Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 13.7s

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, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. associate-*r* 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

   5. div-inv 99.5%

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

   6. pow-sub 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

5. Simplified 99.7%

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

6. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(2 \cdot n\right) \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.4e-27)
   (* (pow k -0.5) (sqrt (* (* 2.0 n) PI)))
   (pow (/ k (pow (* n (* 2.0 PI)) (- 1.0 k))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.4e-27) {
		tmp = pow(k, -0.5) * sqrt(((2.0 * n) * ((double) M_PI)));
	} else {
		tmp = pow((k / pow((n * (2.0 * ((double) M_PI))), (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.4e-27) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(((2.0 * n) * Math.PI));
	} else {
		tmp = Math.pow((k / Math.pow((n * (2.0 * Math.PI)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.4e-27:
		tmp = math.pow(k, -0.5) * math.sqrt(((2.0 * n) * math.pi))
	else:
		tmp = math.pow((k / math.pow((n * (2.0 * math.pi)), (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.4e-27)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(Float64(2.0 * n) * pi)));
	else
		tmp = Float64(k / (Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.4e-27)
		tmp = (k ^ -0.5) * sqrt(((2.0 * n) * pi));
	else
		tmp = (k / ((n * (2.0 * pi)) ^ (1.0 - k))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.4e-27], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.4e-27

   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 99.1%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. sqrt-unprod 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*l* 99.4%

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

      6. sqrt-undiv 74.8%

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

      7. associate-*l* 74.8%

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

      8. *-commutative 74.8%

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

      9. associate-*r* 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. div-inv 74.8%

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

      2. sqrt-prod 99.4%

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

      3. pow1/2 99.4%

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

      4. metadata-eval 99.4%

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

      5. pow-pow 69.3%

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

      6. pow1/3 74.2%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}}} \cdot \sqrt{\frac{1}{k}} \]

      7. *-commutative 74.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}}} \]

      8. inv-pow 74.2%

         \[\leadsto \sqrt{\color{blue}{{k}^{-1}}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

      9. sqrt-pow1 74.3%

         \[\leadsto \color{blue}{{k}^{\left(\frac{-1}{2}\right)}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

      10. metadata-eval 74.3%

          \[\leadsto {k}^{\color{blue}{-0.5}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

      11. pow1/3 69.3%

          \[\leadsto {k}^{-0.5} \cdot \color{blue}{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      12. pow-pow 99.4%

          \[\leadsto {k}^{-0.5} \cdot \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      13. metadata-eval 99.4%

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

      14. pow1/2 99.4%

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

      15. *-commutative 99.4%

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

      16. associate-*l* 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   if 1.4e-27 < 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-*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 99.6%

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

      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. Step-by-step derivation

      1. div-inv 99.6%

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

      2. div-inv 99.6%

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

      3. metadata-eval 99.6%

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

      4. pow1/2 99.6%

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

      5. pow-flip 99.6%

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

      6. metadata-eval 99.6%

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

   5. Applied egg-rr 99.6%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(2 \cdot n\right) \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(n * Pi), $MachinePrecision]), $MachinePrecision], N[(0.5 - N[(k * 0.5), $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. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l/ 99.5%

