

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

       13. exp-prod 66.2%

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

   12. Simplified 66.2%

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

   13. Taylor expanded in y.im around 0 63.4%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-7}:\\ \;\;\;\;{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re} \cdot \cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right)\\ \mathbf{elif}\;y.re \leq 9000000:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{-12} \lor \neg \left(y.im \leq 9.6 \cdot 10^{+76}\right):\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.75e-12) (not (<= y.im 9.6e+76)))
   (exp (* (atan2 x.im x.re) (- y.im)))
   (/ (pow (hypot x.im x.re) y.re) (+ (* (atan2 x.im x.re) y.im) 1.0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.75e-12) || !(y_46_im <= 9.6e+76)) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = pow(hypot(x_46_im, x_46_re), y_46_re) / ((atan2(x_46_im, x_46_re) * y_46_im) + 1.0);
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.75e-12) || !(y_46_im <= 9.6e+76)) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = Math.pow(Math.hypot(x_46_im, x_46_re), y_46_re) / ((Math.atan2(x_46_im, x_46_re) * y_46_im) + 1.0);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.75e-12) or not (y_46_im <= 9.6e+76):
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = math.pow(math.hypot(x_46_im, x_46_re), y_46_re) / ((math.atan2(x_46_im, x_46_re) * y_46_im) + 1.0)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.75e-12) || !(y_46_im <= 9.6e+76))
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = Float64((hypot(x_46_im, x_46_re) ^ y_46_re) / Float64(Float64(atan(x_46_im, x_46_re) * y_46_im) + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.75e-12) || ~((y_46_im <= 9.6e+76)))
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = (hypot(x_46_im, x_46_re) ^ y_46_re) / ((atan2(x_46_im, x_46_re) * y_46_im) + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.75e-12], N[Not[LessEqual[y$46$im, 9.6e+76]], $MachinePrecision]], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[(N[Power[N[Sqrt[x$46$im ^ 2 + x$46$re ^ 2], $MachinePrecision], y$46$re], $MachinePrecision] / N[(N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * y$46$im), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.75e-12 or 9.5999999999999999e76 < y.im

