Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.0%

Time: 17.2s

Alternatives: 14

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: 99.4% 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: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.1 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k} \cdot {\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7.1e-51)
   (/ 1.0 (* (sqrt (/ k PI)) (sqrt (/ 0.5 n))))
   (sqrt (* (/ 1.0 k) (pow (* (* 2.0 n) PI) (- 1.0 k))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 7.1e-51) {
		tmp = 1.0 / (sqrt((k / ((double) M_PI))) * sqrt((0.5 / n)));
	} else {
		tmp = sqrt(((1.0 / k) * pow(((2.0 * n) * ((double) M_PI)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 7.1e-51) {
		tmp = 1.0 / (Math.sqrt((k / Math.PI)) * Math.sqrt((0.5 / n)));
	} else {
		tmp = Math.sqrt(((1.0 / k) * Math.pow(((2.0 * n) * Math.PI), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 7.1e-51:
		tmp = 1.0 / (math.sqrt((k / math.pi)) * math.sqrt((0.5 / n)))
	else:
		tmp = math.sqrt(((1.0 / k) * math.pow(((2.0 * n) * math.pi), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 7.1e-51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(k / pi)) * sqrt(Float64(0.5 / n))));
	else
		tmp = sqrt(Float64(Float64(1.0 / k) * (Float64(Float64(2.0 * n) * pi) ^ Float64(1.0 - k))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 7.1e-51)
		tmp = 1.0 / (sqrt((k / pi)) * sqrt((0.5 / n)));
	else
		tmp = sqrt(((1.0 / k) * (((2.0 * n) * pi) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7.1e-51], N[(1.0 / N[(N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.5 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[(1.0 / k), $MachinePrecision] * N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.1e-51

   1. Initial program 98.6%

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

      1. unpow-prod-down 99.0%

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

      2. unpow-prod-down 98.6%

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

      3. div-sub 98.6%

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

      4. metadata-eval 98.6%

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

      5. pow-sub 98.5%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

      6. pow1/2 98.5%

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

      7. frac-times 98.5%

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

      8. *-un-lft-identity 98.5%

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

      9. associate-*l* 98.5%

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

      10. associate-*l* 98.5%

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

      11. div-inv 98.5%

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

      12. metadata-eval 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. *-commutative 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*r* 98.5%

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

      6. *-commutative 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in k around 0 98.6%

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

      1. *-un-lft-identity 98.6%

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

      2. clear-num 98.5%

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

      3. inv-pow 98.5%

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

      4. *-commutative 98.5%

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

      5. sqrt-undiv 75.3%

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

   8. Applied egg-rr 75.3%

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

      1. unpow-1 75.3%

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

      2. associate-/r* 75.3%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]

   10. Simplified 75.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]
   11. Step-by-step derivation

       1. div-inv 75.3%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{k}{\pi} \cdot \frac{1}{2 \cdot n}}}} \]

       2. sqrt-prod 98.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{1}{2 \cdot n}}}} \]

   12. Applied egg-rr 98.5%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{1}{2 \cdot n}}}} \]
   13. Step-by-step derivation

       1. associate-/r* 99.3%

          \[\leadsto \frac{1}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{n}}}} \]

       2. metadata-eval 99.3%

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

   14. Simplified 99.3%

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

   if 7.1e-51 < k

   1. Initial program 99.6%

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

      1. unpow-prod-down 60.5%

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

      2. unpow-prod-down 99.6%

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

      3. div-sub 99.6%

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

      4. metadata-eval 99.6%

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

      5. pow-sub 99.9%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

      6. pow1/2 99.9%

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

      7. frac-times 99.9%

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

      8. *-un-lft-identity 99.9%

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

      9. associate-*l* 99.9%

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

      10. associate-*l* 99.9%

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

      11. div-inv 99.9%

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

      12. metadata-eval 99.9%

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

   3. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-*r* 99.9%

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

      3. *-commutative 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. *-commutative 99.9%

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

   5. Simplified 99.9%

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

      1. div-inv 99.8%

         \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot \frac{1}{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}}} \]

