Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 16.4s

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: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-/r/ 99.6%

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

   2. associate-*r* 99.6%

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

   3. div-sub 99.6%

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

   4. metadata-eval 99.6%

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

   5. div-inv 99.6%

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

   6. metadata-eval 99.6%

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

3. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{\frac{k}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot {\left(\frac{0.5}{\pi \cdot n}\right)}^{\left(1 - k\right)}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8e-34)
   (/ (sqrt (* PI n)) (sqrt (/ k 2.0)))
   (/ 1.0 (sqrt (* k (pow (/ 0.5 (* PI n)) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8e-34) {
		tmp = sqrt((((double) M_PI) * n)) / sqrt((k / 2.0));
	} else {
		tmp = 1.0 / sqrt((k * pow((0.5 / (((double) M_PI) * n)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 8e-34) {
		tmp = Math.sqrt((Math.PI * n)) / Math.sqrt((k / 2.0));
	} else {
		tmp = 1.0 / Math.sqrt((k * Math.pow((0.5 / (Math.PI * n)), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 8e-34:
		tmp = math.sqrt((math.pi * n)) / math.sqrt((k / 2.0))
	else:
		tmp = 1.0 / math.sqrt((k * math.pow((0.5 / (math.pi * n)), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8e-34)
		tmp = Float64(sqrt(Float64(pi * n)) / sqrt(Float64(k / 2.0)));
	else
		tmp = Float64(1.0 / sqrt(Float64(k * (Float64(0.5 / Float64(pi * n)) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 8e-34)
		tmp = sqrt((pi * n)) / sqrt((k / 2.0));
	else
		tmp = 1.0 / sqrt((k * ((0.5 / (pi * n)) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8e-34], N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k * N[Power[N[(0.5 / N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.99999999999999942e-34

   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. add-sqr-sqrt 99.0%

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

      2. sqrt-unprod 69.9%

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

      3. *-commutative 69.9%

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

      4. div-inv 69.9%

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

      5. *-commutative 69.9%

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

      6. div-inv 69.9%

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

      7. frac-times 69.9%

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

   3. Applied egg-rr 69.9%

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

   5. Simplified 70.1%

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

      1. pow-sub 70.1%

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

      2. pow1 70.1%

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

      3. associate-*r* 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-*r* 70.1%

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

      6. div-inv 70.1%

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

      7. associate-*r* 70.1%

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

      8. *-commutative 70.1%

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

      9. associate-*r* 70.1%

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

   7. Applied egg-rr 70.1%

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

      1. associate-*r/ 70.1%

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

      2. *-rgt-identity 70.1%

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

      3. associate-/l* 70.0%

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

      4. associate-/r/ 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-*r* 70.1%

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

      7. *-commutative 70.1%

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

   9. Simplified 70.1%

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

   10. Taylor expanded in k around 0 70.1%

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

       1. associate-*r* 70.1%

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

       2. *-commutative 70.1%

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

       3. associate-*l* 70.1%

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

   12. Simplified 70.1%

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

       1. sqrt-div 99.4%

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

       2. *-commutative 99.4%

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

       3. associate-*r* 99.4%

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

       4. sqrt-unprod 99.2%

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

       5. associate-/l* 99.3%

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

       6. div-inv 99.1%

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

       7. sqrt-undiv 99.2%

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

   14. Applied egg-rr 99.2%

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

       1. associate-*r/ 99.4%

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

       2. *-rgt-identity 99.4%

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

   16. Simplified 99.4%

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

   if 7.99999999999999942e-34 < 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. associate-/r/ 99.6%

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

      2. associate-*r* 99.6%

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

      3. div-sub 99.6%

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

      4. metadata-eval 99.6%

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

      5. div-inv 99.6%

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

      6. metadata-eval 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.0%

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

      3. pow-sub 99.1%

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

      4. metadata-eval 99.1%

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

      5. div-inv 99.1%

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

      6. pow-sub 99.0%

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

      7. associate-*r* 99.0%

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

      8. metadata-eval 99.0%

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

      9. div-sub 99.0%

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

      10. pow-sub 99.1%

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

   5. Applied egg-rr 99.0%

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

      1. pow1/2 99.0%

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

      2. div-inv 99.0%

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

      3. unpow-prod-down 99.0%

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

      4. pow1/2 99.0%

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

      5. pow-flip 99.0%

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

      6. *-commutative 99.0%

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

      7. associate-*r* 99.0%

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

      8. *-commutative 99.0%

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

   7. Applied egg-rr 99.0%

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

      1. unpow1/2 99.0%

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

      2. associate-*r* 99.0%

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

      3. *-commutative 99.0%

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

      4. neg-sub0 99.0%

         \[\leadsto \frac{1}{\sqrt{k} \cdot \sqrt{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{\left(0 - \left(1 - k\right)\right)}}}} \]

