Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 14.3s

Alternatives: 10

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: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \frac{\sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing
3. Step-by-step derivation

   1. associate-*l/ 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. associate-*r* 99.7%

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

6. Simplified 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\pi \cdot n} \cdot \sqrt{\frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.4e-22)
   (* (sqrt (* PI n)) (sqrt (/ 2.0 k)))
   (sqrt (/ (pow (* (* 2.0 PI) n) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.4e-22) {
		tmp = sqrt((((double) M_PI) * n)) * sqrt((2.0 / k));
	} else {
		tmp = sqrt((pow(((2.0 * ((double) M_PI)) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 3.4e-22) {
		tmp = Math.sqrt((Math.PI * n)) * Math.sqrt((2.0 / k));
	} else {
		tmp = Math.sqrt((Math.pow(((2.0 * Math.PI) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 3.4e-22:
		tmp = math.sqrt((math.pi * n)) * math.sqrt((2.0 / k))
	else:
		tmp = math.sqrt((math.pow(((2.0 * math.pi) * n), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 3.4e-22)
		tmp = Float64(sqrt(Float64(pi * n)) * sqrt(Float64(2.0 / k)));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * pi) * n) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 3.4e-22)
		tmp = sqrt((pi * n)) * sqrt((2.0 / k));
	else
		tmp = sqrt(((((2.0 * pi) * n) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.4e-22], N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 3.3999999999999998e-22

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0 75.5%

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

      1. associate-/l* 75.4%

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

   5. Simplified 75.4%

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

      1. sqrt-unprod 75.6%

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

   7. Applied egg-rr 75.6%

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

      1. sqrt-prod 75.4%

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

      2. associate-*r/ 75.5%

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

      3. *-commutative 75.5%

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

      4. *-commutative 75.5%

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

      5. sqrt-div 99.3%

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

      6. div-inv 99.2%

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

      7. associate-*l* 99.0%

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

      8. sqrt-prod 99.3%

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

      9. *-commutative 99.3%

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

      10. *-un-lft-identity 99.3%

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

      11. *-commutative 99.3%

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

      12. sqrt-prod 99.0%

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

      13. associate-*l* 99.2%

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

      14. div-inv 99.3%

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

      15. sqrt-div 75.5%

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

      16. sqrt-prod 75.7%

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

      17. associate-/l* 75.6%

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

   9. Applied egg-rr 75.6%

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

       1. *-lft-identity 75.6%

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

       2. *-commutative 75.6%

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

       3. associate-*r/ 75.7%

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

       4. *-commutative 75.7%

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

       5. associate-/l* 75.6%

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

       6. associate-*l* 75.6%

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

       7. associate-*l/ 75.6%

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

       8. associate-/l* 75.5%

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

   11. Simplified 75.5%

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

       1. associate-*r* 75.6%

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

       2. sqrt-prod 99.4%

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

       3. *-commutative 99.4%

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

   13. Applied egg-rr 99.4%

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

   if 3.3999999999999998e-22 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. associate-*r* 99.8%

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

      5. div-sub 99.8%

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

      6. metadata-eval 99.8%

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

      7. div-inv 99.8%

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

      8. *-commutative 99.8%

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

   4. Applied egg-rr 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{-22}:\\ \;\;\;\;\sqrt{\pi \cdot n} \cdot \sqrt{\frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(2, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8.8e+17)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (sqrt (* n (+ -1.0 (fma 2.0 (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8.8e+17) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = sqrt((n * (-1.0 + fma(2.0, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8.8e+17)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = sqrt(Float64(n * Float64(-1.0 + fma(2.0, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8.8e+17], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(n * N[(-1.0 + N[(2.0 * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.8e17

