Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.9s

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. add-cbrt-cube 76.2%

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

   3. add-sqr-sqrt 76.2%

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

   4. cbrt-prod 99.0%

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

   5. *-un-lft-identity 99.0%

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

   6. associate-/r* 98.9%

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

   7. *-commutative 98.9%

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

   8. associate-*l* 98.9%

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

   9. div-sub 98.9%

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

   10. metadata-eval 98.9%

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

   11. div-inv 98.9%

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

   12. metadata-eval 98.9%

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

3. Applied egg-rr 98.9%

   \[\leadsto \color{blue}{\frac{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - k \cdot 0.5\right)}}{\sqrt[3]{k}}}{\sqrt[3]{\sqrt{k}}}} \]
4. Step-by-step derivation

   1. associate-/l/ 99.0%

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

   2. sqr-pow 99.0%

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

   3. sqr-pow 99.0%

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

   4. *-commutative 99.0%

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

   5. associate-*r* 99.0%

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

   6. *-commutative 99.0%

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

   7. cancel-sign-sub-inv 99.0%

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

   8. metadata-eval 99.0%

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

   9. *-commutative 99.0%

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

5. Simplified 99.0%

   \[\leadsto \color{blue}{\frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(0.5 + k \cdot -0.5\right)}}{\sqrt[3]{\sqrt{k}} \cdot \sqrt[3]{k}}} \]
6. Step-by-step derivation

   1. cbrt-unprod 76.2%

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

   2. pow1/3 74.3%

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

   3. pow1/2 74.3%

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

   4. pow-plus 74.3%

      \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(0.5 + k \cdot -0.5\right)}}{{\color{blue}{\left({k}^{\left(0.5 + 1\right)}\right)}}^{0.3333333333333333}} \]

   5. metadata-eval 74.3%

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

7. Applied egg-rr 74.3%

   \[\leadsto \frac{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(0.5 + k \cdot -0.5\right)}}{\color{blue}{{\left({k}^{1.5}\right)}^{0.3333333333333333}}} \]
8. Step-by-step derivation

   1. associate-*r* 74.3%

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

   2. *-commutative 74.3%

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

   3. pow-prod-up 74.5%

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

   4. pow1/2 74.5%

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

   5. *-un-lft-identity 74.5%

      \[\leadsto \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{\color{blue}{1 \cdot {\left({k}^{1.5}\right)}^{0.3333333333333333}}} \]

   6. pow-pow 99.7%

      \[\leadsto \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{1 \cdot \color{blue}{{k}^{\left(1.5 \cdot 0.3333333333333333\right)}}} \]

   7. metadata-eval 99.7%

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

   8. pow1/2 99.7%

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

   9. times-frac 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*r* 99.7%

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

   12. *-commutative 99.7%

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

9. Applied egg-rr 99.7%

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

    1. times-frac 99.7%

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

    2. *-commutative 99.7%

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

    3. times-frac 99.7%

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

    4. /-rgt-identity 99.7%

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

    5. *-commutative 99.7%

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

    6. associate-*l* 99.7%

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

    7. *-commutative 99.7%

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

    8. *-commutative 99.7%

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

    9. associate-*l* 99.7%

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

    10. *-commutative 99.7%

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

11. Simplified 99.7%

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

12. Final simplification 99.7%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

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)));
}

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)));
}

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)))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 85.2%