      2. add-exp-log 99.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot \frac{1}{\color{blue}{e^{\log \left({\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}\right)}}} \]

      3. rec-exp 99.2%

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

      4. log-prod 99.2%

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

      5. pow1/2 99.2%

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

      6. metadata-eval 99.2%

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

      7. pow-flip 99.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot e^{-\left(\log \left({\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}\right) + \log \color{blue}{\left(\frac{1}{{k}^{-0.5}}\right)}\right)} \]

      8. log-prod 99.2%

         \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot e^{-\color{blue}{\log \left({\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)} \cdot \frac{1}{{k}^{-0.5}}\right)}} \]

      9. div-inv 99.2%

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

      10. rec-exp 99.2%

          \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot \color{blue}{\frac{1}{e^{\log \left(\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}}{{k}^{-0.5}}\right)}}} \]

   7. Applied egg-rr 99.7%

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

      1. metadata-eval 99.7%

         \[\leadsto {\left(\sqrt{\pi \cdot \left(2 \cdot n\right)}\right)}^{\left(1 - k\right)} \cdot {k}^{\color{blue}{\left(-0.25 \cdot 2\right)}} \]

      2. pow-pow 99.6%

         \[\leadsto {\left(\sqrt{\pi \cdot \left(2 \cdot n\right)}\right)}^{\left(1 - k\right)} \cdot \color{blue}{{\left({k}^{-0.25}\right)}^{2}} \]

      3. sqrt-pow2 99.6%

         \[\leadsto \color{blue}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot {\left({k}^{-0.25}\right)}^{2} \]

      4. associate-*r* 99.6%

         \[\leadsto {\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{-0.25}\right)}^{2} \]

      5. *-commutative 99.6%

         \[\leadsto {\left(\color{blue}{\left(2 \cdot \pi\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{-0.25}\right)}^{2} \]

      6. *-commutative 99.6%

         \[\leadsto \color{blue}{{\left({k}^{-0.25}\right)}^{2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]

      7. add-sqr-sqrt 99.5%

         \[\leadsto \color{blue}{\sqrt{{\left({k}^{-0.25}\right)}^{2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \cdot \sqrt{{\left({k}^{-0.25}\right)}^{2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}} \]

      8. sqrt-unprod 99.6%

         \[\leadsto \color{blue}{\sqrt{\left({\left({k}^{-0.25}\right)}^{2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right) \cdot \left({\left({k}^{-0.25}\right)}^{2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)}} \]

   9. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.1 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{k} \cdot {\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot n\right) \cdot \pi\\ \frac{\sqrt{t_0}}{{t_0}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 n) PI)))
   (/ (sqrt t_0) (* (pow t_0 (* 0.5 k)) (sqrt k)))))

C

double code(double k, double n) {
	double t_0 = (2.0 * n) * ((double) M_PI);
	return sqrt(t_0) / (pow(t_0, (0.5 * k)) * sqrt(k));
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * n) * Math.PI;
	return Math.sqrt(t_0) / (Math.pow(t_0, (0.5 * k)) * Math.sqrt(k));
}

Python

def code(k, n):
	t_0 = (2.0 * n) * math.pi
	return math.sqrt(t_0) / (math.pow(t_0, (0.5 * k)) * math.sqrt(k))

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * n) * pi)
	return Float64(sqrt(t_0) / Float64((t_0 ^ Float64(0.5 * k)) * sqrt(k)))
end

MATLAB

function tmp = code(k, n)
	t_0 = (2.0 * n) * pi;
	tmp = sqrt(t_0) / ((t_0 ^ (0.5 * k)) * sqrt(k));
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]}, N[(N[Sqrt[t$95$0], $MachinePrecision] / N[(N[Power[t$95$0, N[(0.5 * k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Final simplification 99.3%

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

Alternative 3: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ {\left(\sqrt{\left(2 \cdot n\right) \cdot \pi}\right)}^{\left(1 - k\right)} \cdot {k}^{-0.5} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow (sqrt (* (* 2.0 n) PI)) (- 1.0 k)) (pow k -0.5)))

