Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 15.1s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] / N[Power[t$95$0, N[(k * 0.5), $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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r/ 99.5%

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

   2. clear-num 99.6%

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

   3. *-commutative 99.6%

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

   4. associate-*r* 99.6%

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

   5. div-sub 99.6%

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

   6. metadata-eval 99.6%

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

   7. pow-sub 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*l* 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/r* 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*l* 99.7%

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

   5. associate-*r* 99.7%

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

   6. *-commutative 99.7%

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

   7. associate-*l* 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 51.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{+264}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 9e+264)
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))
   (cbrt (pow (* (/ n k) (* PI 2.0)) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 9e+264) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	} else {
		tmp = cbrt(pow(((n / k) * (((double) M_PI) * 2.0)), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 9e+264) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	} else {
		tmp = Math.cbrt(Math.pow(((n / k) * (Math.PI * 2.0)), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 9e+264)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	else
		tmp = cbrt((Float64(Float64(n / k) * Float64(pi * 2.0)) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 9e+264], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(n / k), $MachinePrecision] * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.00000000000000006e264

   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. Add Preprocessing

   3. Taylor expanded in k around 0 54.2%

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

      1. associate-*l/ 54.2%

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

      2. *-un-lft-identity 54.2%

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

      3. sqrt-unprod 54.3%

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

      4. *-commutative 54.3%

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

      5. *-commutative 54.3%

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

   5. Applied egg-rr 54.3%

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

   if 9.00000000000000006e264 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 3.4%

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

      1. associate-*l/ 3.4%

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

      2. *-un-lft-identity 3.4%

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

      3. sqrt-unprod 3.4%

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

      4. *-commutative 3.4%

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

      5. *-commutative 3.4%

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

      6. sqrt-undiv 3.3%

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

   5. Applied egg-rr 3.3%

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

      1. div-inv 3.3%

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

      2. associate-*r* 3.3%

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

      3. *-commutative 3.3%

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

      4. associate-*l* 3.3%

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

      5. *-commutative 3.3%

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

   7. Applied egg-rr 3.3%

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

      1. add-cbrt-cube 37.0%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\sqrt{n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)} \cdot \sqrt{n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)}\right) \cdot \sqrt{n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)}}} \]

      2. add-sqr-sqrt 37.0%

         \[\leadsto \sqrt[3]{\color{blue}{\left(n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)\right)} \cdot \sqrt{n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)}} \]

      3. pow1 37.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)\right)}^{1}} \cdot \sqrt{n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)}} \]

      4. pow1/2 37.0%

         \[\leadsto \sqrt[3]{{\left(n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)\right)}^{0.5}}} \]

      5. pow-prod-up 37.0%

         \[\leadsto \sqrt[3]{\color{blue}{{\left(n \cdot \left(\left(\pi \cdot 2\right) \cdot \frac{1}{k}\right)\right)}^{\left(1 + 0.5\right)}}} \]

      6. *-commutative 37.0%

         \[\leadsto \sqrt[3]{{\left(n \cdot \color{blue}{\left(\frac{1}{k} \cdot \left(\pi \cdot 2\right)\right)}\right)}^{\left(1 + 0.5\right)}} \]

      7. associate-*r* 37.0%

         \[\leadsto \sqrt[3]{{\color{blue}{\left(\left(n \cdot \frac{1}{k}\right) \cdot \left(\pi \cdot 2\right)\right)}}^{\left(1 + 0.5\right)}} \]

      8. div-inv 37.0%

         \[\leadsto \sqrt[3]{{\left(\color{blue}{\frac{n}{k}} \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 + 0.5\right)}} \]

      9. *-commutative 37.0%

         \[\leadsto \sqrt[3]{{\left(\frac{n}{k} \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)}^{\left(1 + 0.5\right)}} \]

      10. metadata-eval 37.0%

          \[\leadsto \sqrt[3]{{\left(\frac{n}{k} \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{1.5}}} \]

   9. Applied egg-rr 37.0%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{n}{k} \cdot \left(2 \cdot \pi\right)\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9 \cdot 10^{+264}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(2.0 * 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. *-commutative 99.6%

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

   6. associate-*l* 99.6%

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

   7. associate-*r/ 99.6%

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

   8. *-commutative 99.6%

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

   9. associate-/l* 99.6%

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

   10. metadata-eval 99.6%

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

   11. /-rgt-identity 99.6%

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

   12. div-sub 99.6%

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

   13. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 4: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in k around 0 50.8%

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

   1. associate-*l/ 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. sqrt-unprod 50.9%

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

   4. *-commutative 50.9%

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

   5. *-commutative 50.9%

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

   6. sqrt-undiv 40.3%

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

5. Applied egg-rr 40.3%

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

   1. div-inv 40.3%

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

   2. associate-*r* 40.3%

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

   3. *-commutative 40.3%

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

   4. associate-*l* 40.3%

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

   5. *-commutative 40.3%

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

7. Applied egg-rr 40.3%

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

   1. *-commutative 40.3%

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

   2. sqrt-prod 50.9%

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

   3. un-div-inv 50.9%

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

   4. *-commutative 50.9%

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

   5. *-un-lft-identity 50.9%

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

   6. times-frac 50.9%

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

   7. metadata-eval 50.9%

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

9. Applied egg-rr 50.9%

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

10. Final simplification 50.9%

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

Alternative 5: 50.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * N[(Pi * 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. Add Preprocessing