   1. Initial program 37.8%

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

      1. exp-diff 30.5%

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

      2. exp-to-pow 30.5%

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

      3. hypot-define 30.5%

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

      4. *-commutative 30.5%

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

      5. exp-prod 29.4%

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

      6. fma-define 29.4%

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

      7. hypot-define 53.2%

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

      8. *-commutative 53.2%

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

   3. Simplified 53.2%

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

      1. add-cube-cbrt 53.2%

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

      2. exp-prod 53.2%

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

      3. pow2 53.2%

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

   6. Applied egg-rr 53.2%

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

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

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

      1. unpow2 27.3%

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

      2. unpow2 27.3%

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

      3. hypot-undefine 50.3%

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

   9. Simplified 50.3%

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

   10. Taylor expanded in y.im around 0 48.6%

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

       1. +-commutative 48.6%

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

       2. unpow2 48.6%

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

       3. unpow2 48.6%

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

       4. hypot-undefine 54.3%

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

       5. hypot-undefine 48.6%

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

       6. unpow2 48.6%

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

       7. unpow2 48.6%

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

       8. +-commutative 48.6%

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

       9. unpow2 48.6%

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

       10. unpow2 48.6%

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

       11. hypot-undefine 54.3%

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

       12. *-commutative 54.3%

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

       13. exp-prod 51.7%

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

   12. Simplified 51.7%

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

   13. Taylor expanded in y.re around 0 58.4%

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

       1. rec-exp 58.4%

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

       2. distribute-rgt-neg-in 58.4%

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

   15. Simplified 58.4%

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

   if -1.75e-12 < y.im < 9.5999999999999999e76

   1. Initial program 37.4%

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

      1. exp-diff 36.7%

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

      2. exp-to-pow 36.7%

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

      3. hypot-define 36.7%

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

      4. *-commutative 36.7%

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

      5. exp-prod 36.1%

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

      6. fma-define 36.1%

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

      7. hypot-define 83.3%

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

      8. *-commutative 83.3%

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

   3. Simplified 83.3%

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

      1. add-cube-cbrt 83.3%

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

      2. exp-prod 83.3%

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

      3. pow2 83.3%

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

   6. Applied egg-rr 83.3%

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

   7. Taylor expanded in y.re around 0 39.5%

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

      1. unpow2 39.5%

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

      2. unpow2 39.5%

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

      3. hypot-undefine 88.3%

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

   9. Simplified 88.3%

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

   10. Taylor expanded in y.im around 0 68.8%

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

       1. +-commutative 68.8%

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

       2. unpow2 68.8%

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

       3. unpow2 68.8%

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

       4. hypot-undefine 89.6%

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

       5. hypot-undefine 68.8%

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

       6. unpow2 68.8%

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

       7. unpow2 68.8%

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

       8. +-commutative 68.8%

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

       9. unpow2 68.8%

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

       10. unpow2 68.8%

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

       11. hypot-undefine 89.6%

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

       12. *-commutative 89.6%

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

       13. exp-prod 89.6%

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

   12. Simplified 89.6%

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

   13. Taylor expanded in y.im around 0 89.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{-12} \lor \neg \left(y.im \leq 9.6 \cdot 10^{+76}\right):\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}}{\tan^{-1}_* \frac{x.im}{x.re} \cdot y.im + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 60.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{-12} \lor \neg \left(y.im \leq 3.1 \cdot 10^{-67}\right):\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.75e-12) (not (<= y.im 3.1e-67)))
   (exp (* (atan2 x.im x.re) (- y.im)))
   (+ 1.0 (log (pow (hypot x.re x.im) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.75e-12) || !(y_46_im <= 3.1e-67)) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = 1.0 + log(pow(hypot(x_46_re, x_46_im), y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.75e-12) || !(y_46_im <= 3.1e-67)) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = 1.0 + Math.log(Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.75e-12) or not (y_46_im <= 3.1e-67):
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = 1.0 + math.log(math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.75e-12) || !(y_46_im <= 3.1e-67))
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = Float64(1.0 + log((hypot(x_46_re, x_46_im) ^ y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.75e-12) || ~((y_46_im <= 3.1e-67)))
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = 1.0 + log((hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.75e-12], N[Not[LessEqual[y$46$im, 3.1e-67]], $MachinePrecision]], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[Log[N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.75e-12 or 3.1000000000000003e-67 < y.im