C

double code(double k, double n) {
	return pow(sqrt(((2.0 * n) * ((double) M_PI))), (1.0 - k)) * pow(k, -0.5);
}

Java

public static double code(double k, double n) {
	return Math.pow(Math.sqrt(((2.0 * n) * Math.PI)), (1.0 - k)) * Math.pow(k, -0.5);
}

Python

def code(k, n):
	return math.pow(math.sqrt(((2.0 * n) * math.pi)), (1.0 - k)) * math.pow(k, -0.5)

Julia

function code(k, n)
	return Float64((sqrt(Float64(Float64(2.0 * n) * pi)) ^ Float64(1.0 - k)) * (k ^ -0.5))
end

MATLAB

function tmp = code(k, n)
	tmp = (sqrt(((2.0 * n) * pi)) ^ (1.0 - k)) * (k ^ -0.5);
end

Wolfram

code[k_, n_] := N[(N[Power[N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

   1. div-inv 99.3%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot \frac{1}{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}}} \]

   2. add-exp-log 95.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot \frac{1}{\color{blue}{e^{\log \left({\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}\right)}}} \]

   3. rec-exp 95.7%

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

   4. log-prod 95.7%

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

   5. pow1/2 95.7%

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

   6. metadata-eval 95.7%

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

   7. pow-flip 95.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot e^{-\left(\log \left({\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}\right) + \log \color{blue}{\left(\frac{1}{{k}^{-0.5}}\right)}\right)} \]

   8. log-prod 95.7%

      \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot e^{-\color{blue}{\log \left({\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)} \cdot \frac{1}{{k}^{-0.5}}\right)}} \]

   9. div-inv 95.7%

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

   10. rec-exp 95.7%

       \[\leadsto \sqrt{\left(2 \cdot n\right) \cdot \pi} \cdot \color{blue}{\frac{1}{e^{\log \left(\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(0.5 \cdot k\right)}}{{k}^{-0.5}}\right)}}} \]

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(\sqrt{\pi \cdot \left(2 \cdot n\right)}\right)}^{\left(1 - k\right)} \cdot {k}^{-0.5}} \]

8. Final simplification 99.2%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (* (pow k -0.5) (pow (* n (* 2.0 PI)) (/ (- 1.0 k) 2.0))))

C

double code(double k, double n) {
	return pow(k, -0.5) * pow((n * (2.0 * ((double) M_PI))), ((1.0 - k) / 2.0));
}

Java

public static double code(double k, double n) {
	return Math.pow(k, -0.5) * Math.pow((n * (2.0 * Math.PI)), ((1.0 - k) / 2.0));
}

Python

def code(k, n):
	return math.pow(k, -0.5) * math.pow((n * (2.0 * math.pi)), ((1.0 - k) / 2.0))

Julia

function code(k, n)
	return Float64((k ^ -0.5) * (Float64(n * Float64(2.0 * pi)) ^ Float64(Float64(1.0 - k) / 2.0)))
end

MATLAB

function tmp = code(k, n)
	tmp = (k ^ -0.5) * ((n * (2.0 * pi)) ^ ((1.0 - k) / 2.0));
end

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

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

   1. expm1-log1p-u 95.7%

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

   2. expm1-udef 76.9%

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

   3. pow1/2 76.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\frac{1}{\color{blue}{{k}^{0.5}}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. pow-flip 76.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left(\color{blue}{{k}^{\left(-0.5\right)}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   5. metadata-eval 76.9%

      \[\leadsto \left(e^{\mathsf{log1p}\left({k}^{\color{blue}{-0.5}}\right)} - 1\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

3. Applied egg-rr 76.9%

   \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left({k}^{-0.5}\right)} - 1\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
4. Step-by-step derivation

   1. expm1-def 95.7%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({k}^{-0.5}\right)\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   2. expm1-log1p 99.2%