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

   8. associate-/l* 99.5%

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

   9. metadata-eval 99.5%

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

   10. /-rgt-identity 99.5%

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

   11. div-sub 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. pow1/2 99.5%

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

   5. pow-flip 99.5%

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

   6. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

6. Final simplification 99.5%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot n\right) \cdot \pi\\ \mathbf{if}\;k \leq 2.55 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 n) PI)))
   (if (<= k 2.55e-27)
     (* (pow k -0.5) (sqrt t_0))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = (2.0 * n) * ((double) M_PI);
	double tmp;
	if (k <= 2.55e-27) {
		tmp = pow(k, -0.5) * sqrt(t_0);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = (2.0 * n) * Math.PI;
	double tmp;
	if (k <= 2.55e-27) {
		tmp = Math.pow(k, -0.5) * Math.sqrt(t_0);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = (2.0 * n) * math.pi
	tmp = 0
	if k <= 2.55e-27:
		tmp = math.pow(k, -0.5) * math.sqrt(t_0)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * n) * pi)
	tmp = 0.0
	if (k <= 2.55e-27)
		tmp = Float64((k ^ -0.5) * sqrt(t_0));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = (2.0 * n) * pi;
	tmp = 0.0;
	if (k <= 2.55e-27)
		tmp = (k ^ -0.5) * sqrt(t_0);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision]}, If[LessEqual[k, 2.55e-27], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.55e-27

   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 99.1%

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

      1. associate-*l/ 99.2%

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

      2. *-un-lft-identity 99.2%

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

      3. sqrt-unprod 99.4%

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

      4. *-commutative 99.4%

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

      5. associate-*l* 99.4%

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

      6. sqrt-undiv 74.8%

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

      7. associate-*l* 74.8%

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

      8. *-commutative 74.8%

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

      9. associate-*r* 74.8%

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

   4. Applied egg-rr 74.8%

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

      1. div-inv 74.8%

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

      2. sqrt-prod 99.4%

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

      3. pow1/2 99.4%

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

      4. metadata-eval 99.4%

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

      5. pow-pow 69.3%

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

      6. pow1/3 74.2%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}}} \cdot \sqrt{\frac{1}{k}} \]

      7. *-commutative 74.2%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}}} \]

      8. inv-pow 74.2%

         \[\leadsto \sqrt{\color{blue}{{k}^{-1}}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

      9. sqrt-pow1 74.3%

         \[\leadsto \color{blue}{{k}^{\left(\frac{-1}{2}\right)}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

      10. metadata-eval 74.3%

          \[\leadsto {k}^{\color{blue}{-0.5}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

      11. pow1/3 69.3%

          \[\leadsto {k}^{-0.5} \cdot \color{blue}{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}\right)}^{0.3333333333333333}} \]

      12. pow-pow 99.4%

          \[\leadsto {k}^{-0.5} \cdot \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

      13. metadata-eval 99.4%

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

      14. pow1/2 99.4%

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

      15. *-commutative 99.4%

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

      16. associate-*l* 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. *-commutative 99.4%

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

   8. Simplified 99.4%

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

   if 2.55e-27 < 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 99.6%

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

      3. *-commutative 99.6%

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

      4. div-inv 99.6%

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

      5. *-commutative 99.6%

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

      6. div-inv 99.6%

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

      7. frac-times 99.6%

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

   3. Applied egg-rr 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.55 \cdot 10^{-27}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(2 \cdot n\right) \cdot \pi}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(2.0 * N[(n * Pi), $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/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l/ 99.5%

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

   8. associate-/l* 99.5%

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

   9. metadata-eval 99.5%

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

   10. /-rgt-identity 99.5%

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

   11. div-sub 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 6: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * n), $MachinePrecision] * Pi), $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 51.2%

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

   1. associate-*l/ 51.2%

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

   2. *-un-lft-identity 51.2%

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

   3. sqrt-unprod 51.3%

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

   4. *-commutative 51.3%

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

   5. associate-*l* 51.3%

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

   6. sqrt-undiv 40.0%

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

   7. associate-*l* 40.0%

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

   8. *-commutative 40.0%

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

   9. associate-*r* 40.0%

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

4. Applied egg-rr 40.0%

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

   1. div-inv 40.0%

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

   2. sqrt-prod 51.3%

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

   3. pow1/2 51.3%

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

   4. metadata-eval 51.3%

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

   5. pow-pow 36.3%

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

   6. pow1/3 38.6%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}}} \cdot \sqrt{\frac{1}{k}} \]

   7. *-commutative 38.6%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}}} \]

   8. inv-pow 38.6%

      \[\leadsto \sqrt{\color{blue}{{k}^{-1}}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