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

   3. *-commutative 85.2%

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

   4. div-inv 85.2%

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

   5. *-commutative 85.2%

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

   6. div-inv 85.2%

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

   7. frac-times 85.2%

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

3. Applied egg-rr 85.2%

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

5. Simplified 85.3%

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

   1. clear-num 85.3%

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

   2. sqrt-div 86.0%

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

   3. metadata-eval 86.0%

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

   4. associate-*r* 86.0%

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

   5. *-commutative 86.0%

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

7. Applied egg-rr 86.0%

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

   1. inv-pow 86.0%

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

   2. sqrt-pow2 86.1%

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

   3. associate-*r* 86.1%

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

   4. *-commutative 86.1%

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

   5. *-commutative 86.1%

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

   6. *-commutative 86.1%

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

   7. metadata-eval 86.1%

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

9. Applied egg-rr 86.1%

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

    1. rem-square-sqrt 86.0%

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

    2. rem-square-sqrt 86.1%

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

    3. *-commutative 86.1%

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

    4. associate-*l* 86.1%

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

    5. *-commutative 86.1%

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

11. Simplified 86.1%

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

    1. add-sqr-sqrt 85.9%

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

    2. sqrt-unprod 85.3%

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

    3. pow-prod-up 85.3%

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

    4. metadata-eval 85.3%

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

    5. inv-pow 85.3%

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

    6. *-commutative 85.3%

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

    7. pow-sub 85.5%

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

    8. pow1 85.5%

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

    9. clear-num 85.5%

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

    10. associate-/l/ 85.5%

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

13. Applied egg-rr 99.7%

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

    1. *-commutative 99.7%

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

    2. *-commutative 99.7%

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

15. Simplified 99.7%

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

16. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   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 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.4%

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

   2. associate-*r* 99.4%

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

   3. *-commutative 99.4%

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

   4. associate-*l* 99.4%

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

   5. div-inv 99.4%

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

   6. metadata-eval 99.4%

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

   7. pow1/2 99.4%

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

   8. pow-flip 99.4%

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

   9. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   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 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

3. Simplified 99.4%

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

4. Final simplification 99.4%

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

Alternative 5: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 85.2%

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

   3. *-commutative 85.2%

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

   4. div-inv 85.2%

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

   5. *-commutative 85.2%

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

   6. div-inv 85.2%

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

   7. frac-times 85.2%

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

3. Applied egg-rr 85.2%

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

5. Simplified 85.3%

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

   1. clear-num 85.3%

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

   2. sqrt-div 86.0%

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

   3. metadata-eval 86.0%

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

   4. associate-*r* 86.0%

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

   5. *-commutative 86.0%

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

7. Applied egg-rr 86.0%

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

   1. inv-pow 86.0%

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

   2. sqrt-pow2 86.1%

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

   3. associate-*r* 86.1%

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

   4. *-commutative 86.1%

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

   5. *-commutative 86.1%

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

   6. *-commutative 86.1%

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

   7. metadata-eval 86.1%

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

9. Applied egg-rr 86.1%

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

    1. rem-square-sqrt 86.0%

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

    2. rem-square-sqrt 86.1%

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

    3. *-commutative 86.1%

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

    4. associate-*l* 86.1%

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

    5. *-commutative 86.1%

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

11. Simplified 86.1%

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

    1. sqr-pow 86.0%

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

13. Applied egg-rr 86.0%

    \[\leadsto \color{blue}{{\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k - 1\right)}\right)}^{-0.25} \cdot {\left(k \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k - 1\right)}\right)}^{-0.25}} \]
14. Step-by-step derivation