3. Taylor expanded in k around 0 50.8%

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

   1. associate-*l/ 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. sqrt-unprod 50.9%

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

   4. *-commutative 50.9%

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

   5. *-commutative 50.9%

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

5. Applied egg-rr 50.9%

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

6. Final simplification 50.9%

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

Alternative 6: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(N[(0.5 / Pi), $MachinePrecision] * 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. Add Preprocessing

4. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{{\left(\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(\frac{1 - k}{4}\right)}}{{k}^{0.25}}\right)}^{2}} \]

5. Taylor expanded in k around 0 40.2%

   \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.25}\right)}}^{2} \]
6. Step-by-step derivation

   1. associate-*r/ 40.2%

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

   2. *-commutative 40.2%

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

   3. associate-*r* 40.2%

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

   4. *-commutative 40.2%

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

   5. associate-*l* 40.2%

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

   6. associate-/l* 40.2%

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

7. Simplified 40.2%

   \[\leadsto {\color{blue}{\left({\left(\frac{\pi}{\frac{k}{2 \cdot n}}\right)}^{0.25}\right)}}^{2} \]
8. Step-by-step derivation

   1. pow-pow 40.3%

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

   2. metadata-eval 40.3%

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

   3. pow1/2 40.3%

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

   4. clear-num 40.3%

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

   5. sqrt-div 41.4%

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

   6. metadata-eval 41.4%

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

   7. *-un-lft-identity 41.4%

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

   8. times-frac 41.4%

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

   9. metadata-eval 41.4%

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

9. Applied egg-rr 41.4%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{0.5 \cdot \frac{k}{n}}{\pi}}}} \]
10. Step-by-step derivation

    1. associate-/l* 40.3%

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

    2. associate-/r/ 41.3%

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

11. Simplified 41.3%

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

12. Final simplification 41.3%

    \[\leadsto \frac{1}{\sqrt{\frac{0.5}{\pi} \cdot \frac{k}{n}}} \]
13. Add Preprocessing

Alternative 7: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(k / N[(n * N[(Pi / 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. Add Preprocessing

3. Taylor expanded in k around 0 50.8%

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

   1. associate-*l/ 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. sqrt-unprod 50.9%

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

   4. *-commutative 50.9%

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

   5. *-commutative 50.9%

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

   6. sqrt-undiv 40.3%

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

5. Applied egg-rr 40.3%

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

   1. div-inv 40.3%

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

   2. associate-*r* 40.3%

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

   3. *-commutative 40.3%

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

   4. associate-*l* 40.3%

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

   5. *-commutative 40.3%

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

7. Applied egg-rr 40.3%

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

   1. associate-*r* 40.3%

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

   2. sqrt-prod 50.8%

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

   3. associate-*r* 50.8%

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

   4. *-commutative 50.8%

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

   5. *-commutative 50.8%

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

   6. sqrt-div 50.8%

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

   7. metadata-eval 50.8%

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

   8. div-inv 50.9%

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

   9. clear-num 50.8%

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

   10. sqrt-undiv 41.4%

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

   11. *-commutative 41.4%

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

   12. *-commutative 41.4%

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

   13. associate-*r* 41.4%

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

   14. *-commutative 41.4%

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

9. Applied egg-rr 41.4%

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

    1. *-commutative 41.4%

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

    2. *-commutative 41.4%

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

    3. associate-*r* 41.4%

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

    4. /-rgt-identity 41.4%

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

    5. associate-/r/ 41.3%

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

    6. associate-/r* 41.3%

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

    7. metadata-eval 41.3%

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

    8. associate-/r/ 41.4%

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

11. Simplified 41.4%

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

12. Final simplification 41.4%

    \[\leadsto \frac{1}{\sqrt{\frac{k}{n \cdot \frac{\pi}{0.5}}}} \]
13. Add Preprocessing

Alternative 8: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.8%

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

   1. associate-*l/ 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. sqrt-unprod 50.9%

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

   4. *-commutative 50.9%

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

   5. *-commutative 50.9%

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

   6. sqrt-undiv 40.3%

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

5. Applied egg-rr 40.3%

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

6. Taylor expanded in n around 0 40.3%

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

   1. associate-/l* 40.3%

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

8. Simplified 40.3%

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

   1. associate-/r/ 40.3%

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

10. Applied egg-rr 40.3%

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

11. Final simplification 40.3%

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

Alternative 9: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 50.8%

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

   1. associate-*l/ 50.8%

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

   2. *-un-lft-identity 50.8%

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

   3. sqrt-unprod 50.9%

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

   4. *-commutative 50.9%

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

   5. *-commutative 50.9%

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

   6. sqrt-undiv 40.3%

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

5. Applied egg-rr 40.3%

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

6. Final simplification 40.3%

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

Reproduce

herbie shell --seed 2024011 
(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))))