Migdal et al, Equation (51)

Percentage Accurate: 31.3% → 69.4%

Time: 27.8s

Precision: binary64

Cost: 72400

• 81.6% of points produce a very large (infinite) output. You may want to add a precondition.

Math

\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

↓

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t_0}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-182}:\\ \;\;\;\;e^{\sqrt{{\log \left(\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\right)}^{2}}}\\ \mathbf{elif}\;t_0 \leq 0.01:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k} + -2 \cdot \left(n \cdot \left(\pi \cdot \mathsf{log1p}\left(t_0 + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))

↓

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 PI) n))
        (t_1
         (sqrt
          (pow
           (pow (log1p (expm1 (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))) 3.0)
           0.3333333333333333))))
   (if (<= t_0 -5e+113)
     (sqrt (pow (pow (log1p (expm1 (/ t_0 k))) 3.0) 0.3333333333333333))
     (if (<= t_0 -5e-211)
       t_1
       (if (<= t_0 1e-182)
         (exp (sqrt (pow (log (sqrt (/ (pow t_0 (- 1.0 k)) k))) 2.0)))
         (if (<= t_0 0.01)
           (sqrt
            (+
             (* 2.0 (/ (* PI n) k))
             (* -2.0 (* n (* PI (log1p (+ t_0 -1.0)))))))
           t_1))))))

C

double code(double k, double n) {
	return (1.0 / sqrt(k)) * pow(((2.0 * ((double) M_PI)) * n), ((1.0 - k) / 2.0));
}

↓

double code(double k, double n) {
	double t_0 = (2.0 * ((double) M_PI)) * n;
	double t_1 = sqrt(pow(pow(log1p(expm1((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k))), 3.0), 0.3333333333333333));
	double tmp;
	if (t_0 <= -5e+113) {
		tmp = sqrt(pow(pow(log1p(expm1((t_0 / k))), 3.0), 0.3333333333333333));
	} else if (t_0 <= -5e-211) {
		tmp = t_1;
	} else if (t_0 <= 1e-182) {
		tmp = exp(sqrt(pow(log(sqrt((pow(t_0, (1.0 - k)) / k))), 2.0)));
	} else if (t_0 <= 0.01) {
		tmp = sqrt(((2.0 * ((((double) M_PI) * n) / k)) + (-2.0 * (n * (((double) M_PI) * log1p((t_0 + -1.0)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	return (1.0 / Math.sqrt(k)) * Math.pow(((2.0 * Math.PI) * n), ((1.0 - k) / 2.0));
}

↓

public static double code(double k, double n) {
	double t_0 = (2.0 * Math.PI) * n;
	double t_1 = Math.sqrt(Math.pow(Math.pow(Math.log1p(Math.expm1((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k))), 3.0), 0.3333333333333333));
	double tmp;
	if (t_0 <= -5e+113) {
		tmp = Math.sqrt(Math.pow(Math.pow(Math.log1p(Math.expm1((t_0 / k))), 3.0), 0.3333333333333333));
	} else if (t_0 <= -5e-211) {
		tmp = t_1;
	} else if (t_0 <= 1e-182) {
		tmp = Math.exp(Math.sqrt(Math.pow(Math.log(Math.sqrt((Math.pow(t_0, (1.0 - k)) / k))), 2.0)));
	} else if (t_0 <= 0.01) {
		tmp = Math.sqrt(((2.0 * ((Math.PI * n) / k)) + (-2.0 * (n * (Math.PI * Math.log1p((t_0 + -1.0)))))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(k, n):
	return (1.0 / math.sqrt(k)) * math.pow(((2.0 * math.pi) * n), ((1.0 - k) / 2.0))

↓

def code(k, n):
	t_0 = (2.0 * math.pi) * n
	t_1 = math.sqrt(math.pow(math.pow(math.log1p(math.expm1((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))), 3.0), 0.3333333333333333))
	tmp = 0
	if t_0 <= -5e+113:
		tmp = math.sqrt(math.pow(math.pow(math.log1p(math.expm1((t_0 / k))), 3.0), 0.3333333333333333))
	elif t_0 <= -5e-211:
		tmp = t_1
	elif t_0 <= 1e-182:
		tmp = math.exp(math.sqrt(math.pow(math.log(math.sqrt((math.pow(t_0, (1.0 - k)) / k))), 2.0)))
	elif t_0 <= 0.01:
		tmp = math.sqrt(((2.0 * ((math.pi * n) / k)) + (-2.0 * (n * (math.pi * math.log1p((t_0 + -1.0)))))))
	else:
		tmp = t_1
	return tmp