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 90.9%

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

      1. associate-*l/ 91.1%

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

      2. *-un-lft-identity 91.1%

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

      3. *-commutative 91.1%

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

      4. *-commutative 91.1%

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

      5. sqrt-prod 91.3%

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

   5. Applied egg-rr 91.3%

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

      1. *-commutative 91.3%

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

      2. associate-*l* 91.3%

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

   7. Simplified 91.3%

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

   if 8.8e17 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.5%

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

      1. associate-/l* 2.5%

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

   5. Simplified 2.5%

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

      1. sqrt-unprod 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. sqrt-prod 2.5%

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

      2. associate-*r/ 2.5%

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

      3. *-commutative 2.5%

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

      4. *-commutative 2.5%

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

      5. sqrt-div 2.6%

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

      6. div-inv 2.6%

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

      7. associate-*l* 2.6%

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

      8. sqrt-prod 2.6%

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

      9. *-commutative 2.6%

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

      10. *-un-lft-identity 2.6%

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

      11. *-commutative 2.6%

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

      12. sqrt-prod 2.6%

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

      13. associate-*l* 2.6%

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

      14. div-inv 2.6%

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

      15. sqrt-div 2.5%

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

      16. sqrt-prod 2.5%

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

      17. associate-/l* 2.5%

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

   9. Applied egg-rr 2.5%

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

       1. *-lft-identity 2.5%

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

       2. *-commutative 2.5%

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

       3. associate-*r/ 2.5%

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

       4. *-commutative 2.5%

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

       5. associate-/l* 2.5%

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

       6. associate-*l* 2.5%

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

       7. associate-*l/ 2.5%

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

       8. associate-/l* 2.5%

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

   11. Simplified 2.5%

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

       1. expm1-log1p-u 2.5%

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

       2. expm1-undefine 50.4%

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

       3. associate-*r/ 50.4%

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

   13. Applied egg-rr 50.4%

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

       1. sub-neg 50.4%

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

       2. metadata-eval 50.4%

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

       3. +-commutative 50.4%

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

       4. log1p-undefine 50.4%

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

       5. rem-exp-log 50.4%

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

       6. +-commutative 50.4%

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

       7. *-commutative 50.4%

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

       8. associate-*r/ 50.4%

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

       9. fma-define 50.4%

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

   15. Simplified 50.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{n \cdot \left(-1 + \mathsf{fma}\left(2, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

