Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 16.0s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 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 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{\pi}{\frac{0.5}{n}}} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.1e-26)
   (* (sqrt (/ PI (/ 0.5 n))) (pow k -0.5))
   (sqrt (/ (pow (* PI (* 2.0 n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.1e-26) {
		tmp = sqrt((((double) M_PI) / (0.5 / n))) * pow(k, -0.5);
	} 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 <= 2.1e-26) {
		tmp = Math.sqrt((Math.PI / (0.5 / n))) * Math.pow(k, -0.5);
	} 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 <= 2.1e-26:
		tmp = math.sqrt((math.pi / (0.5 / n))) * math.pow(k, -0.5)
	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 <= 2.1e-26)
		tmp = Float64(sqrt(Float64(pi / Float64(0.5 / n))) * (k ^ -0.5));
	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 <= 2.1e-26)
		tmp = sqrt((pi / (0.5 / n))) * (k ^ -0.5);
	else
		tmp = sqrt((((pi * (2.0 * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.1e-26], N[(N[Sqrt[N[(Pi / N[(0.5 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $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 < 2.10000000000000008e-26

   1. Initial program 99.3%

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

   2. Taylor expanded in k around 0 99.2%

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

      1. *-commutative 99.2%

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

      2. div-inv 99.2%

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

      3. sqrt-unprod 99.3%

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

      4. *-commutative 99.3%

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

      5. *-commutative 99.3%

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

      6. sqrt-undiv 76.5%

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

      7. associate-*r* 76.5%

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

      8. *-commutative 76.5%

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

      9. associate-*r* 76.5%

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

   4. Applied egg-rr 76.5%

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

      1. sqrt-div 99.3%

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

      2. div-inv 99.3%

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

      3. metadata-eval 99.3%

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

      4. sqrt-div 99.3%

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

      5. associate-*r* 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*l* 99.3%

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

      8. inv-pow 99.3%

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

      9. sqrt-pow1 99.3%

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

      10. metadata-eval 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-*l* 99.3%

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

   8. Simplified 99.3%

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

      1. associate-*r* 99.3%

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

      2. metadata-eval 99.3%

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

      3. div-inv 99.3%

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

      4. associate-/l* 99.4%

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

   10. Applied egg-rr 99.4%

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

   if 2.10000000000000008e-26 < k

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.7%

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

      2. sqrt-unprod 99.1%

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

      3. *-commutative 99.1%

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

      4. div-inv 99.1%

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

      5. *-commutative 99.1%

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

      6. div-inv 99.1%

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

      7. frac-times 99.1%

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

   3. Applied egg-rr 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.1 \cdot 10^{-26}:\\ \;\;\;\;\sqrt{\frac{\pi}{\frac{0.5}{n}}} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3: 50.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi / N[(0.5 / n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Power[k, -0.5], $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. Taylor expanded in k around 0 47.8%

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

   1. *-commutative 47.8%

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

   2. div-inv 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-undiv 37.8%

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

   7. associate-*r* 37.8%

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

   8. *-commutative 37.8%

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

   9. associate-*r* 37.8%

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

4. Applied egg-rr 37.8%

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

   1. sqrt-div 47.9%

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

   2. div-inv 47.9%

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

   3. metadata-eval 47.9%

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

   4. sqrt-div 47.9%

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

   5. associate-*r* 47.9%

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

   6. *-commutative 47.9%

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

   7. associate-*l* 47.9%

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

   8. inv-pow 47.9%

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

   9. sqrt-pow1 47.9%

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

   10. metadata-eval 47.9%

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

6. Applied egg-rr 47.9%

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

   1. *-commutative 47.9%

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

   2. associate-*l* 47.9%

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

8. Simplified 47.9%

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

   1. associate-*r* 47.9%

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

   2. metadata-eval 47.9%

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

   3. div-inv 47.9%

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

   4. associate-/l* 47.9%

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

10. Applied egg-rr 47.9%

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

11. Final simplification 47.9%

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

Alternative 4: 50.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in k around 0 47.8%

