Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.4%

Time: 22.4s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

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));
}

Python

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

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

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
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]

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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));
}

Python

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

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

MATLAB

function tmp = code(k, n)
	tmp = (1.0 / sqrt(k)) * (((2.0 * pi) * n) ^ ((1.0 - k) / 2.0));
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]

Alternative 1: 99.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{2 \cdot n}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{k}}} \cdot \sqrt{\frac{\pi}{k}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = sqrt(((2.0 * n) / ((2.0 * (n * pi)) ^ k))) * sqrt((pi / k));
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. unpow-prod-down 72.7%

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

   4. unpow-prod-down 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

   1. *-un-lft-identity 99.7%

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

   2. *-commutative 99.7%

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

   3. sqrt-unprod 99.7%

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

   4. sqrt-undiv 90.7%

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

8. Applied egg-rr 90.7%

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

   1. *-rgt-identity 90.7%

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

   2. times-frac 90.7%

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

   3. *-commutative 90.7%

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

   4. associate-*r* 90.7%

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

10. Simplified 90.7%

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

    1. *-commutative 90.7%

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

    2. sqrt-prod 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-*r* 99.7%

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

    7. *-commutative 99.7%

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

12. Applied egg-rr 99.7%

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

13. Final simplification 99.7%

    \[\leadsto \sqrt{\frac{2 \cdot n}{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{k}}} \cdot \sqrt{\frac{\pi}{k}} \]
14. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.45e-45)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.45e-45) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.45e-45) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.45e-45:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((2.0 * n))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.45e-45)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.45e-45)
		tmp = sqrt((pi / k)) * sqrt((2.0 * n));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.45e-45], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4499999999999999e-45

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.4%

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

      2. *-un-lft-identity 99.4%

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

      3. unpow-prod-down 98.9%

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

      4. unpow-prod-down 99.4%

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

      5. div-sub 99.4%

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

      6. metadata-eval 99.4%

         \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

      7. pow-div 99.4%

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

      8. pow1/2 99.4%

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

      9. associate-/l/ 99.4%

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

      10. *-commutative 99.4%

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

      11. associate-*l* 99.4%

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

      12. div-inv 99.4%

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

      13. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
   5. Step-by-step derivation

      1. unpow1/2 99.4%

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

   6. Simplified 99.4%

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

      1. sqrt-unprod 99.4%

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

      2. sqrt-undiv 77.5%

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

   8. Applied egg-rr 77.5%

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

   9. Taylor expanded in k around 0 77.5%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{n \cdot \pi}{k}}} \]
   10. Step-by-step derivation

       1. associate-/l* 77.5%

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

   11. Simplified 77.5%

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

       1. associate-*r* 77.5%

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

       2. sqrt-prod 99.5%

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

       3. *-commutative 99.5%

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

   13. Applied egg-rr 99.5%

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

   if 2.4499999999999999e-45 < k

   1. Initial program 99.7%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.7%

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

      3. frac-times 99.6%

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

      4. metadata-eval 99.6%

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

      5. add-sqr-sqrt 99.6%

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

   4. Applied egg-rr 99.6%

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

   6. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{k} \cdot {\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}\right)}^{0.5}} \]
   7. Step-by-step derivation

      1. unpow1/2 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-lft-identity 99.7%

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

      4. *-commutative 99.7%

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

      5. associate-*r* 99.7%

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

      6. *-commutative 99.7%

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

      7. associate-*r* 99.7%

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

   8. Simplified 99.7%

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.2e+238)
   (* (sqrt (/ PI k)) (sqrt (* 2.0 n)))
   (cbrt (pow (* 2.0 (/ n (/ k PI))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.2e+238) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((2.0 * n));
	} else {
		tmp = cbrt(pow((2.0 * (n / (k / ((double) M_PI)))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.2e+238) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((2.0 * n));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (n / (k / Math.PI))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.2e+238)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(2.0 * n)));
	else
		tmp = cbrt((Float64(2.0 * Float64(n / Float64(k / pi))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.2e+238], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(n / N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.2e238