   1. Initial program 38.4%

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

      1. exp-diff 31.5%

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

      2. exp-to-pow 31.5%

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

      3. hypot-define 31.5%

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

      4. *-commutative 31.5%

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

      5. exp-prod 29.8%

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

      6. fma-define 29.8%

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

      7. hypot-define 56.9%

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

      8. *-commutative 56.9%

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

   3. Simplified 56.9%

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

      1. add-cube-cbrt 56.9%

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

      2. exp-prod 56.9%

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

      3. pow2 56.9%

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

   6. Applied egg-rr 56.9%

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

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

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

      1. unpow2 27.3%

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

      2. unpow2 27.3%

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

      3. hypot-undefine 53.7%

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

   9. Simplified 53.7%

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

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

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

       1. +-commutative 50.5%

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

       2. unpow2 50.5%

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

       3. unpow2 50.5%

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

       4. hypot-undefine 58.5%

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

       5. hypot-undefine 50.5%

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

       6. unpow2 50.5%

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

       7. unpow2 50.5%

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

       8. +-commutative 50.5%

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

       9. unpow2 50.5%

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

       10. unpow2 50.5%

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

       11. hypot-undefine 58.5%

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

       12. *-commutative 58.5%

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

       13. exp-prod 56.4%

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

   12. Simplified 56.4%

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

   13. Taylor expanded in y.re around 0 57.5%

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

       1. rec-exp 57.5%

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

       2. distribute-rgt-neg-in 57.5%

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

   15. Simplified 57.5%

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

   if -1.75e-12 < y.im < 3.1000000000000003e-67

   1. Initial program 36.8%

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

      1. cancel-sign-sub-inv 36.8%

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

      2. fma-define 36.8%

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

      3. hypot-define 36.8%

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

      4. distribute-lft-neg-in 36.8%

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

      5. distribute-rgt-neg-out 36.8%

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

      6. fma-define 36.8%

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

      7. hypot-define 84.7%

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

      8. *-commutative 84.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in y.im around 0 63.1%

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

      1. unpow2 63.1%

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

      2. unpow2 63.1%

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

      3. hypot-undefine 84.6%

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

   7. Simplified 84.6%

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

   8. Taylor expanded in y.re around 0 37.4%

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

      1. unpow2 37.4%

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

      2. unpow2 37.4%

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

      3. hypot-undefine 45.7%

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

   10. Simplified 45.7%

       \[\leadsto \color{blue}{1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \]
   11. Step-by-step derivation

       1. log-pow 68.4%

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

   12. Applied egg-rr 68.4%

       \[\leadsto 1 + \color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]
   13. Step-by-step derivation

       1. hypot-undefine 47.2%

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

       2. unpow2 47.2%

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

       3. unpow2 47.2%

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

       4. +-commutative 47.2%

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

       5. unpow2 47.2%

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

       6. unpow2 47.2%

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

       7. hypot-define 68.4%

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

   14. Simplified 68.4%

       \[\leadsto 1 + \color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.75 \cdot 10^{-12} \lor \neg \left(y.im \leq 3.1 \cdot 10^{-67}\right):\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1700000:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 330:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1700000.0)
   (* (cos (* y.re (atan2 x.im x.re))) (pow x.im y.re))
   (if (<= y.re 330.0)
     (exp (* (atan2 x.im x.re) (- y.im)))
     (+ 1.0 (log (pow (hypot x.re x.im) y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1700000.0) {
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * pow(x_46_im, y_46_re);
	} else if (y_46_re <= 330.0) {
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = 1.0 + log(pow(hypot(x_46_re, x_46_im), y_46_re));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1700000.0) {
		tmp = Math.cos((y_46_re * Math.atan2(x_46_im, x_46_re))) * Math.pow(x_46_im, y_46_re);
	} else if (y_46_re <= 330.0) {
		tmp = Math.exp((Math.atan2(x_46_im, x_46_re) * -y_46_im));
	} else {
		tmp = 1.0 + Math.log(Math.pow(Math.hypot(x_46_re, x_46_im), y_46_re));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1700000.0:
		tmp = math.cos((y_46_re * math.atan2(x_46_im, x_46_re))) * math.pow(x_46_im, y_46_re)
	elif y_46_re <= 330.0:
		tmp = math.exp((math.atan2(x_46_im, x_46_re) * -y_46_im))
	else:
		tmp = 1.0 + math.log(math.pow(math.hypot(x_46_re, x_46_im), y_46_re))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1700000.0)
		tmp = Float64(cos(Float64(y_46_re * atan(x_46_im, x_46_re))) * (x_46_im ^ y_46_re));
	elseif (y_46_re <= 330.0)
		tmp = exp(Float64(atan(x_46_im, x_46_re) * Float64(-y_46_im)));
	else
		tmp = Float64(1.0 + log((hypot(x_46_re, x_46_im) ^ y_46_re)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1700000.0)
		tmp = cos((y_46_re * atan2(x_46_im, x_46_re))) * (x_46_im ^ y_46_re);
	elseif (y_46_re <= 330.0)
		tmp = exp((atan2(x_46_im, x_46_re) * -y_46_im));
	else
		tmp = 1.0 + log((hypot(x_46_re, x_46_im) ^ y_46_re));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1700000.0], N[(N[Cos[N[(y$46$re * N[ArcTan[x$46$im / x$46$re], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[x$46$im, y$46$re], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 330.0], N[Exp[N[(N[ArcTan[x$46$im / x$46$re], $MachinePrecision] * (-y$46$im)), $MachinePrecision]], $MachinePrecision], N[(1.0 + N[Log[N[Power[N[Sqrt[x$46$re ^ 2 + x$46$im ^ 2], $MachinePrecision], y$46$re], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.7e6