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

5. Simplified 99.2%

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

6. Final simplification 99.2%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6e-51)
   (/ 1.0 (* (sqrt (/ k PI)) (sqrt (/ 0.5 n))))
   (sqrt (/ (pow (* (* 2.0 n) PI) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6e-51) {
		tmp = 1.0 / (sqrt((k / ((double) M_PI))) * sqrt((0.5 / n)));
	} else {
		tmp = sqrt((pow(((2.0 * n) * ((double) M_PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6e-51) {
		tmp = 1.0 / (Math.sqrt((k / Math.PI)) * Math.sqrt((0.5 / n)));
	} else {
		tmp = Math.sqrt((Math.pow(((2.0 * n) * Math.PI), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6e-51:
		tmp = 1.0 / (math.sqrt((k / math.pi)) * math.sqrt((0.5 / n)))
	else:
		tmp = math.sqrt((math.pow(((2.0 * n) * math.pi), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6e-51)
		tmp = Float64(1.0 / Float64(sqrt(Float64(k / pi)) * sqrt(Float64(0.5 / n))));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * n) * pi) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6e-51)
		tmp = 1.0 / (sqrt((k / pi)) * sqrt((0.5 / n)));
	else
		tmp = sqrt(((((2.0 * n) * pi) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6e-51], N[(1.0 / N[(N[Sqrt[N[(k / Pi), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(0.5 / n), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.00000000000000005e-51

   1. Initial program 98.6%

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

      1. unpow-prod-down 99.0%

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

      2. unpow-prod-down 98.6%

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

      3. div-sub 98.6%

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

      4. metadata-eval 98.6%

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

      5. pow-sub 98.5%

         \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

      6. pow1/2 98.5%

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

      7. frac-times 98.5%

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

      8. *-un-lft-identity 98.5%

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

      9. associate-*l* 98.5%

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

      10. associate-*l* 98.5%

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

      11. div-inv 98.5%

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

      12. metadata-eval 98.5%

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

   3. Applied egg-rr 98.5%

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

      1. *-commutative 98.5%

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

      2. associate-*r* 98.5%

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

      3. *-commutative 98.5%

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

      4. *-commutative 98.5%

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

      5. associate-*r* 98.5%

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

      6. *-commutative 98.5%

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

   5. Simplified 98.5%

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

   6. Taylor expanded in k around 0 98.6%

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

      1. *-un-lft-identity 98.6%

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

      2. clear-num 98.5%

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

      3. inv-pow 98.5%

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

      4. *-commutative 98.5%

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

      5. sqrt-undiv 75.3%

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

   8. Applied egg-rr 75.3%

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

      1. unpow-1 75.3%

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

      2. associate-/r* 75.3%

         \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]

   10. Simplified 75.3%

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]
   11. Step-by-step derivation

       1. div-inv 75.3%

          \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{k}{\pi} \cdot \frac{1}{2 \cdot n}}}} \]

       2. sqrt-prod 98.5%

          \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{1}{2 \cdot n}}}} \]

   12. Applied egg-rr 98.5%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{1}{2 \cdot n}}}} \]
   13. Step-by-step derivation

       1. associate-/r* 99.3%

          \[\leadsto \frac{1}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{n}}}} \]

       2. metadata-eval 99.3%

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

   14. Simplified 99.3%

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

   if 6.00000000000000005e-51 < k

   1. Initial program 99.6%

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

      1. *-commutative 99.6%

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

      2. div-sub 99.6%

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

      3. metadata-eval 99.6%

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

      4. div-inv 99.6%

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

      5. expm1-log1p-u 99.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\right)\right)} \]

      6. expm1-udef 92.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}\right)} - 1} \]

   3. Applied egg-rr 92.3%

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

      1. expm1-def 99.0%

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

      2. expm1-log1p 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

   5. Simplified 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-51}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{0.5}{n}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 6: 99.4% 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 99.2%

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

   1. associate-*l/ 99.2%

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

   2. *-lft-identity 99.2%

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

   3. sqr-pow 99.0%

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

   4. pow-sqr 99.2%

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

   5. *-commutative 99.2%

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

   6. associate-*l/ 99.2%

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

   7. associate-/l* 99.2%

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

   8. metadata-eval 99.2%

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

   9. /-rgt-identity 99.2%

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

   10. div-sub 99.2%

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

   11. metadata-eval 99.2%

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

3. Simplified 99.2%

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

4. Final simplification 99.2%

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

Alternative 7: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in k around 0 53.2%