   1. Initial program 99.5%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. unpow-prod-down 73.4%

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

      4. unpow-prod-down 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

         \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

      7. pow-div 99.7%

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

      8. pow1/2 99.7%

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

      9. associate-/l/ 99.7%

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

      10. *-commutative 99.7%

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

      11. associate-*l* 99.7%

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

      12. div-inv 99.7%

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

      13. metadata-eval 99.7%

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

   4. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
   5. Step-by-step derivation

      1. unpow1/2 99.7%

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

   6. Simplified 99.7%

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

      1. sqrt-unprod 99.7%

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

      2. sqrt-undiv 89.7%

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

   8. Applied egg-rr 89.7%

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

   9. Taylor expanded in k around 0 43.3%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{n \cdot \pi}{k}}} \]
   10. Step-by-step derivation

       1. associate-/l* 43.3%

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

   11. Simplified 43.3%

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

       1. associate-*r* 43.3%

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

       2. sqrt-prod 53.3%

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

       3. *-commutative 53.3%

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

   13. Applied egg-rr 53.3%

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

   if 1.2e238 < k

   1. Initial program 100.0%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*l/ 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. unpow-prod-down 66.7%

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

      4. unpow-prod-down 100.0%

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

      5. div-sub 100.0%

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

      6. metadata-eval 100.0%

         \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

      7. pow-div 100.0%

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

      8. pow1/2 100.0%

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

      9. associate-/l/ 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*l* 100.0%

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

      12. div-inv 100.0%

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

      13. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
   5. Step-by-step derivation

      1. unpow1/2 100.0%

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

   6. Simplified 100.0%

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

      1. sqrt-unprod 100.0%

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

      2. sqrt-undiv 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in k around 0 3.1%

      \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{n \cdot \pi}{k}}} \]
   10. Step-by-step derivation

       1. associate-/l* 3.1%

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

   11. Simplified 3.1%

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

       1. add-cbrt-cube 27.0%

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

       2. pow1/3 27.0%

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

   13. Applied egg-rr 27.0%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \left(2 \cdot \frac{\pi}{k}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   14. Step-by-step derivation

       1. unpow1/3 27.0%

          \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(2 \cdot \frac{\pi}{k}\right)\right)}^{1.5}}} \]

       2. associate-*r* 27.0%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(n \cdot 2\right) \cdot \frac{\pi}{k}\right)}}^{1.5}} \]

       3. *-commutative 27.0%

          \[\leadsto \sqrt[3]{{\left(\color{blue}{\left(2 \cdot n\right)} \cdot \frac{\pi}{k}\right)}^{1.5}} \]

       4. associate-*l* 27.0%

          \[\leadsto \sqrt[3]{{\color{blue}{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}}^{1.5}} \]

       5. associate-/l* 27.0%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{1.5}} \]

       6. *-commutative 27.0%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{1.5}} \]

       7. associate-*r/ 27.0%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{1.5}} \]

       8. *-commutative 27.0%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(\frac{n}{k} \cdot \pi\right)}\right)}^{1.5}} \]

       9. associate-/r/ 27.0%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\frac{n}{\frac{k}{\pi}}}\right)}^{1.5}} \]

   15. Simplified 27.0%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.2 \cdot 10^{+238}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* n (* 2.0 PI)) (- 0.5 (/ k 2.0))) (sqrt k)))

C

double code(double k, double n) {
	return pow((n * (2.0 * ((double) M_PI))), (0.5 - (k / 2.0))) / sqrt(k);
}

Java

public static double code(double k, double n) {
	return Math.pow((n * (2.0 * Math.PI)), (0.5 - (k / 2.0))) / Math.sqrt(k);
}

Python

def code(k, n):
	return math.pow((n * (2.0 * math.pi)), (0.5 - (k / 2.0))) / math.sqrt(k)

Julia

function code(k, n)
	return Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(0.5 - Float64(k / 2.0))) / sqrt(k))
end

MATLAB

function tmp = code(k, n)
	tmp = ((n * (2.0 * pi)) ^ (0.5 - (k / 2.0))) / sqrt(k);
end

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(k / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

      \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

3. Simplified 99.5%

   \[\leadsto \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
4. Add Preprocessing

5. Final simplification 99.5%

   \[\leadsto \frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]
6. Add Preprocessing

