Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 23.5s

Alternatives: 12

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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. associate-*r* 99.4%

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

   3. associate-/r/ 99.5%

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

   4. add-sqr-sqrt 99.3%

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

   5. sqrt-unprod 99.5%

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

   6. associate-*r* 99.5%

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

   7. *-commutative 99.5%

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

   8. associate-*r* 99.5%

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

   9. *-commutative 99.5%

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

   10. pow-prod-up 99.5%

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

3. Applied egg-rr 99.5%

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

   1. inv-pow 99.5%

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

   2. div-inv 99.5%

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

   3. unpow-prod-down 99.4%

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

   4. inv-pow 99.4%

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

   5. pow1/2 99.4%

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

   6. pow-flip 99.4%

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

   7. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. unpow-1 99.4%

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

   2. associate-*r/ 99.4%

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

   3. *-commutative 99.4%

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

   4. *-lft-identity 99.4%

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

   5. *-commutative 99.4%

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

   6. associate-*l* 99.4%

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

   7. *-commutative 99.4%

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

7. Simplified 99.4%

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

   1. pow-sub 99.6%

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

   2. pow1 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

9. Applied egg-rr 99.6%

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

10. Final simplification 99.6%

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

Alternative 2: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-60}:\\ \;\;\;\;{k}^{-0.5} \cdot {\left(\frac{\frac{0.5}{n}}{\pi}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.15e-60)
   (* (pow k -0.5) (pow (/ (/ 0.5 n) PI) -0.5))
   (sqrt (/ (pow (* 2.0 (* n PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.15e-60) {
		tmp = pow(k, -0.5) * pow(((0.5 / n) / ((double) M_PI)), -0.5);
	} 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 <= 2.15e-60) {
		tmp = Math.pow(k, -0.5) * Math.pow(((0.5 / n) / Math.PI), -0.5);
	} 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 <= 2.15e-60:
		tmp = math.pow(k, -0.5) * math.pow(((0.5 / n) / math.pi), -0.5)
	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 <= 2.15e-60)
		tmp = Float64((k ^ -0.5) * (Float64(Float64(0.5 / n) / pi) ^ -0.5));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(n * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.15e-60)
		tmp = (k ^ -0.5) * (((0.5 / n) / pi) ^ -0.5);
	else
		tmp = sqrt((((2.0 * (n * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.15e-60], N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(N[(0.5 / n), $MachinePrecision] / Pi), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.15e-60

   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. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. div-inv 99.4%

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

      5. add-sqr-sqrt 99.0%

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

      6. sqrt-unprod 70.2%

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

      7. frac-times 70.2%

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

   3. Applied egg-rr 70.5%

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

   4. Taylor expanded in k around 0 70.5%

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

      1. *-commutative 70.5%

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

      2. associate-*l* 70.5%

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

   6. Simplified 70.5%

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

   7. Taylor expanded in n around 0 70.5%

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

      1. associate-*r/ 70.5%

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

      2. associate-/l* 70.4%

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

      3. associate-/r/ 70.4%

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

   9. Simplified 70.4%

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

       1. associate-*l/ 70.5%

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

       2. clear-num 70.4%

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

       3. unpow-1 70.4%

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

       4. sqrt-pow1 71.3%

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

       5. div-inv 71.2%

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

       6. metadata-eval 71.2%

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

       7. unpow-prod-down 99.4%

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

       8. associate-*r* 99.4%

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

   11. Applied egg-rr 99.4%

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

       1. associate-/r* 99.4%

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

       2. associate-/r* 99.4%

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

       3. metadata-eval 99.4%

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

   13. Simplified 99.4%

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

   if 2.15e-60 < 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. *-commutative 99.6%

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

      3. associate-*r* 99.6%

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

      4. div-inv 99.6%

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

      5. add-sqr-sqrt 99.5%

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

      6. sqrt-unprod 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.7%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.15 \cdot 10^{-60}:\\ \;\;\;\;{k}^{-0.5} \cdot {\left(\frac{\frac{0.5}{n}}{\pi}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. *-commutative 99.5%