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

   1. *-un-lft-identity 53.2%

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

   2. clear-num 53.2%

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

   3. inv-pow 53.2%

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

   4. *-commutative 53.2%

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

   5. sqrt-undiv 43.2%

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

8. Applied egg-rr 43.2%

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

   1. unpow-1 43.2%

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

   2. associate-/r* 43.1%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]

10. Simplified 43.1%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]
11. Step-by-step derivation

    1. div-inv 43.1%

       \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{k}{\pi} \cdot \frac{1}{2 \cdot n}}}} \]

    2. sqrt-prod 53.1%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{1}{2 \cdot n}}}} \]

12. Applied egg-rr 53.1%

    \[\leadsto \frac{1}{\color{blue}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\frac{1}{2 \cdot n}}}} \]
13. Step-by-step derivation

    1. associate-/r* 53.5%

       \[\leadsto \frac{1}{\sqrt{\frac{k}{\pi}} \cdot \sqrt{\color{blue}{\frac{\frac{1}{2}}{n}}}} \]

    2. metadata-eval 53.5%

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

14. Simplified 53.5%

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

15. Final simplification 53.5%

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

Alternative 8: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in k around 0 53.2%

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

   1. *-commutative 53.2%

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

   2. *-un-lft-identity 53.2%

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

   3. *-commutative 53.2%

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

   4. times-frac 53.2%

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

   5. /-rgt-identity 53.2%

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

   6. pow1/2 53.2%

      \[\leadsto \frac{1}{\color{blue}{{k}^{0.5}}} \cdot \sqrt{\pi \cdot \left(2 \cdot n\right)} \]

   7. pow-flip 53.3%

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

   8. metadata-eval 53.3%

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

8. Applied egg-rr 53.3%

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

9. Final simplification 53.3%

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

Alternative 9: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in k around 0 53.2%

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

   1. *-un-lft-identity 53.2%

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

   2. div-inv 53.2%

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

   3. associate-*l* 53.2%

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

   4. sqrt-prod 53.1%

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

   5. *-commutative 53.1%

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

   6. *-commutative 53.1%

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

   7. expm1-log1p-u 50.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}} \cdot \left(\sqrt{n \cdot \pi} \cdot \sqrt{2}\right)\right)\right)} \]

   8. expm1-udef 49.9%

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

8. Applied egg-rr 39.6%

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

   1. expm1-def 39.8%

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

   2. expm1-log1p 42.0%

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

   3. *-commutative 42.0%

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

   4. associate-*r* 42.0%

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

   5. *-commutative 42.0%

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

   6. associate-/l* 41.9%

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

   7. *-commutative 41.9%

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

10. Simplified 41.9%

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

    1. associate-/r/ 41.9%

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

    2. sqrt-prod 53.2%

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

    3. *-commutative 53.2%

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

12. Applied egg-rr 53.2%

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

13. Final simplification 53.2%

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

Alternative 10: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

2. Taylor expanded in k around 0 53.1%

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

   1. *-commutative 53.1%

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

   2. *-commutative 53.1%

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

   3. sqrt-prod 53.2%

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

   4. associate-*l* 53.2%

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

   5. *-commutative 53.2%

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

   6. div-inv 53.2%

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

4. Applied egg-rr 53.2%

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

5. Final simplification 53.2%

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

Alternative 11: 38.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in k around 0 53.2%

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

   1. *-un-lft-identity 53.2%

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

   2. clear-num 53.2%

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

   3. inv-pow 53.2%

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

   4. *-commutative 53.2%

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

   5. sqrt-undiv 43.2%

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

8. Applied egg-rr 43.2%

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

   1. unpow-1 43.2%

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

   2. associate-/r* 43.1%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]