   9. sqrt-pow1 38.7%

      \[\leadsto \color{blue}{{k}^{\left(\frac{-1}{2}\right)}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

   10. metadata-eval 38.7%

       \[\leadsto {k}^{\color{blue}{-0.5}} \cdot \sqrt[3]{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}} \]

   11. pow1/3 36.3%

       \[\leadsto {k}^{-0.5} \cdot \color{blue}{{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{1.5}\right)}^{0.3333333333333333}} \]

   12. pow-pow 51.3%

       \[\leadsto {k}^{-0.5} \cdot \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   13. metadata-eval 51.3%

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

   14. pow1/2 51.3%

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

   15. *-commutative 51.3%

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

   16. associate-*l* 51.3%

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

6. Applied egg-rr 51.3%

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

   1. *-commutative 51.3%

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

8. Simplified 51.3%

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

9. Final simplification 51.3%

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

Alternative 7: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(n * N[(2.0 * Pi), $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 51.2%

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

   1. associate-*l/ 51.2%

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

   2. *-un-lft-identity 51.2%

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

   3. sqrt-unprod 51.3%

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

   4. *-commutative 51.3%

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

   5. *-commutative 51.3%

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

   6. associate-*r* 51.3%

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

4. Applied egg-rr 51.3%

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

5. Final simplification 51.3%

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

Alternative 8: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Power[N[(k / N[(N[(2.0 * n), $MachinePrecision] * 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. Step-by-step derivation

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. sqr-pow 99.3%

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

   4. pow-sqr 99.5%

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

   5. associate-*l* 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*l/ 99.5%

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

   8. associate-/l* 99.5%

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

   9. metadata-eval 99.5%

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

   10. /-rgt-identity 99.5%

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

   11. div-sub 99.5%

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

   12. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. div-inv 99.5%

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

   3. metadata-eval 99.5%

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

   4. pow1/2 99.5%

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

   5. pow-flip 99.5%

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

   6. metadata-eval 99.5%

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

5. Applied egg-rr 99.5%

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

7. Applied egg-rr 89.3%

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

8. Taylor expanded in k around 0 41.0%

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

   1. associate-*r* 41.0%

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

   2. *-commutative 41.0%

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

   3. *-commutative 41.0%

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

10. Simplified 41.0%

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

11. Final simplification 41.0%

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

Alternative 9: 37.6% 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. Taylor expanded in k around 0 51.2%

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

   1. associate-*l/ 51.2%

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

   2. *-un-lft-identity 51.2%

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

   3. sqrt-unprod 51.3%

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

   4. *-commutative 51.3%

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

   5. associate-*l* 51.3%

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

   6. sqrt-undiv 40.0%

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

   7. associate-*l* 40.0%

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

   8. *-commutative 40.0%

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

   9. associate-*r* 40.0%

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

4. Applied egg-rr 40.0%

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

5. Taylor expanded in n around 0 40.0%

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

   1. associate-/l* 40.0%

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

7. Simplified 40.0%

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

8. Final simplification 40.0%

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

Alternative 10: 37.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(N[(2.0 * Pi), $MachinePrecision] / 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. Taylor expanded in k around 0 51.2%

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

   1. associate-*l/ 51.2%

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

   2. *-un-lft-identity 51.2%

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

   3. sqrt-unprod 51.3%

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

   4. *-commutative 51.3%

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

   5. associate-*l* 51.3%

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

   6. sqrt-undiv 40.0%

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

   7. associate-*l* 40.0%

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

   8. *-commutative 40.0%

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

   9. associate-*r* 40.0%

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

4. Applied egg-rr 40.0%

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

   1. associate-/l* 40.0%

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

   2. associate-/r/ 40.1%

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

6. Applied egg-rr 40.1%

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

7. Final simplification 40.1%

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

Reproduce

herbie shell --seed 2023318 
(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))))