   1. Initial program 32.8%

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

      1. cancel-sign-sub-inv 32.8%

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

      2. fma-define 32.8%

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

      3. hypot-define 32.8%

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

      4. distribute-lft-neg-in 32.8%

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

      5. distribute-rgt-neg-out 32.8%

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

      6. fma-define 32.8%

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

      7. hypot-define 79.7%

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

      8. *-commutative 79.7%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in y.im around 0 78.2%

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

      1. unpow2 78.2%

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

      2. unpow2 78.2%

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

      3. hypot-undefine 78.2%

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

   7. Simplified 78.2%

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

   8. Taylor expanded in x.re around 0 61.3%

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

      1. *-commutative 61.3%

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

   10. Simplified 61.3%

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

   if -1.7e6 < y.re < 330

   1. Initial program 44.5%

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

      1. exp-diff 44.5%

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

      2. exp-to-pow 44.5%

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

      3. hypot-define 44.5%

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

      4. *-commutative 44.5%

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

      5. exp-prod 44.3%

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

      6. fma-define 44.3%

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

      7. hypot-define 88.2%

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

      8. *-commutative 88.2%

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

   3. Simplified 88.2%

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

      1. add-cube-cbrt 88.2%

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

      2. exp-prod 88.2%

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

      3. pow2 88.2%

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

   6. Applied egg-rr 88.2%

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

   7. Taylor expanded in y.re around 0 44.1%

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

      1. unpow2 44.1%

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

      2. unpow2 44.1%

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

      3. hypot-undefine 87.3%

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

   9. Simplified 87.3%

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

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

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

       1. +-commutative 53.3%

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

       2. unpow2 53.3%

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

       3. unpow2 53.3%

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

       4. hypot-undefine 83.5%

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

       5. hypot-undefine 53.3%

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

       6. unpow2 53.3%

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

       7. unpow2 53.3%

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

       8. +-commutative 53.3%

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

       9. unpow2 53.3%

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

       10. unpow2 53.3%

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

       11. hypot-undefine 83.5%

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

       12. *-commutative 83.5%

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

       13. exp-prod 80.3%

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

   12. Simplified 80.3%

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

   13. Taylor expanded in y.re around 0 82.6%

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

       1. rec-exp 82.6%

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

       2. distribute-rgt-neg-in 82.6%

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

   15. Simplified 82.6%

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

   if 330 < y.re

   1. Initial program 30.0%

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

      1. cancel-sign-sub-inv 30.0%

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

      2. fma-define 30.0%

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

      3. hypot-define 30.0%

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

      4. distribute-lft-neg-in 30.0%

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

      5. distribute-rgt-neg-out 30.0%

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

      6. fma-define 30.0%

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

      7. hypot-define 55.7%

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

      8. *-commutative 55.7%

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

   3. Simplified 55.7%

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

   5. Taylor expanded in y.im around 0 55.9%

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

      1. unpow2 55.9%

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

      2. unpow2 55.9%

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

      3. hypot-undefine 55.9%

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

   7. Simplified 55.9%

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

   8. Taylor expanded in y.re around 0 37.1%

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

      1. unpow2 37.1%

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

      2. unpow2 37.1%

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

      3. hypot-undefine 4.6%

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

   10. Simplified 4.6%

       \[\leadsto \color{blue}{1 + y.re \cdot \log \left(\mathsf{hypot}\left(x.im, x.re\right)\right)} \]
   11. Step-by-step derivation