Julia

function code(k, n)
	return Float64(Float64(1.0 / sqrt(k)) * (Float64(Float64(2.0 * pi) * n) ^ Float64(Float64(1.0 - k) / 2.0)))
end

↓

function code(k, n)
	t_0 = Float64(Float64(2.0 * pi) * n)
	t_1 = sqrt(((log1p(expm1(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k))) ^ 3.0) ^ 0.3333333333333333))
	tmp = 0.0
	if (t_0 <= -5e+113)
		tmp = sqrt(((log1p(expm1(Float64(t_0 / k))) ^ 3.0) ^ 0.3333333333333333));
	elseif (t_0 <= -5e-211)
		tmp = t_1;
	elseif (t_0 <= 1e-182)
		tmp = exp(sqrt((log(sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k))) ^ 2.0)));
	elseif (t_0 <= 0.01)
		tmp = sqrt(Float64(Float64(2.0 * Float64(Float64(pi * n) / k)) + Float64(-2.0 * Float64(n * Float64(pi * log1p(Float64(t_0 + -1.0)))))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[k_, n_] := N[(N[(1.0 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

↓

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Power[N[Power[N[Log[1 + N[(Exp[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$0, -5e+113], N[Sqrt[N[Power[N[Power[N[Log[1 + N[(Exp[N[(t$95$0 / k), $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision], 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, -5e-211], t$95$1, If[LessEqual[t$95$0, 1e-182], N[Exp[N[Sqrt[N[Power[N[Log[N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], If[LessEqual[t$95$0, 0.01], N[Sqrt[N[(N[(2.0 * N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] + N[(-2.0 * N[(n * N[(Pi * N[Log[1 + N[(t$95$0 + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 (*.f64 2 (PI.f64)) n) < -5e113

   1. Initial program 0.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Applied egg-rr 16.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1} \]
      Step-by-step derivation

      [Start] 0.3%	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      *-commutative [=>] 0.3%	\[ \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}}} \]
      *-commutative [=>] 0.3%	\[ {\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      associate-*r* [<=] 0.3%	\[ {\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      div-inv [<=] 0.3%	\[ \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      expm1-log1p-u [=>] 0.3%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]
      expm1-udef [=>] 0.3%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)} - 1} \]

   3. Simplified 16.1%

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}} \]
      Step-by-step derivation

      [Start] 16.1%	\[ e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1 \]
      expm1-def [=>] 16.1%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]
      expm1-log1p [=>] 16.1%	\[ \color{blue}{\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}} \]
      *-commutative [<=] 16.1%	\[ \sqrt{\frac{{\left(2 \cdot \color{blue}{\left(n \cdot \pi\right)}\right)}^{\left(1 - k\right)}}{k}} \]
      associate-*r* [=>] 16.1%	\[ \sqrt{\frac{{\color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}^{\left(1 - k\right)}}{k}} \]
      *-commutative [=>] 16.1%	\[ \sqrt{\frac{{\left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}^{\left(1 - k\right)}}{k}} \]

   4. Applied egg-rr 74.2%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}}} \]
      Step-by-step derivation

      [Start] 16.1%	\[ \sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}} \]
      associate-*l* [=>] 16.1%	\[ \sqrt{\frac{{\color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}^{\left(1 - k\right)}}{k}} \]
      *-commutative [<=] 16.1%	\[ \sqrt{\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(1 - k\right)}}{k}} \]
      associate-*r* [<=] 16.1%	\[ \sqrt{\frac{{\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(1 - k\right)}}{k}} \]
      add-cbrt-cube [=>] 28.6%	\[ \sqrt{\color{blue}{\sqrt[3]{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k} \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right) \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}}} \]
      pow1/3 [=>] 74.2%	\[ \sqrt{\color{blue}{{\left(\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k} \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right) \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{0.3333333333333333}}} \]
      pow3 [=>] 74.2%	\[ \sqrt{{\color{blue}{\left({\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}}^{0.3333333333333333}} \]
      associate-*r* [=>] 74.2%	\[ \sqrt{{\left({\left(\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [=>] 74.2%	\[ \sqrt{{\left({\left(\frac{{\color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. Applied egg-rr 74.2%

      \[\leadsto \sqrt{{\left({\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(\left(n \cdot \pi\right) \cdot 2\right)}^{\left(1 - k\right)}}{k}\right)\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]
      Step-by-step derivation