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

   4. associate-*l* 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 4: 54.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.15e+122)
   (/ (sqrt (* (* n 2.0) PI)) (sqrt k))
   (pow (pow (* 2.0 (/ n (/ k PI))) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.15e+122) {
		tmp = sqrt(((n * 2.0) * ((double) M_PI))) / sqrt(k);
	} else {
		tmp = pow(pow((2.0 * (n / (k / ((double) M_PI)))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.15e+122) {
		tmp = Math.sqrt(((n * 2.0) * Math.PI)) / Math.sqrt(k);
	} else {
		tmp = Math.pow(Math.pow((2.0 * (n / (k / Math.PI))), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.15e+122:
		tmp = math.sqrt(((n * 2.0) * math.pi)) / math.sqrt(k)
	else:
		tmp = math.pow(math.pow((2.0 * (n / (k / math.pi))), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.15e+122)
		tmp = Float64(sqrt(Float64(Float64(n * 2.0) * pi)) / sqrt(k));
	else
		tmp = (Float64(2.0 * Float64(n / Float64(k / pi))) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.15e+122)
		tmp = sqrt(((n * 2.0) * pi)) / sqrt(k);
	else
		tmp = ((2.0 * (n / (k / pi))) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.15e+122], N[(N[Sqrt[N[(N[(n * 2.0), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(n / N[(k / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.15e122

   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. *-commutative 99.2%

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

      2. *-commutative 99.2%

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

      3. associate-*r* 99.2%

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

      4. div-inv 99.4%

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

      5. add-sqr-sqrt 99.1%

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

      6. sqrt-unprod 83.0%

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

      7. frac-times 83.0%

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

   3. Applied egg-rr 83.2%

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

   4. Taylor expanded in k around 0 54.2%

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

      1. *-commutative 54.2%

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

      2. associate-*l* 54.2%

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

   6. Simplified 54.2%

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

      1. sqrt-div 70.4%

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

      2. associate-*r* 70.4%

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

      3. *-commutative 70.4%

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

   8. Applied egg-rr 70.4%

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

      1. associate-*r* 70.4%

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

      2. *-commutative 70.4%

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

   10. Simplified 70.4%

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

   if 1.15e122 < 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. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-*r* 100.0%

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

      4. div-inv 100.0%

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

      5. add-sqr-sqrt 100.0%

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

      6. sqrt-unprod 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   4. Taylor expanded in k around 0 2.8%

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

      1. *-commutative 2.8%

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

      2. associate-*l* 2.8%

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

   6. Simplified 2.8%

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

   7. Taylor expanded in n around 0 2.8%

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

      1. associate-*r/ 2.8%

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

      2. associate-/l* 2.7%

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

      3. associate-/r/ 2.8%

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

   9. Simplified 2.8%

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

       1. pow1/2 2.8%

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

       2. associate-*l/ 2.8%

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

       3. metadata-eval 2.8%

          \[\leadsto {\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       4. pow-pow 8.4%

          \[\leadsto \color{blue}{{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       5. sqr-pow 8.4%

          \[\leadsto \color{blue}{{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \cdot {\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       6. pow-prod-down 25.6%

          \[\leadsto \color{blue}{{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1.5} \cdot {\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{1.5}\right)}^{\left(\frac{0.3333333333333333}{2}\right)}} \]

       7. pow-prod-up 25.6%

          \[\leadsto {\color{blue}{\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}^{\left(1.5 + 1.5\right)}\right)}}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       8. *-un-lft-identity 25.6%

          \[\leadsto {\left({\left(\frac{2 \cdot \left(n \cdot \pi\right)}{\color{blue}{1 \cdot k}}\right)}^{\left(1.5 + 1.5\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       9. times-frac 25.6%

          \[\leadsto {\left({\color{blue}{\left(\frac{2}{1} \cdot \frac{n \cdot \pi}{k}\right)}}^{\left(1.5 + 1.5\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       10. metadata-eval 25.6%

           \[\leadsto {\left({\left(\color{blue}{2} \cdot \frac{n \cdot \pi}{k}\right)}^{\left(1.5 + 1.5\right)}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       11. metadata-eval 25.6%

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

       12. metadata-eval 25.6%

           \[\leadsto {\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   11. Applied egg-rr 25.6%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{3}\right)}^{0.16666666666666666}} \]
   12. Step-by-step derivation

       1. associate-/l* 25.6%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\frac{n}{\frac{k}{\pi}}}\right)}^{3}\right)}^{0.16666666666666666} \]