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

   1. *-commutative 47.8%

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

   2. div-inv 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-undiv 37.8%

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

   7. associate-*r* 37.8%

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

   8. *-commutative 37.8%

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

   9. associate-*r* 37.8%

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

4. Applied egg-rr 37.8%

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

5. Taylor expanded in n around 0 37.8%

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

   1. associate-*l/ 37.8%

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

   2. associate-*r* 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. *-commutative 37.8%

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

   6. associate-*r/ 37.8%

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

7. Simplified 37.8%

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

   1. associate-*r* 37.8%

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

   2. associate-*r/ 37.8%

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

   3. metadata-eval 37.8%

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

   4. div-inv 37.8%

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

   5. associate-/r* 37.8%

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

   6. *-commutative 37.8%

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

   7. clear-num 37.7%

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

   8. metadata-eval 37.7%

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

   9. add-sqr-sqrt 37.7%

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

   10. frac-times 37.7%

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

   11. sqrt-unprod 38.2%

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

   12. add-sqr-sqrt 38.5%

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

   13. sqrt-div 47.9%

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

   14. associate-/r/ 47.9%

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

9. Applied egg-rr 47.9%

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

10. Final simplification 47.9%

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

Alternative 5: 50.1% 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. Taylor expanded in k around 0 47.8%

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

   1. associate-*l/ 47.8%

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

   2. *-un-lft-identity 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. associate-*r* 47.9%

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

   7. *-commutative 47.9%

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

   8. associate-*r* 47.9%

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

4. Applied egg-rr 47.9%

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

5. Final simplification 47.9%

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

Alternative 6: 38.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(N[(k / n), $MachinePrecision] * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision], -0.5], $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. Taylor expanded in k around 0 47.8%

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

   1. *-commutative 47.8%

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

   2. div-inv 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-undiv 37.8%

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

   7. associate-*r* 37.8%

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

   8. *-commutative 37.8%

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

   9. associate-*r* 37.8%

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

4. Applied egg-rr 37.8%

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

5. Taylor expanded in n around 0 37.8%

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

   1. associate-*l/ 37.8%

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

   2. associate-*r* 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. *-commutative 37.8%

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

   6. associate-*r/ 37.8%

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

7. Simplified 37.8%

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

   1. associate-*r* 37.8%

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

   2. associate-*r/ 37.8%

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

   3. metadata-eval 37.8%

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

   4. div-inv 37.8%

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

   5. associate-/r* 37.8%

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

   6. *-commutative 37.8%

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

   7. clear-num 37.7%

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

   8. metadata-eval 37.7%

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

   9. add-sqr-sqrt 37.7%

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

   10. frac-times 37.7%

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

   11. sqrt-unprod 38.2%

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

   12. add-sqr-sqrt 38.5%

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

   13. inv-pow 38.5%

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

   14. sqrt-pow2 38.5%

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

9. Applied egg-rr 38.5%

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

    1. associate-*r/ 38.5%

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

    2. *-commutative 38.5%

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

    3. *-commutative 38.5%

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

    4. times-frac 38.5%

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

11. Simplified 38.5%

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

12. Final simplification 38.5%

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

Alternative 7: 38.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(N[(0.5 * k), $MachinePrecision] / N[(Pi * n), $MachinePrecision]), $MachinePrecision], -0.5], $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. Taylor expanded in k around 0 47.8%

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

   1. *-commutative 47.8%

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

   2. div-inv 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-undiv 37.8%

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

   7. associate-*r* 37.8%

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

   8. *-commutative 37.8%

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

   9. associate-*r* 37.8%

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

4. Applied egg-rr 37.8%

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

5. Taylor expanded in n around 0 37.8%

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

   1. associate-*l/ 37.8%

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

   2. associate-*r* 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. *-commutative 37.8%