      [Start] 74.2%	\[ \sqrt{{\left({\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]
      rem-cbrt-cube [<=] 74.2%	\[ \sqrt{{\left({\color{blue}{\left(\sqrt[3]{{\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}}\right)}}^{3}\right)}^{0.3333333333333333}} \]
      sqr-pow [=>] 65.5%	\[ \sqrt{{\left({\left(\sqrt[3]{\color{blue}{{\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      associate-*r* [=>] 65.5%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\color{blue}{\left(\left(n \cdot 2\right) \cdot \pi\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [<=] 65.5%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\color{blue}{\left(\pi \cdot \left(n \cdot 2\right)\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      metadata-eval [=>] 65.5%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\color{blue}{1.5}} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      associate-*r* [=>] 65.5%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5} \cdot {\left(\frac{{\color{blue}{\left(\left(n \cdot 2\right) \cdot \pi\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [<=] 65.5%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5} \cdot {\left(\frac{{\color{blue}{\left(\pi \cdot \left(n \cdot 2\right)\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      metadata-eval [=>] 65.5%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5} \cdot {\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\color{blue}{1.5}}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      cbrt-prod [=>] 65.5%	\[ \sqrt{{\left({\color{blue}{\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}} \cdot \sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}\right)}}^{3}\right)}^{0.3333333333333333}} \]

   6. Taylor expanded in k around 0 95.7%

      \[\leadsto \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{2 \cdot \left(n \cdot \pi\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. Simplified 95.7%

      \[\leadsto \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
      Step-by-step derivation

      [Start] 95.7%	\[ \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [=>] 95.7%	\[ \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{\left(n \cdot \pi\right) \cdot 2}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
      associate-*r* [<=] 95.7%	\[ \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\color{blue}{n \cdot \left(\pi \cdot 2\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [<=] 95.7%	\[ \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{n \cdot \color{blue}{\left(2 \cdot \pi\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}} \]

   if -5e113 < (*.f64 (*.f64 2 (PI.f64)) n) < -5.0000000000000002e-211 or 0.0100000000000000002 < (*.f64 (*.f64 2 (PI.f64)) n)

   1. Initial program 29.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Applied egg-rr 26.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1} \]
      Step-by-step derivation

      [Start] 29.1%	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      *-commutative [=>] 29.1%	\[ \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}}} \]
      *-commutative [=>] 29.1%	\[ {\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      associate-*r* [<=] 29.1%	\[ {\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      div-inv [<=] 29.1%	\[ \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      expm1-log1p-u [=>] 28.1%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]
      expm1-udef [=>] 28.1%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)} - 1} \]

   3. Simplified 27.6%

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}} \]
      Step-by-step derivation

      [Start] 26.9%	\[ e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1 \]
      expm1-def [=>] 26.9%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]
      expm1-log1p [=>] 27.6%	\[ \color{blue}{\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}} \]
      *-commutative [<=] 27.6%	\[ \sqrt{\frac{{\left(2 \cdot \color{blue}{\left(n \cdot \pi\right)}\right)}^{\left(1 - k\right)}}{k}} \]
      associate-*r* [=>] 27.6%	\[ \sqrt{\frac{{\color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}^{\left(1 - k\right)}}{k}} \]
      *-commutative [=>] 27.6%	\[ \sqrt{\frac{{\left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}^{\left(1 - k\right)}}{k}} \]

   4. Applied egg-rr 69.3%

      \[\leadsto \sqrt{\color{blue}{{\left({\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}}} \]
      Step-by-step derivation