   13. Simplified 25.6%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{3}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.15 \cdot 10^{+122}:\\ \;\;\;\;\frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(2 \cdot \frac{n}{\frac{k}{\pi}}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 5: 49.6% 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 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

7. Taylor expanded in n around 0 40.7%

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

   1. associate-*r/ 40.7%

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

   2. associate-/l* 40.7%

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

   3. associate-/r/ 40.7%

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

9. Simplified 40.7%

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

    1. sqrt-prod 52.7%

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

11. Applied egg-rr 52.7%

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

12. Final simplification 52.7%

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

Alternative 6: 49.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

   1. sqrt-div 52.7%

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

   2. associate-*r* 52.7%

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

   3. *-commutative 52.7%

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

8. Applied egg-rr 52.7%

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

   1. associate-*r* 52.7%

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

   2. *-commutative 52.7%

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

10. Simplified 52.7%

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

11. Final simplification 52.7%

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

Alternative 7: 38.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

7. Taylor expanded in n around 0 40.7%

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

   1. associate-*r/ 40.7%

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

   2. associate-/l* 40.7%

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

   3. associate-/r/ 40.7%

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

9. Simplified 40.7%

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

    1. associate-*l/ 40.7%

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

    2. clear-num 40.7%

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

    3. unpow-1 40.7%

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

    4. sqrt-pow1 41.1%

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

    5. associate-*r* 41.1%

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

    6. metadata-eval 41.1%

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

11. Applied egg-rr 41.1%

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

    1. *-rgt-identity 41.1%

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

    2. associate-*r/ 41.0%

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

    3. associate-/r* 41.0%

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

    4. associate-/r* 41.0%

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

    5. metadata-eval 41.0%

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

13. Simplified 41.0%

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

14. Final simplification 41.0%

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

Alternative 8: 38.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

7. Taylor expanded in n around 0 40.7%

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

   1. associate-*r/ 40.7%

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

   2. associate-/l* 40.7%

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

   3. associate-/r/ 40.7%

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

9. Simplified 40.7%

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

    1. associate-*l/ 40.7%

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

    2. clear-num 40.7%

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

    3. unpow-1 40.7%

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

    4. sqrt-pow1 41.1%

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

    5. associate-*r* 41.1%

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

    6. metadata-eval 41.1%

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

11. Applied egg-rr 41.1%

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

12. Final simplification 41.1%

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

Alternative 9: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

7. Taylor expanded in n around 0 40.7%

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

   1. associate-*r/ 40.7%

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

   2. associate-/l* 40.7%

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

   3. associate-/r/ 40.7%

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

9. Simplified 40.7%

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

10. Taylor expanded in k around 0 40.7%

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

    1. associate-/l* 40.7%

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

12. Simplified 40.7%

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

13. Final simplification 40.7%

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

Alternative 10: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

   1. clear-num 40.7%

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

   2. inv-pow 40.7%

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

   3. associate-*r* 40.7%

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

   4. *-commutative 40.7%

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

8. Applied egg-rr 40.7%

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

   1. unpow-1 40.7%

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

   2. clear-num 40.7%

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

   3. associate-*l/ 40.7%

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

   4. associate-*r* 40.7%

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

10. Applied egg-rr 40.7%

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

11. Final simplification 40.7%

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

Alternative 11: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

7. Taylor expanded in n around 0 40.7%

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

   1. associate-*r/ 40.7%

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

   2. associate-/l* 40.7%

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

   3. associate-/r/ 40.7%

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

9. Simplified 40.7%

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

    1. associate-*l/ 40.7%

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

    2. associate-/l* 40.7%

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

11. Applied egg-rr 40.7%

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

12. Final simplification 40.7%

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

Alternative 12: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. *-commutative 99.4%

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

   2. *-commutative 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-inv 99.5%

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

   5. add-sqr-sqrt 99.3%

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

   6. sqrt-unprod 87.4%

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

   7. frac-times 87.4%

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

3. Applied egg-rr 87.6%

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

4. Taylor expanded in k around 0 40.7%

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

   1. *-commutative 40.7%

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

   2. associate-*l* 40.7%

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

6. Simplified 40.7%

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

7. Final simplification 40.7%

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

Reproduce

herbie shell --seed 2023187 
(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))))