      5. associate--r- 99.0%

         \[\leadsto \frac{1}{\sqrt{k} \cdot \sqrt{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{\left(\left(0 - 1\right) + k\right)}}}} \]

      6. metadata-eval 99.0%

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

   9. Simplified 99.0%

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

   10. Taylor expanded in n around 0 98.4%

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

   12. Simplified 99.0%

       \[\leadsto \frac{1}{\color{blue}{\sqrt{k \cdot {\left(\frac{0.5}{n \cdot \pi}\right)}^{\left(1 - k\right)}}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{\frac{k}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{k \cdot {\left(\frac{0.5}{\pi \cdot n}\right)}^{\left(1 - k\right)}}}\\ \end{array} \]

Alternative 3: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{\frac{k}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7e-34)
   (/ (sqrt (* PI n)) (sqrt (/ k 2.0)))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 7e-34) {
		tmp = sqrt((((double) M_PI) * n)) / sqrt((k / 2.0));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 7e-34) {
		tmp = Math.sqrt((Math.PI * n)) / Math.sqrt((k / 2.0));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 7e-34:
		tmp = math.sqrt((math.pi * n)) / math.sqrt((k / 2.0))
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 7e-34)
		tmp = Float64(sqrt(Float64(pi * n)) / sqrt(Float64(k / 2.0)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 7e-34)
		tmp = sqrt((pi * n)) / sqrt((k / 2.0));
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7e-34], N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7e-34

   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. add-sqr-sqrt 99.0%

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

      2. sqrt-unprod 69.9%

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

      3. *-commutative 69.9%

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

      4. div-inv 69.9%

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

      5. *-commutative 69.9%

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

      6. div-inv 69.9%

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

      7. frac-times 69.9%

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

   3. Applied egg-rr 69.9%

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

   5. Simplified 70.1%

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

      1. pow-sub 70.1%

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

      2. pow1 70.1%

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

      3. associate-*r* 70.1%

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

      4. *-commutative 70.1%

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

      5. associate-*r* 70.1%

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

      6. div-inv 70.1%

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

      7. associate-*r* 70.1%

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

      8. *-commutative 70.1%

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

      9. associate-*r* 70.1%

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

   7. Applied egg-rr 70.1%

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

      1. associate-*r/ 70.1%

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

      2. *-rgt-identity 70.1%

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

      3. associate-/l* 70.0%

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

      4. associate-/r/ 70.1%

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

      5. *-commutative 70.1%

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

      6. associate-*r* 70.1%

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

      7. *-commutative 70.1%

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

   9. Simplified 70.1%

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

   10. Taylor expanded in k around 0 70.1%

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

       1. associate-*r* 70.1%

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

       2. *-commutative 70.1%

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

       3. associate-*l* 70.1%

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

   12. Simplified 70.1%

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

       1. sqrt-div 99.4%

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

       2. *-commutative 99.4%

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

       3. associate-*r* 99.4%

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

       4. sqrt-unprod 99.2%

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

       5. associate-/l* 99.3%

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

       6. div-inv 99.1%

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

       7. sqrt-undiv 99.2%

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

   14. Applied egg-rr 99.2%

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

       1. associate-*r/ 99.4%

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

       2. *-rgt-identity 99.4%

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

   16. Simplified 99.4%

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

   if 7e-34 < 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. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. div-inv 99.0%

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

      5. *-commutative 99.0%

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

      6. div-inv 99.0%

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

      7. frac-times 99.0%

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

   3. Applied egg-rr 99.0%

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

   5. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot n}}{\sqrt{\frac{k}{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 99.4% 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 99.5%

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

   1. associate-*l/ 99.6%

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

   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. sqr-pow 99.4%

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

   4. pow-sqr 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. associate-/l* 99.6%

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

   9. metadata-eval 99.6%

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

   10. /-rgt-identity 99.6%

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

   11. div-sub 99.6%

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

   12. 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. Final simplification 99.6%