      [Start] 27.6%	\[ \sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}} \]
      associate-*l* [=>] 27.6%	\[ \sqrt{\frac{{\color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}^{\left(1 - k\right)}}{k}} \]
      *-commutative [<=] 27.6%	\[ \sqrt{\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(1 - k\right)}}{k}} \]
      associate-*r* [<=] 27.6%	\[ \sqrt{\frac{{\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(1 - k\right)}}{k}} \]
      add-cbrt-cube [=>] 29.9%	\[ \sqrt{\color{blue}{\sqrt[3]{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k} \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right) \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}}} \]
      pow1/3 [=>] 69.3%	\[ \sqrt{\color{blue}{{\left(\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k} \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right) \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{0.3333333333333333}}} \]
      pow3 [=>] 69.3%	\[ \sqrt{{\color{blue}{\left({\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}}^{0.3333333333333333}} \]
      associate-*r* [=>] 69.3%	\[ \sqrt{{\left({\left(\frac{{\color{blue}{\left(\left(2 \cdot \pi\right) \cdot n\right)}}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [=>] 69.3%	\[ \sqrt{{\left({\left(\frac{{\color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. Applied egg-rr 75.5%

      \[\leadsto \sqrt{{\left({\color{blue}{\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(\left(n \cdot \pi\right) \cdot 2\right)}^{\left(1 - k\right)}}{k}\right)\right)\right)}}^{3}\right)}^{0.3333333333333333}} \]
      Step-by-step derivation

      [Start] 69.3%	\[ \sqrt{{\left({\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}} \]
      rem-cbrt-cube [<=] 69.3%	\[ \sqrt{{\left({\color{blue}{\left(\sqrt[3]{{\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{3}}\right)}}^{3}\right)}^{0.3333333333333333}} \]
      sqr-pow [=>] 58.2%	\[ \sqrt{{\left({\left(\sqrt[3]{\color{blue}{{\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      associate-*r* [=>] 58.2%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\color{blue}{\left(\left(n \cdot 2\right) \cdot \pi\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [<=] 58.2%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\color{blue}{\left(\pi \cdot \left(n \cdot 2\right)\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      metadata-eval [=>] 58.2%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\color{blue}{1.5}} \cdot {\left(\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      associate-*r* [=>] 58.2%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5} \cdot {\left(\frac{{\color{blue}{\left(\left(n \cdot 2\right) \cdot \pi\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      *-commutative [<=] 58.2%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5} \cdot {\left(\frac{{\color{blue}{\left(\pi \cdot \left(n \cdot 2\right)\right)}}^{\left(1 - k\right)}}{k}\right)}^{\left(\frac{3}{2}\right)}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      metadata-eval [=>] 58.2%	\[ \sqrt{{\left({\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5} \cdot {\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{\color{blue}{1.5}}}\right)}^{3}\right)}^{0.3333333333333333}} \]
      cbrt-prod [=>] 58.2%	\[ \sqrt{{\left({\color{blue}{\left(\sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}} \cdot \sqrt[3]{{\left(\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)}^{1.5}}\right)}}^{3}\right)}^{0.3333333333333333}} \]

   if -5.0000000000000002e-211 < (*.f64 (*.f64 2 (PI.f64)) n) < 1e-182

   1. Initial program 43.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Applied egg-rr 43.3%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)}} \]
      Step-by-step derivation

      [Start] 43.4%	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      *-commutative [=>] 43.4%	\[ \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}}} \]
      *-commutative [=>] 43.4%	\[ {\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      associate-*r* [<=] 43.4%	\[ {\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      div-inv [<=] 43.4%	\[ \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      add-exp-log [=>] 42.6%	\[ \color{blue}{e^{\log \left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)}} \]
      add-sqr-sqrt [=>] 42.6%	\[ e^{\log \color{blue}{\left(\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)}} \]
      sqrt-unprod [=>] 42.6%	\[ e^{\log \color{blue}{\left(\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)}} \]
      frac-times [=>] 42.6%	\[ e^{\log \left(\sqrt{\color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k} \cdot \sqrt{k}}}}\right)} \]

   3. Applied egg-rr 65.4%

      \[\leadsto e^{\color{blue}{\sqrt{{\log \left(\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\right)}^{2}}}} \]
      Step-by-step derivation