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

   6. associate-*r/ 37.8%

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

7. Simplified 37.8%

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

   1. associate-*r* 37.8%

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

   2. associate-*r/ 37.8%

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

   3. metadata-eval 37.8%

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

   4. div-inv 37.8%

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

   5. associate-/r* 37.8%

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

   6. *-commutative 37.8%

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

   7. clear-num 37.7%

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

   8. metadata-eval 37.7%

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

   9. add-sqr-sqrt 37.7%

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

   10. frac-times 37.7%

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

   11. sqrt-unprod 38.2%

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

   12. add-sqr-sqrt 38.5%

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

   13. inv-pow 38.5%

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

   14. sqrt-pow2 38.5%

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

9. Applied egg-rr 38.5%

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

    1. associate-*r/ 38.5%

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

    2. *-commutative 38.5%

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

    3. *-commutative 38.5%

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

11. Simplified 38.5%

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

12. Final simplification 38.5%

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

Alternative 8: 37.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(Pi / N[(k / n), $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. Taylor expanded in k around 0 47.8%

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

   1. *-commutative 47.8%

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

   2. div-inv 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-undiv 37.8%

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

   7. associate-*r* 37.8%

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

   8. *-commutative 37.8%

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

   9. associate-*r* 37.8%

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

4. Applied egg-rr 37.8%

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

5. Taylor expanded in n around 0 37.8%

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

   1. associate-*l/ 37.8%

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

   2. associate-*r* 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. *-commutative 37.8%

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

   6. associate-*r/ 37.8%

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

7. Simplified 37.8%

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

8. Taylor expanded in n around 0 37.8%

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

   1. *-commutative 37.8%

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

   2. associate-/l* 37.8%

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

10. Simplified 37.8%

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

11. Final simplification 37.8%

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

Alternative 9: 37.8% 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 99.5%

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

2. Taylor expanded in k around 0 47.8%

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

   1. *-commutative 47.8%

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

   2. div-inv 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-undiv 37.8%

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

   7. associate-*r* 37.8%

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

   8. *-commutative 37.8%

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

   9. associate-*r* 37.8%

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

4. Applied egg-rr 37.8%

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

5. Taylor expanded in n around 0 37.8%

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

   1. associate-*l/ 37.8%

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

   2. associate-*r* 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. *-commutative 37.8%

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

   6. associate-*r/ 37.8%

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

7. Simplified 37.8%

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

8. Final simplification 37.8%

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

Alternative 10: 37.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n / 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. Taylor expanded in k around 0 47.8%

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

   1. *-commutative 47.8%

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

   2. div-inv 47.8%

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

   3. sqrt-unprod 47.9%

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

   4. *-commutative 47.9%

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

   5. *-commutative 47.9%

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

   6. sqrt-undiv 37.8%

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

   7. associate-*r* 37.8%

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

   8. *-commutative 37.8%

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

   9. associate-*r* 37.8%

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

4. Applied egg-rr 37.8%

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

5. Taylor expanded in n around 0 37.8%

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

   1. associate-*l/ 37.8%

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

   2. associate-*r* 37.8%

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

   3. associate-*r/ 37.8%

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

   4. *-commutative 37.8%

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

   5. *-commutative 37.8%

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

   6. associate-*r/ 37.8%

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

7. Simplified 37.8%

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

   1. associate-*r* 37.8%

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

   2. associate-*r/ 37.8%

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

   3. metadata-eval 37.8%

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

   4. div-inv 37.8%

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

   5. associate-/r* 37.8%

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

   6. *-commutative 37.8%

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

   7. *-commutative 37.8%

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

   8. associate-/l* 37.8%

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

   9. *-commutative 37.8%

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

   10. *-un-lft-identity 37.8%

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

   11. times-frac 37.8%

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

   12. metadata-eval 37.8%

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

9. Applied egg-rr 37.8%

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

10. Final simplification 37.8%

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

Reproduce

herbie shell --seed 2023307 
(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))))