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

Alternative 5: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 86.1%

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

   3. *-commutative 86.1%

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

   4. div-inv 86.1%

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

   5. *-commutative 86.1%

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

   6. div-inv 86.1%

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

   7. frac-times 86.0%

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

3. Applied egg-rr 86.1%

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

5. Simplified 86.1%

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

   1. pow-sub 86.3%

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

   2. pow1 86.3%

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

   3. associate-*r* 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*r* 86.3%

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

   6. div-inv 86.3%

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

   7. associate-*r* 86.3%

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

   8. *-commutative 86.3%

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

   9. associate-*r* 86.3%

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

7. Applied egg-rr 86.3%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. associate-/l* 86.3%

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

   4. associate-/r/ 86.3%

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

   5. *-commutative 86.3%

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

   6. associate-*r* 86.3%

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

   7. *-commutative 86.3%

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

9. Simplified 86.3%

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

10. Taylor expanded in k around 0 37.0%

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

    1. associate-*r* 37.0%

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

    2. *-commutative 37.0%

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

    3. associate-*l* 37.0%

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

12. Simplified 37.0%

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

    1. associate-/l* 37.0%

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

    2. sqrt-div 50.0%

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

    3. *-un-lft-identity 50.0%

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

    4. times-frac 50.0%

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

    5. metadata-eval 50.0%

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

14. Applied egg-rr 50.0%

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

15. Final simplification 50.0%

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

Alternative 6: 50.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 86.1%

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

   3. *-commutative 86.1%

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

   4. div-inv 86.1%

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

   5. *-commutative 86.1%

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

   6. div-inv 86.1%

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

   7. frac-times 86.0%

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

3. Applied egg-rr 86.1%

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

5. Simplified 86.1%

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

   1. pow-sub 86.3%

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

   2. pow1 86.3%

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

   3. associate-*r* 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*r* 86.3%

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

   6. div-inv 86.3%

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

   7. associate-*r* 86.3%

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

   8. *-commutative 86.3%

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

   9. associate-*r* 86.3%

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

7. Applied egg-rr 86.3%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. associate-/l* 86.3%

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

   4. associate-/r/ 86.3%

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

   5. *-commutative 86.3%

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

   6. associate-*r* 86.3%

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

   7. *-commutative 86.3%

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

9. Simplified 86.3%

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

10. Taylor expanded in k around 0 37.0%

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

    1. associate-*r* 37.0%

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

    2. *-commutative 37.0%

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

    3. associate-*l* 37.0%

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

12. Simplified 37.0%

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

    1. sqrt-div 50.1%

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

    2. *-commutative 50.1%

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

    3. associate-*r* 50.1%

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

    4. sqrt-unprod 50.0%

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

    5. associate-/l* 50.0%

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

    6. div-inv 50.0%

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

    7. sqrt-undiv 50.0%

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

14. Applied egg-rr 50.0%

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

    1. associate-*r/ 50.1%

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

    2. *-rgt-identity 50.1%

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

16. Simplified 50.1%

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

17. Final simplification 50.1%

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

Alternative 7: 50.6% 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 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 86.1%