      [Start] 43.3%	\[ e^{\log \left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} \]
      add-sqr-sqrt [=>] 36.3%	\[ e^{\color{blue}{\sqrt{\log \left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} \cdot \sqrt{\log \left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)}}} \]
      sqrt-unprod [=>] 65.4%	\[ e^{\color{blue}{\sqrt{\log \left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right) \cdot \log \left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)}}} \]
      pow2 [=>] 65.4%	\[ e^{\sqrt{\color{blue}{{\log \left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)}^{2}}}} \]

   if 1e-182 < (*.f64 (*.f64 2 (PI.f64)) n) < 0.0100000000000000002

   1. Initial program 35.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. Applied egg-rr 33.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1} \]
      Step-by-step derivation

      [Start] 35.1%	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      *-commutative [=>] 35.1%	\[ \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}}} \]
      *-commutative [=>] 35.1%	\[ {\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      associate-*r* [<=] 35.1%	\[ {\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \]
      div-inv [<=] 35.1%	\[ \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      expm1-log1p-u [=>] 34.4%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)\right)} \]
      expm1-udef [=>] 33.4%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\right)} - 1} \]

   3. Simplified 35.6%

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(\left(n \cdot 2\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}} \]
      Step-by-step derivation

      [Start] 33.9%	\[ e^{\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)} - 1 \]
      expm1-def [=>] 34.9%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\right)\right)} \]
      expm1-log1p [=>] 35.6%	\[ \color{blue}{\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}} \]
      *-commutative [<=] 35.6%	\[ \sqrt{\frac{{\left(2 \cdot \color{blue}{\left(n \cdot \pi\right)}\right)}^{\left(1 - k\right)}}{k}} \]
      associate-*r* [=>] 35.6%	\[ \sqrt{\frac{{\color{blue}{\left(\left(2 \cdot n\right) \cdot \pi\right)}}^{\left(1 - k\right)}}{k}} \]
      *-commutative [=>] 35.6%	\[ \sqrt{\frac{{\left(\color{blue}{\left(n \cdot 2\right)} \cdot \pi\right)}^{\left(1 - k\right)}}{k}} \]

   4. Taylor expanded in k around 0 15.0%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \pi\right)\right)}} \]

   5. Applied egg-rr 84.4%

      \[\leadsto \sqrt{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\color{blue}{\mathsf{log1p}\left(n \cdot \left(\pi \cdot 2\right) - 1\right)} \cdot \pi\right)\right)} \]
      Step-by-step derivation

      [Start] 15.0%	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\log \left(2 \cdot \left(n \cdot \pi\right)\right) \cdot \pi\right)\right)} \]
      log1p-expm1-u [=>] 84.4%	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(2 \cdot \left(n \cdot \pi\right)\right)\right)\right)} \cdot \pi\right)\right)} \]
      expm1-udef [=>] 84.4%	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\mathsf{log1p}\left(\color{blue}{e^{\log \left(2 \cdot \left(n \cdot \pi\right)\right)} - 1}\right) \cdot \pi\right)\right)} \]
      add-exp-log [<=] 84.4%	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\mathsf{log1p}\left(\color{blue}{2 \cdot \left(n \cdot \pi\right)} - 1\right) \cdot \pi\right)\right)} \]
      *-commutative [<=] 84.4%	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\mathsf{log1p}\left(\color{blue}{\left(n \cdot \pi\right) \cdot 2} - 1\right) \cdot \pi\right)\right)} \]
      associate-*l* [=>] 84.4%	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k} + -2 \cdot \left(n \cdot \left(\mathsf{log1p}\left(\color{blue}{n \cdot \left(\pi \cdot 2\right)} - 1\right) \cdot \pi\right)\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(2 \cdot \pi\right) \cdot n \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{\left(2 \cdot \pi\right) \cdot n}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;\left(2 \cdot \pi\right) \cdot n \leq -5 \cdot 10^{-211}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;\left(2 \cdot \pi\right) \cdot n \leq 10^{-182}:\\ \;\;\;\;e^{\sqrt{{\log \left(\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\right)}^{2}}}\\ \mathbf{elif}\;\left(2 \cdot \pi\right) \cdot n \leq 0.01:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k} + -2 \cdot \left(n \cdot \left(\pi \cdot \mathsf{log1p}\left(\left(2 \cdot \pi\right) \cdot n + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	69.4%
Cost	72400

