Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.6%

Time: 24.3s

Alternatives: 16

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: 99.5% 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: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ {k}^{-0.5} \cdot \frac{\sqrt{t\_0}}{{t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. div-sub 99.5%

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

   2. metadata-eval 99.5%

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

   3. pow-div 99.7%

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

   4. pow1/2 99.7%

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

   5. associate-*r/ 99.7%

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

   6. pow1/2 99.7%

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

   7. pow-flip 99.7%

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

   8. metadata-eval 99.7%

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

   9. associate-*l* 99.7%

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

   10. associate-*l* 99.7%

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

   11. div-inv 99.7%

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

   12. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l* 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. associate-*l* 99.7%

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

   4. *-lft-identity 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. *-commutative 99.7%

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

   8. associate-*l* 99.7%

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

   9. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k} \cdot {t\_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $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. Add Preprocessing
3. 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. *-un-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. unpow-prod-down 72.3%

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

   4. unpow-prod-down 99.5%

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

   5. div-sub 99.5%

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

   6. metadata-eval 99.5%

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

   7. pow-div 99.7%

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

   8. pow1/2 99.7%

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

   9. associate-/l/ 99.7%

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

   10. associate-*l* 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. *-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. *-commutative 99.7%

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

6. Simplified 99.7%

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

7. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \frac{\sqrt{t\_0}}{\sqrt{k \cdot {t\_0}^{k}}} \end{array} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $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 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. pow-sub 87.4%

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

   2. pow1 87.4%

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

8. Applied egg-rr 87.4%

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

   1. associate-/l/ 87.4%

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

   2. *-commutative 87.4%

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

   3. *-commutative 87.4%

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

   4. sqrt-div 99.7%

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

   5. associate-*r* 99.7%

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

   6. associate-*r* 99.7%

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

   7. *-commutative 99.7%

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

   8. *-commutative 99.7%

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

10. Applied egg-rr 99.7%

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

    1. associate-*r* 99.7%

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

    2. *-commutative 99.7%

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

    3. associate-*r* 99.7%

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

    4. *-commutative 99.7%

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

12. Simplified 99.7%

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

13. Final simplification 99.7%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{t\_0}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 5.5e-66)
     (/ (sqrt t_0) (sqrt k))
     (/ 1.0 (sqrt (/ k (pow t_0 (- 1.0 k))))))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 5.5e-66) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = 1.0 / sqrt((k / pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n * 2.0);
	double tmp;
	if (k <= 5.5e-66) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow(t_0, (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	tmp = 0
	if k <= 5.5e-66:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = 1.0 / math.sqrt((k / math.pow(t_0, (1.0 - k))))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	tmp = 0.0
	if (k <= 5.5e-66)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (t_0 ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = 0.0;
	if (k <= 5.5e-66)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = 1.0 / sqrt((k / (t_0 ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.5e-66], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.50000000000000053e-66

   1. Initial program 99.4%

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

      1. associate-/r/ 99.3%

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

      2. associate-*l* 99.3%

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

      3. div-sub 99.3%

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

      4. metadata-eval 99.3%

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

      5. sub-neg 99.3%

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

      6. div-inv 99.3%

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

      7. metadata-eval 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in k around 0 99.4%

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

      1. clear-num 99.4%

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

      2. add-sqr-sqrt 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. times-frac 99.1%

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

      5. sqrt-unprod 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*r* 99.3%

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

      8. *-commutative 99.3%

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

      9. pow1/2 99.3%

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

      10. sqrt-pow1 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-*r* 99.3%

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

      13. *-commutative 99.3%

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

      14. metadata-eval 99.3%

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

   7. Applied egg-rr 99.2%

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

      1. /-rgt-identity 99.2%

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

      2. associate-*r/ 99.2%

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

      3. pow-sqr 99.5%

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

      4. metadata-eval 99.5%

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

      5. unpow1/2 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.5%

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

   9. Simplified 99.5%

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

   if 5.50000000000000053e-66 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. *-commutative 99.0%