Alternative 5: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 47.1%

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

   1. associate-*l/ 47.2%

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

   2. *-un-lft-identity 47.2%

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

   3. *-commutative 47.2%

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

   4. *-commutative 47.2%

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

   5. sqrt-prod 47.3%

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

5. Applied egg-rr 47.3%

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

   1. *-commutative 47.3%

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

   2. associate-*l* 47.3%

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

7. Simplified 47.3%

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

8. Final simplification 47.3%

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

Alternative 6: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 36.7%

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

   1. associate-/l* 36.6%

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

5. Simplified 36.6%

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

   1. sqrt-unprod 36.8%

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

7. Applied egg-rr 36.8%

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

   1. sqrt-prod 36.6%

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

   2. associate-*r/ 36.7%

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

   3. *-commutative 36.7%

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

   4. *-commutative 36.7%

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

   5. sqrt-div 47.2%

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

   6. div-inv 47.2%

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

   7. associate-*l* 47.1%

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

   8. sqrt-prod 47.2%

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

   9. *-commutative 47.2%

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

   10. *-un-lft-identity 47.2%

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

   11. *-commutative 47.2%

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

   12. sqrt-prod 47.1%

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

   13. associate-*l* 47.2%

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

   14. div-inv 47.2%

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

   15. sqrt-div 36.7%

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

   16. sqrt-prod 36.8%

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

   17. associate-/l* 36.8%

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

9. Applied egg-rr 36.8%

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

    1. *-lft-identity 36.8%

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

    2. *-commutative 36.8%

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

    3. associate-*r/ 36.8%

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

    4. *-commutative 36.8%

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

    5. associate-/l* 36.8%

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

    6. associate-*l* 36.8%

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

    7. associate-*l/ 36.8%

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

    8. associate-/l* 36.7%

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

11. Simplified 36.7%

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

    1. associate-*r* 36.7%

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

    2. sqrt-prod 47.3%

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

    3. *-commutative 47.3%

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

13. Applied egg-rr 47.3%

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

Alternative 7: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 36.7%

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

   1. associate-/l* 36.6%

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

5. Simplified 36.6%

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

   1. sqrt-unprod 36.8%

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

7. Applied egg-rr 36.8%

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

   1. pow1/2 36.8%

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

   2. associate-*l* 36.8%

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

   3. unpow-prod-down 47.2%

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

   4. pow1/2 47.2%

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

   5. *-commutative 47.2%

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

9. Applied egg-rr 47.2%

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

    1. unpow1/2 47.2%

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

11. Simplified 47.2%

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

Alternative 8: 38.2% accurate, 2.0× speedup

Math

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

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(Float64(2.0 * Float64(pi * n)) / k))
end

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 47.1%

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

   1. associate-*l/ 47.2%

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

   2. *-un-lft-identity 47.2%

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

   3. *-commutative 47.2%

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

   4. *-commutative 47.2%

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

   5. sqrt-prod 47.3%

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

   6. associate-*l* 47.3%

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

   7. sqrt-undiv 36.8%

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

   8. associate-*l* 36.8%

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

5. Applied egg-rr 36.8%

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

Alternative 9: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 36.7%

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

   1. associate-/l* 36.6%

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

5. Simplified 36.6%

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

   1. sqrt-unprod 36.8%

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

7. Applied egg-rr 36.8%

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

   1. sqrt-prod 36.6%

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

   2. associate-*r/ 36.7%

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

   3. *-commutative 36.7%

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

   4. *-commutative 36.7%

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

   5. sqrt-div 47.2%

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

   6. div-inv 47.2%

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

   7. associate-*l* 47.1%

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

   8. sqrt-prod 47.2%

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

   9. *-commutative 47.2%

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

   10. *-un-lft-identity 47.2%

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

   11. *-commutative 47.2%

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

   12. sqrt-prod 47.1%

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

   13. associate-*l* 47.2%

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

   14. div-inv 47.2%

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

   15. sqrt-div 36.7%

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

   16. sqrt-prod 36.8%

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

   17. associate-/l* 36.8%

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

9. Applied egg-rr 36.8%

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

    1. *-lft-identity 36.8%

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

    2. associate-*r* 36.8%

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

    3. *-commutative 36.8%

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

    4. associate-*l* 36.8%

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

11. Simplified 36.8%

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

Alternative 10: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in k around 0 36.7%

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

   1. associate-/l* 36.6%

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

5. Simplified 36.6%

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

   1. sqrt-unprod 36.8%

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

7. Applied egg-rr 36.8%

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

   1. sqrt-prod 36.6%

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

   2. associate-*r/ 36.7%

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

   3. *-commutative 36.7%

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

   4. *-commutative 36.7%

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

   5. sqrt-div 47.2%

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

   6. div-inv 47.2%

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

   7. associate-*l* 47.1%

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

   8. sqrt-prod 47.2%

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

   9. *-commutative 47.2%

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

   10. *-un-lft-identity 47.2%

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

   11. *-commutative 47.2%

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

   12. sqrt-prod 47.1%

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

   13. associate-*l* 47.2%

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

   14. div-inv 47.2%

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

   15. sqrt-div 36.7%

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

   16. sqrt-prod 36.8%

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

   17. associate-/l* 36.8%

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

9. Applied egg-rr 36.8%

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

    1. *-lft-identity 36.8%

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

    2. *-commutative 36.8%

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

    3. associate-*r/ 36.8%

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

    4. *-commutative 36.8%

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

    5. associate-/l* 36.8%

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

    6. associate-*l* 36.8%

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

    7. associate-*l/ 36.8%

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

    8. associate-/l* 36.7%

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

11. Simplified 36.7%

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

Reproduce

herbie shell --seed 2024106 
(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))))