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

   3. *-commutative 86.1%

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

   4. div-inv 86.1%

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

   5. *-commutative 86.1%

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

   6. div-inv 86.1%

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

   7. frac-times 86.0%

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

3. Applied egg-rr 86.1%

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

5. Simplified 86.1%

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

   1. pow-sub 86.3%

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

   2. pow1 86.3%

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

   3. associate-*r* 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*r* 86.3%

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

   6. div-inv 86.3%

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

   7. associate-*r* 86.3%

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

   8. *-commutative 86.3%

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

   9. associate-*r* 86.3%

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

7. Applied egg-rr 86.3%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. associate-/l* 86.3%

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

   4. associate-/r/ 86.3%

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

   5. *-commutative 86.3%

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

   6. associate-*r* 86.3%

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

   7. *-commutative 86.3%

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

9. Simplified 86.3%

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

10. Taylor expanded in k around 0 37.0%

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

    1. associate-*r* 37.0%

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

    2. *-commutative 37.0%

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

    3. associate-*l* 37.0%

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

12. Simplified 37.0%

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

    1. sqrt-div 50.1%

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

    2. *-commutative 50.1%

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

    3. associate-*r* 50.1%

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

    4. sqrt-unprod 50.0%

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

    5. *-commutative 50.0%

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

    6. sqrt-prod 50.0%

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

    7. associate-*r* 50.0%

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

    8. sqrt-prod 49.9%

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

    9. sqrt-prod 50.1%

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

    10. *-commutative 50.1%

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

    11. *-commutative 50.1%

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

14. Applied egg-rr 50.1%

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

15. Final simplification 50.1%

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

Alternative 8: 39.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 86.1%

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

   3. *-commutative 86.1%

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

   4. div-inv 86.1%

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

   5. *-commutative 86.1%

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

   6. div-inv 86.1%

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

   7. frac-times 86.0%

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

3. Applied egg-rr 86.1%

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

5. Simplified 86.1%

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

   1. pow-sub 86.3%

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

   2. pow1 86.3%

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

   3. associate-*r* 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*r* 86.3%

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

   6. div-inv 86.3%

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

   7. associate-*r* 86.3%

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

   8. *-commutative 86.3%

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

   9. associate-*r* 86.3%

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

7. Applied egg-rr 86.3%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. associate-/l* 86.3%

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

   4. associate-/r/ 86.3%

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

   5. *-commutative 86.3%

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

   6. associate-*r* 86.3%

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

   7. *-commutative 86.3%

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

9. Simplified 86.3%

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

10. Taylor expanded in k around 0 37.0%

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

    1. associate-*r* 37.0%

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

    2. *-commutative 37.0%

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

    3. associate-*l* 37.0%

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

12. Simplified 37.0%

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

    1. clear-num 37.0%

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

    2. sqrt-div 38.2%

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

    3. metadata-eval 38.2%

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

    4. associate-*r* 38.2%

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

    5. *-commutative 38.2%

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

    6. *-commutative 38.2%

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

    7. *-commutative 38.2%

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

14. Applied egg-rr 38.2%

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

15. Final simplification 38.2%

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

Alternative 9: 38.6% 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 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 86.1%

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

   3. *-commutative 86.1%

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

   4. div-inv 86.1%

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

   5. *-commutative 86.1%

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

   6. div-inv 86.1%

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

   7. frac-times 86.0%

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

3. Applied egg-rr 86.1%

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

5. Simplified 86.1%

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

   1. pow-sub 86.3%

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

   2. pow1 86.3%

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

   3. associate-*r* 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*r* 86.3%

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

   6. div-inv 86.3%

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

   7. associate-*r* 86.3%

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

   8. *-commutative 86.3%

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

   9. associate-*r* 86.3%

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

7. Applied egg-rr 86.3%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. associate-/l* 86.3%

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

   4. associate-/r/ 86.3%

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

   5. *-commutative 86.3%

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

   6. associate-*r* 86.3%

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

   7. *-commutative 86.3%

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

9. Simplified 86.3%

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

10. Taylor expanded in k around 0 37.0%

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

    1. associate-*r* 37.0%

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

    2. *-commutative 37.0%

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

    3. associate-*l* 37.0%

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

12. Simplified 37.0%

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

13. Taylor expanded in n around 0 37.0%

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

    1. associate-/l* 37.0%

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

15. Simplified 37.0%

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

16. Final simplification 37.0%

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

Alternative 10: 38.6% accurate, 1.5× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 86.1%

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

   3. *-commutative 86.1%

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

   4. div-inv 86.1%

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

   5. *-commutative 86.1%

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

   6. div-inv 86.1%

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

   7. frac-times 86.0%

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

3. Applied egg-rr 86.1%

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

5. Simplified 86.1%

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

   1. pow-sub 86.3%

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

   2. pow1 86.3%

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

   3. associate-*r* 86.3%

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

   4. *-commutative 86.3%

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

   5. associate-*r* 86.3%

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

   6. div-inv 86.3%

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

   7. associate-*r* 86.3%

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

   8. *-commutative 86.3%

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

   9. associate-*r* 86.3%

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

7. Applied egg-rr 86.3%

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

   1. associate-*r/ 86.3%

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

   2. *-rgt-identity 86.3%

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

   3. associate-/l* 86.3%

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

   4. associate-/r/ 86.3%

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

   5. *-commutative 86.3%

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

   6. associate-*r* 86.3%

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

   7. *-commutative 86.3%

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

9. Simplified 86.3%

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

10. Taylor expanded in k around 0 37.0%

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

    1. associate-*r* 37.0%

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

    2. *-commutative 37.0%

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

    3. associate-*l* 37.0%

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

12. Simplified 37.0%

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

    1. *-commutative 37.0%

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

    2. *-un-lft-identity 37.0%

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

    3. times-frac 37.0%

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

14. Applied egg-rr 37.0%

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

15. Final simplification 37.0%

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

Reproduce

herbie shell --seed 2024024 
(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))))