       1. log-pow 56.0%

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

   12. Applied egg-rr 56.0%

       \[\leadsto 1 + \color{blue}{\log \left({\left(\mathsf{hypot}\left(x.im, x.re\right)\right)}^{y.re}\right)} \]
   13. Step-by-step derivation

       1. hypot-undefine 56.0%

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

       2. unpow2 56.0%

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

       3. unpow2 56.0%

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

       4. +-commutative 56.0%

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

       5. unpow2 56.0%

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

       6. unpow2 56.0%

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

       7. hypot-define 56.0%

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

   14. Simplified 56.0%

       \[\leadsto 1 + \color{blue}{\log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1700000:\\ \;\;\;\;\cos \left(y.re \cdot \tan^{-1}_* \frac{x.im}{x.re}\right) \cdot {x.im}^{y.re}\\ \mathbf{elif}\;y.re \leq 330:\\ \;\;\;\;e^{\tan^{-1}_* \frac{x.im}{x.re} \cdot \left(-y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left({\left(\mathsf{hypot}\left(x.re, x.im\right)\right)}^{y.re}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.2% 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 37.6%

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

   1. exp-diff 34.1%

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

   2. exp-to-pow 34.1%

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

   3. hypot-define 34.1%

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

   4. *-commutative 34.1%

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

   5. exp-prod 33.2%

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

   6. fma-define 33.2%

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

   7. hypot-define 70.5%

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

   8. *-commutative 70.5%

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

3. Simplified 70.5%

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

   1. add-cube-cbrt 70.5%

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

   2. exp-prod 70.5%

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

   3. pow2 70.5%

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

6. Applied egg-rr 70.5%

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

7. Taylor expanded in y.re around 0 34.3%

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

   1. unpow2 34.3%

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

   2. unpow2 34.3%

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

   3. hypot-undefine 72.1%

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

9. Simplified 72.1%

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

10. Taylor expanded in y.im around 0 60.2%

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

    1. +-commutative 60.2%

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

    2. unpow2 60.2%

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

    3. unpow2 60.2%

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

    4. hypot-undefine 74.6%

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

    5. hypot-undefine 60.2%

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

    6. unpow2 60.2%

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

    7. unpow2 60.2%

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

    8. +-commutative 60.2%

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

    9. unpow2 60.2%

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

    10. unpow2 60.2%

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

    11. hypot-undefine 74.6%

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

    12. *-commutative 74.6%

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

    13. exp-prod 73.5%

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

12. Simplified 73.5%

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

13. Taylor expanded in y.re around 0 51.3%

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

    1. rec-exp 51.3%

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

    2. distribute-rgt-neg-in 51.3%

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

15. Simplified 51.3%

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

16. Final simplification 51.3%

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

Alternative 13: 26.1% accurate, 829.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 1.0)

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return 1.0;
}

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
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;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return 1.0

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return 1.0
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 1.0;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 1.0

Derivation

1. Initial program 37.6%

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

   1. cancel-sign-sub-inv 37.6%

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

   2. fma-define 37.6%

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

   3. hypot-define 37.6%

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

   4. distribute-lft-neg-in 37.6%

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

   5. distribute-rgt-neg-out 37.6%

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

   6. fma-define 37.6%

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

   7. hypot-define 77.2%

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

   8. *-commutative 77.2%

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

3. Simplified 77.2%

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

5. Taylor expanded in y.im around 0 52.6%

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

   1. unpow2 52.6%

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

   2. unpow2 52.6%

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

   3. hypot-undefine 61.0%

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

7. Simplified 61.0%

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

8. Taylor expanded in y.re around 0 26.6%

   \[\leadsto \color{blue}{1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024146 
(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)))))