Alternative 5: 49.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = sqrt((pi / k)) * sqrt((2.0 * n));
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. unpow-prod-down 72.7%

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

   4. unpow-prod-down 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

   1. sqrt-unprod 99.7%

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

   2. sqrt-undiv 90.7%

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

8. Applied egg-rr 90.7%

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

9. Taylor expanded in k around 0 39.5%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{n \cdot \pi}{k}}} \]
10. Step-by-step derivation

    1. associate-/l* 39.5%

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

11. Simplified 39.5%

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

    1. associate-*r* 39.5%

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

    2. sqrt-prod 48.6%

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

    3. *-commutative 48.6%

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

13. Applied egg-rr 48.6%

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

14. Final simplification 48.6%

    \[\leadsto \sqrt{\frac{\pi}{k}} \cdot \sqrt{2 \cdot n} \]
15. Add Preprocessing

Alternative 6: 38.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k \cdot \left(\frac{0.5}{n} \cdot \frac{1}{\pi}\right)}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ 1.0 (sqrt (* k (* (/ 0.5 n) (/ 1.0 PI))))))

C

double code(double k, double n) {
	return 1.0 / sqrt((k * ((0.5 / n) * (1.0 / ((double) M_PI)))));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt((k * ((0.5 / n) * (1.0 / Math.PI))));
}

Python

def code(k, n):
	return 1.0 / math.sqrt((k * ((0.5 / n) * (1.0 / math.pi))))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(k * Float64(Float64(0.5 / n) * Float64(1.0 / pi)))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt((k * ((0.5 / n) * (1.0 / pi))));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(k * N[(N[(0.5 / n), $MachinePrecision] * N[(1.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. unpow-prod-down 72.7%

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

   4. unpow-prod-down 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

   3. sqrt-unprod 99.7%

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

   4. sqrt-undiv 91.3%

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

8. Applied egg-rr 91.3%

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

   1. unpow-1 91.3%

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

   2. associate-/l* 91.2%

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

   3. *-commutative 91.2%

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

   4. associate-*r* 91.2%

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

   5. *-commutative 91.2%

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

   6. associate-*r* 91.2%

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

10. Simplified 91.2%

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

11. Taylor expanded in k around 0 40.1%

    \[\leadsto \frac{1}{\sqrt{k \cdot \color{blue}{\frac{0.5}{n \cdot \pi}}}} \]
12. Step-by-step derivation

    1. associate-/r* 40.1%

       \[\leadsto \frac{1}{\sqrt{k \cdot \color{blue}{\frac{\frac{0.5}{n}}{\pi}}}} \]

    2. div-inv 40.1%

       \[\leadsto \frac{1}{\sqrt{k \cdot \color{blue}{\left(\frac{0.5}{n} \cdot \frac{1}{\pi}\right)}}} \]

13. Applied egg-rr 40.1%

    \[\leadsto \frac{1}{\sqrt{k \cdot \color{blue}{\left(\frac{0.5}{n} \cdot \frac{1}{\pi}\right)}}} \]

14. Final simplification 40.1%

    \[\leadsto \frac{1}{\sqrt{k \cdot \left(\frac{0.5}{n} \cdot \frac{1}{\pi}\right)}} \]
15. Add Preprocessing

Alternative 7: 38.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k \cdot \frac{0.5}{n \cdot \pi}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (* k (/ 0.5 (* n PI))))))

C

double code(double k, double n) {
	return 1.0 / sqrt((k * (0.5 / (n * ((double) M_PI)))));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt((k * (0.5 / (n * Math.PI))));
}

Python

def code(k, n):
	return 1.0 / math.sqrt((k * (0.5 / (n * math.pi))))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(k * Float64(0.5 / Float64(n * pi)))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt((k * (0.5 / (n * pi))));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(k * N[(0.5 / N[(n * Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. unpow-prod-down 72.7%

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

   4. unpow-prod-down 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

   3. sqrt-unprod 99.7%

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

   4. sqrt-undiv 91.3%

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

8. Applied egg-rr 91.3%

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

   1. unpow-1 91.3%

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

   2. associate-/l* 91.2%

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

   3. *-commutative 91.2%

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

   4. associate-*r* 91.2%

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

   5. *-commutative 91.2%

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

   6. associate-*r* 91.2%

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

10. Simplified 91.2%

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

11. Taylor expanded in k around 0 40.1%

    \[\leadsto \frac{1}{\sqrt{k \cdot \color{blue}{\frac{0.5}{n \cdot \pi}}}} \]