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := \sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t_0}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 10^{-182}:\\ \;\;\;\;e^{\sqrt{{\log \left(\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\right)}^{2}}}\\ \mathbf{elif}\;t_0 \leq 0.01:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k} + -2 \cdot \left(n \cdot \left(\pi \cdot \mathsf{log1p}\left(t_0 + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	67.4%
Cost	65612

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := \frac{{t_0}^{\left(1 - k\right)}}{k}\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t_0}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq -1 \cdot 10^{-81}:\\ \;\;\;\;\sqrt{{\left({\left(\left|\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right|\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq 10^{-182}:\\ \;\;\;\;e^{\sqrt{{\log \left(\sqrt{t_1}\right)}^{2}}}\\ \mathbf{elif}\;t_0 \leq 10^{-27}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k} + -2 \cdot \left(n \cdot \left(\pi \cdot \mathsf{log1p}\left(t_0 + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_1}^{0.5}\\ \end{array} \]

Alternative 3
Accuracy	65.7%
Cost	53324

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := \frac{{t_0}^{\left(1 - k\right)}}{k}\\ \mathbf{if}\;t_0 \leq -1.2 \cdot 10^{+27}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t_0}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{{\left({t_1}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq 10^{-27}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k} + -2 \cdot \left(n \cdot \left(\pi \cdot \mathsf{log1p}\left(t_0 + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_1}^{0.5}\\ \end{array} \]

Alternative 4
Accuracy	66.5%
Cost	53324

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \mathbf{if}\;t_0 \leq -5 \cdot 10^{+113}:\\ \;\;\;\;\sqrt{{\left({\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{t_0}{k}\right)\right)\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{{\left({\left(\left|\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}\right|\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq 10^{-27}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k} + -2 \cdot \left(n \cdot \left(\pi \cdot \mathsf{log1p}\left(t_0 + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{t_0}^{\left(1 - k\right)}}{k}\right)}^{0.5}\\ \end{array} \]

Alternative 5
Accuracy	64.9%
Cost	52804

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := {t_0}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot t_1 \leq 4 \cdot 10^{+265}:\\ \;\;\;\;t_1 \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{\left({\left(\frac{{t_0}^{\left(1 - k\right)}}{k}\right)}^{3}\right)}^{0.3333333333333333}}\\ \end{array} \]

Alternative 6
Accuracy	62.3%
Cost	52740

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := {t_0}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot t_1 \leq 4 \cdot 10^{+265}:\\ \;\;\;\;t_1 \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt{{\left(\frac{{t_0}^{\left(1 - k\right)}}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 7
Accuracy	64.3%
Cost	46600

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := \frac{{t_0}^{\left(1 - k\right)}}{k}\\ \mathbf{if}\;t_0 \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\sqrt{{\left({t_1}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;t_0 \leq 10^{-27}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi \cdot n}{k} + -2 \cdot \left(n \cdot \left(\pi \cdot \mathsf{log1p}\left(t_0 + -1\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{t_1}^{0.5}\\ \end{array} \]

Alternative 8
Accuracy	52.9%
Cost	40068

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ t_1 := {t_0}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot t_1 \leq 4 \cdot 10^{+265}:\\ \;\;\;\;t_1 \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{{t_0}^{\left(1 - k\right)}}{k}\right)}^{0.5}\\ \end{array} \]

Alternative 9
Accuracy	58.8%
Cost	26704

\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \mathbf{if}\;k \leq -1750:\\ \;\;\;\;{\left(\frac{{t_0}^{\left(1 - k\right)}}{k}\right)}^{0.5}\\ \mathbf{elif}\;k \leq 1.55 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{{\left({\left(\frac{\pi \cdot \left(2 \cdot n\right)}{k}\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;k \leq 1.8 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right|}\\ \mathbf{elif}\;k \leq 1.3 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\sqrt{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left|t_0\right|\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 10
Accuracy	53.5%
Cost	26644