10. Simplified 43.1%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]
11. Step-by-step derivation

    1. pow1/2 43.1%

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

    2. pow-flip 43.2%

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

    3. associate-/l/ 43.2%

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

    4. *-commutative 43.2%

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

    5. metadata-eval 43.2%

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

12. Applied egg-rr 43.2%

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

    1. associate-/r* 43.2%

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

    2. *-lft-identity 43.2%

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

    3. associate-*l/ 43.2%

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

    4. *-commutative 43.2%

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

    5. associate-/r* 43.2%

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

    6. metadata-eval 43.2%

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

14. Simplified 43.2%

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

15. Final simplification 43.2%

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

Alternative 12: 38.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in k around 0 53.2%

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

   1. *-un-lft-identity 53.2%

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

   2. clear-num 53.2%

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

   3. inv-pow 53.2%

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

   4. *-commutative 53.2%

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

   5. sqrt-undiv 43.2%

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

8. Applied egg-rr 43.2%

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

   1. unpow-1 43.2%

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

   2. associate-/r* 43.1%

      \[\leadsto \frac{1}{\sqrt{\color{blue}{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]

10. Simplified 43.1%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{k}{\pi}}{2 \cdot n}}}} \]
11. Step-by-step derivation

    1. pow1/2 43.1%

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

    2. pow-flip 43.2%

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

    3. associate-/l/ 43.2%

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

    4. *-commutative 43.2%

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

    5. metadata-eval 43.2%

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

12. Applied egg-rr 43.2%

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

13. Final simplification 43.2%

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

Alternative 13: 38.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in k around 0 53.2%

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

   1. *-un-lft-identity 53.2%

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

   2. div-inv 53.2%

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

   3. associate-*l* 53.2%

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

   4. sqrt-prod 53.1%

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

   5. *-commutative 53.1%

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

   6. *-commutative 53.1%

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

   7. expm1-log1p-u 50.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}} \cdot \left(\sqrt{n \cdot \pi} \cdot \sqrt{2}\right)\right)\right)} \]

   8. expm1-udef 49.9%

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

8. Applied egg-rr 39.6%

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

   1. expm1-def 39.8%

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

   2. expm1-log1p 42.0%

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

   3. *-commutative 42.0%

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

   4. associate-*r* 42.0%

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

   5. *-commutative 42.0%

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

   6. associate-/l* 41.9%

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

   7. *-commutative 41.9%

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

10. Simplified 41.9%

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

11. Taylor expanded in k around 0 42.0%

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

    1. associate-/l* 41.9%

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

    2. associate-/r/ 42.0%

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

13. Simplified 42.0%

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

14. Final simplification 42.0%

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

Alternative 14: 38.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. unpow-prod-down 77.0%

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

   2. unpow-prod-down 99.2%

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

   3. div-sub 99.2%

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

   4. metadata-eval 99.2%

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

   5. pow-sub 99.3%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{0.5}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}} \]

   6. pow1/2 99.3%

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

   7. frac-times 99.3%

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

   8. *-un-lft-identity 99.3%

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

   9. associate-*l* 99.3%

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

   10. associate-*l* 99.3%

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

   11. div-inv 99.3%

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

   12. metadata-eval 99.3%

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

3. Applied egg-rr 99.3%

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

   1. *-commutative 99.3%

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

   2. associate-*r* 99.3%

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

   3. *-commutative 99.3%

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

   4. *-commutative 99.3%

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

   5. associate-*r* 99.3%

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

   6. *-commutative 99.3%

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

5. Simplified 99.3%

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

6. Taylor expanded in k around 0 53.2%

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

   1. *-un-lft-identity 53.2%

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

   2. div-inv 53.2%

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

   3. associate-*l* 53.2%

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

   4. sqrt-prod 53.1%

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

   5. *-commutative 53.1%

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

   6. *-commutative 53.1%

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

   7. expm1-log1p-u 50.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\sqrt{k}} \cdot \left(\sqrt{n \cdot \pi} \cdot \sqrt{2}\right)\right)\right)} \]

   8. expm1-udef 49.9%

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

8. Applied egg-rr 39.6%

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

   1. expm1-def 39.8%

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

   2. expm1-log1p 42.0%

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

   3. associate-/l* 42.0%

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

10. Simplified 42.0%

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

11. Final simplification 42.0%

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

Reproduce

herbie shell --seed 2023293 
(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))))