    1. pow-sqr 86.1%

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

    2. metadata-eval 86.1%

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

    3. *-commutative 86.1%

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

    4. sub-neg 86.1%

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

    5. metadata-eval 86.1%

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

15. Simplified 86.1%

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

16. Final simplification 86.1%

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

Alternative 6: 88.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 85.2%

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

   3. *-commutative 85.2%

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

   4. div-inv 85.2%

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

   5. *-commutative 85.2%

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

   6. div-inv 85.2%

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

   7. frac-times 85.2%

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

3. Applied egg-rr 85.2%

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

5. Simplified 85.3%

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

   1. clear-num 85.3%

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

   2. sqrt-div 86.0%

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

   3. metadata-eval 86.0%

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

   4. associate-*r* 86.0%

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

   5. *-commutative 86.0%

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

7. Applied egg-rr 86.0%

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

   1. inv-pow 86.0%

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

   2. sqrt-pow2 86.1%

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

   3. associate-*r* 86.1%

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

   4. *-commutative 86.1%

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

   5. *-commutative 86.1%

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

   6. *-commutative 86.1%

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

   7. metadata-eval 86.1%

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

9. Applied egg-rr 86.1%

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

    1. rem-square-sqrt 86.0%

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

    2. rem-square-sqrt 86.1%

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

    3. *-commutative 86.1%

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

    4. associate-*l* 86.1%

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

    5. *-commutative 86.1%

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

11. Simplified 86.1%

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

12. Final simplification 86.1%

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

Alternative 7: 88.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 85.2%

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

   3. *-commutative 85.2%

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

   4. div-inv 85.2%

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

   5. *-commutative 85.2%

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

   6. div-inv 85.2%

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

   7. frac-times 85.2%

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

3. Applied egg-rr 85.2%

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

5. Simplified 85.3%

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

6. Final simplification 85.3%

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

Alternative 8: 38.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 85.2%

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

   3. *-commutative 85.2%

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

   4. div-inv 85.2%

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

   5. *-commutative 85.2%

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

   6. div-inv 85.2%

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

   7. frac-times 85.2%

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

3. Applied egg-rr 85.2%

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

5. Simplified 85.3%

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

   1. clear-num 85.3%

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

   2. sqrt-div 86.0%

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

   3. metadata-eval 86.0%

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

   4. associate-*r* 86.0%

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

   5. *-commutative 86.0%

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

7. Applied egg-rr 86.0%

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

   1. inv-pow 86.0%

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

   2. sqrt-pow2 86.1%

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

   3. associate-*r* 86.1%

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

   4. *-commutative 86.1%

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

   5. *-commutative 86.1%

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

   6. *-commutative 86.1%

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

   7. metadata-eval 86.1%

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

9. Applied egg-rr 86.1%

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

    1. rem-square-sqrt 86.0%

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

    2. rem-square-sqrt 86.1%

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

    3. *-commutative 86.1%

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

    4. associate-*l* 86.1%

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

    5. *-commutative 86.1%

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

11. Simplified 86.1%

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

12. Taylor expanded in k around 0 32.8%

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

    1. associate-*r* 32.8%

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

    2. *-commutative 32.8%

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

    3. *-commutative 32.8%

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

14. Simplified 32.8%

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

15. Final simplification 32.8%

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

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 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 85.2%

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

   3. *-commutative 85.2%

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

   4. div-inv 85.2%

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

   5. *-commutative 85.2%

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

   6. div-inv 85.2%

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

   7. frac-times 85.2%

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

3. Applied egg-rr 85.2%

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

5. Simplified 85.3%

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

   1. pow-sub 85.5%

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

   2. pow1 85.5%

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

   3. associate-*r* 85.5%

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

   4. *-commutative 85.5%

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

   5. associate-*r* 85.5%

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

   6. *-commutative 85.5%

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

7. Applied egg-rr 85.5%

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

8. Taylor expanded in k around 0 32.0%

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

   1. associate-/l* 31.9%

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

10. Simplified 31.9%

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

11. Final simplification 31.9%

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

Alternative 10: 37.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

function code(k, n)
	return sqrt(Float64(2.0 * Float64(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[(2.0 * N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 85.2%

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

   3. *-commutative 85.2%

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

   4. div-inv 85.2%

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

   5. *-commutative 85.2%

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

   6. div-inv 85.2%

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

   7. frac-times 85.2%

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

3. Applied egg-rr 85.2%

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

5. Simplified 85.3%

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

   1. pow-sub 85.5%

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

   2. pow1 85.5%

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

   3. associate-*r* 85.5%

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

   4. *-commutative 85.5%

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

   5. associate-*r* 85.5%

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

   6. *-commutative 85.5%

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

7. Applied egg-rr 85.5%

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

8. Taylor expanded in k around 0 32.0%

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

9. Final simplification 32.0%

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

Reproduce

herbie shell --seed 2023298 
(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))))