\[\begin{array}{l} t_0 := \log \left(e^{\sqrt{\left(2 \cdot n\right) \cdot \frac{\pi}{k}}}\right)\\ t_1 := \left(2 \cdot \pi\right) \cdot n\\ t_2 := {\left(\frac{{t_1}^{\left(1 - k\right)}}{k}\right)}^{0.5}\\ t_3 := \sqrt{2 \cdot \frac{n}{\log \left(e^{\frac{k}{\pi}}\right)}}\\ \mathbf{if}\;k \leq -560:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -7.3 \cdot 10^{-193}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq -9 \cdot 10^{-237}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -2 \cdot 10^{-309}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;k \leq 1.45 \cdot 10^{-268}:\\ \;\;\;\;\sqrt{t_1} \cdot \sqrt{\frac{1}{k}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 11
Accuracy	55.4%
Cost	26576

\[\begin{array}{l} t_0 := {\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}\right)}^{0.5}\\ \mathbf{if}\;k \leq -380:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq -5 \cdot 10^{-166}:\\ \;\;\;\;\log \left(e^{\sqrt{\left(2 \cdot n\right) \cdot \frac{\pi}{k}}}\right)\\ \mathbf{elif}\;k \leq -1.5 \cdot 10^{-201}:\\ \;\;\;\;\sqrt{2 \cdot \frac{n}{\log \left(e^{\frac{k}{\pi}}\right)}}\\ \mathbf{elif}\;k \leq 1.35 \cdot 10^{-96}:\\ \;\;\;\;\sqrt{\sqrt{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Accuracy	58.4%
Cost	26576

\[\begin{array}{l} t_0 := {\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}\right)}^{0.5}\\ \mathbf{if}\;k \leq -380:\\ \;\;\;\;t_0\\ \mathbf{elif}\;k \leq 7 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{{\left({\left(\frac{\pi \cdot \left(2 \cdot n\right)}{k}\right)}^{3}\right)}^{0.3333333333333333}}\\ \mathbf{elif}\;k \leq 5.5 \cdot 10^{-218}:\\ \;\;\;\;\sqrt{\left|2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right|}\\ \mathbf{elif}\;k \leq 2.1 \cdot 10^{-97}:\\ \;\;\;\;\sqrt{\sqrt{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Accuracy	52.8%
Cost	26249

\[\begin{array}{l} \mathbf{if}\;k \leq -830 \lor \neg \left(k \leq -4.1 \cdot 10^{-236}\right):\\ \;\;\;\;{\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\sqrt{\left(2 \cdot n\right) \cdot \frac{\pi}{k}}}\right)\\ \end{array} \]

Alternative 14
Accuracy	40.1%
Cost	20040

\[\begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot \frac{n}{k}\right)\\ \mathbf{if}\;k \leq -4.1 \cdot 10^{-236}:\\ \;\;\;\;\sqrt[3]{{t_0}^{1.5}}\\ \mathbf{elif}\;k \leq 5.9:\\ \;\;\;\;\sqrt{\left|t_0\right|}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 15
Accuracy	50.2%
Cost	19840

\[{\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}\right)}^{0.5} \]

Alternative 16
Accuracy	21.9%
Cost	19780

\[\begin{array}{l} t_0 := 2 \cdot \left(\pi \cdot \frac{n}{k}\right)\\ \mathbf{if}\;n \leq -2.4 \cdot 10^{+42}:\\ \;\;\;\;\sqrt[3]{{t_0}^{1.5}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left|t_0\right|}\\ \end{array} \]

Alternative 17
Accuracy	20.3%
Cost	19584

\[\sqrt{\left|2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right|} \]

Alternative 18
Accuracy	19.2%
Cost	13248

\[{\left(\left(2 \cdot n\right) \cdot \frac{\pi}{k}\right)}^{0.5} \]

Alternative 19
Accuracy	19.1%
Cost	13248

\[{\left(\frac{\pi}{k \cdot \frac{0.5}{n}}\right)}^{0.5} \]

Alternative 20
Accuracy	12.7%
Cost	13184

\[\sqrt{2 \cdot \frac{n}{\frac{k}{\pi}}} \]

Alternative 21
Accuracy	12.7%
Cost	13184

\[\sqrt{\frac{\pi}{\frac{k}{2 \cdot n}}} \]

Reproduce

herbie shell --seed 2023255 
(FPCore (k n)
  :name "Migdal et al, Equation (51)"
  :precision binary64
  (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))