12. Final simplification 40.1%

    \[\leadsto \frac{1}{\sqrt{k \cdot \frac{0.5}{n \cdot \pi}}} \]
13. Add Preprocessing

Alternative 8: 38.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\frac{k}{n} \cdot \frac{0.5}{\pi}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ 1.0 (sqrt (* (/ k n) (/ 0.5 PI)))))

C

double code(double k, double n) {
	return 1.0 / sqrt(((k / n) * (0.5 / ((double) M_PI))));
}

Java

public static double code(double k, double n) {
	return 1.0 / Math.sqrt(((k / n) * (0.5 / Math.PI)));
}

Python

def code(k, n):
	return 1.0 / math.sqrt(((k / n) * (0.5 / math.pi)))

Julia

function code(k, n)
	return Float64(1.0 / sqrt(Float64(Float64(k / n) * Float64(0.5 / pi))))
end

MATLAB

function tmp = code(k, n)
	tmp = 1.0 / sqrt(((k / n) * (0.5 / pi)));
end

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(N[(k / n), $MachinePrecision] * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. unpow-prod-down 72.7%

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

   4. unpow-prod-down 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

   1. clear-num 99.7%

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

   2. inv-pow 99.7%

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

   3. sqrt-unprod 99.7%

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

   4. sqrt-undiv 91.3%

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

8. Applied egg-rr 91.3%

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

   1. unpow-1 91.3%

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

   2. associate-/l* 91.2%

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

   3. *-commutative 91.2%

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

   4. associate-*r* 91.2%

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

   5. *-commutative 91.2%

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

   6. associate-*r* 91.2%

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

10. Simplified 91.2%

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

11. Taylor expanded in k around 0 40.1%

    \[\leadsto \frac{1}{\sqrt{k \cdot \color{blue}{\frac{0.5}{n \cdot \pi}}}} \]
12. Step-by-step derivation

    1. *-un-lft-identity 40.1%

       \[\leadsto \frac{1}{\color{blue}{1 \cdot \sqrt{k \cdot \frac{0.5}{n \cdot \pi}}}} \]

    2. associate-*r/ 40.1%

       \[\leadsto \frac{1}{1 \cdot \sqrt{\color{blue}{\frac{k \cdot 0.5}{n \cdot \pi}}}} \]

    3. times-frac 40.1%

       \[\leadsto \frac{1}{1 \cdot \sqrt{\color{blue}{\frac{k}{n} \cdot \frac{0.5}{\pi}}}} \]

13. Applied egg-rr 40.1%

    \[\leadsto \frac{1}{\color{blue}{1 \cdot \sqrt{\frac{k}{n} \cdot \frac{0.5}{\pi}}}} \]
14. Step-by-step derivation

    1. *-lft-identity 40.1%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{n} \cdot \frac{0.5}{\pi}}}} \]

15. Simplified 40.1%

    \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{n} \cdot \frac{0.5}{\pi}}}} \]

16. Final simplification 40.1%

    \[\leadsto \frac{1}{\sqrt{\frac{k}{n} \cdot \frac{0.5}{\pi}}} \]
17. Add Preprocessing

Alternative 9: 38.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(k, n)
	tmp = sqrt((2.0 * (n * (pi / k))));
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. unpow-prod-down 72.7%

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

   4. unpow-prod-down 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

      \[\leadsto \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. unpow1/2 99.7%

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

6. Simplified 99.7%

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

   1. sqrt-unprod 99.7%

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

   2. sqrt-undiv 90.7%

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

8. Applied egg-rr 90.7%

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

9. Taylor expanded in k around 0 39.5%

   \[\leadsto \sqrt{\color{blue}{2 \cdot \frac{n \cdot \pi}{k}}} \]
10. Step-by-step derivation

    1. associate-/l* 39.5%

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

11. Simplified 39.5%

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

12. Final simplification 39.5%

    \[\leadsto \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024041 
(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))))