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

      8. div-sub 99.0%

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

      9. metadata-eval 99.0%

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

      10. div-inv 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l* 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. metadata-eval 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*r* 99.0%

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

      7. metadata-eval 99.0%

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

      8. mul-1-neg 99.0%

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

      9. sub-neg 99.0%

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

   6. Simplified 99.0%

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

      1. clear-num 99.0%

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

      2. sqrt-div 99.3%

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

      3. metadata-eval 99.3%

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

   8. Applied egg-rr 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(n \cdot 2\right)\\ \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* n 2.0))))
   (if (<= k 5.5e-66)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (n * 2.0);
	double tmp;
	if (k <= 5.5e-66) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = Math.PI * (n * 2.0);
	double tmp;
	if (k <= 5.5e-66) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = math.pi * (n * 2.0)
	tmp = 0
	if k <= 5.5e-66:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(pi * Float64(n * 2.0))
	tmp = 0.0
	if (k <= 5.5e-66)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = pi * (n * 2.0);
	tmp = 0.0;
	if (k <= 5.5e-66)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.5e-66], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $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 < 5.50000000000000053e-66

   1. Initial program 99.4%

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

      1. associate-/r/ 99.3%

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

      2. associate-*l* 99.3%

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

      3. div-sub 99.3%

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

      4. metadata-eval 99.3%

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

      5. sub-neg 99.3%

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

      6. div-inv 99.3%

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

      7. metadata-eval 99.3%

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

      8. distribute-rgt-neg-in 99.3%

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

      9. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

   5. Taylor expanded in k around 0 99.4%

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

      1. clear-num 99.4%

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

      2. add-sqr-sqrt 99.1%

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

      3. *-un-lft-identity 99.1%

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

      4. times-frac 99.1%

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

      5. sqrt-unprod 99.3%

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

      6. *-commutative 99.3%

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

      7. associate-*r* 99.3%

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

      8. *-commutative 99.3%

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

      9. pow1/2 99.3%

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

      10. sqrt-pow1 99.3%

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

      11. *-commutative 99.3%

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

      12. associate-*r* 99.3%

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

      13. *-commutative 99.3%

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

      14. metadata-eval 99.3%

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

   7. Applied egg-rr 99.2%

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

      1. /-rgt-identity 99.2%

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

      2. associate-*r/ 99.2%

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

      3. pow-sqr 99.5%

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

      4. metadata-eval 99.5%

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

      5. unpow1/2 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*r* 99.5%

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

   9. Simplified 99.5%

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

   if 5.50000000000000053e-66 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.0%

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

      3. *-commutative 99.0%

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

      4. div-sub 99.0%

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

      5. metadata-eval 99.0%

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

      6. div-inv 99.0%

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

      7. *-commutative 99.0%

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

      8. div-sub 99.0%

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

      9. metadata-eval 99.0%

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

      10. div-inv 99.0%

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

   4. Applied egg-rr 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-*l* 99.0%

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

      3. distribute-lft-in 99.0%

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

      4. metadata-eval 99.0%

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

      5. *-commutative 99.0%

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

      6. associate-*r* 99.0%

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

      7. metadata-eval 99.0%

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

      8. mul-1-neg 99.0%

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

      9. sub-neg 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[k, -0.5], $MachinePrecision] * N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(0.5 + 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. 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. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.5%

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

   2. metadata-eval 99.5%

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

   3. div-sub 99.5%

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

   4. unpow-prod-down 72.3%

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

   5. unpow-prod-down 99.5%

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

   6. associate-*l* 99.5%

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

   7. div-sub 99.5%

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

   8. metadata-eval 99.5%

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

   9. sub-neg 99.5%

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

   10. div-inv 99.5%

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

   11. metadata-eval 99.5%

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

   12. distribute-rgt-neg-in 99.5%

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

   13. metadata-eval 99.5%

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

   14. pow1/2 99.5%

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

   15. pow-flip 99.5%

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

   16. metadata-eval 99.5%

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

6. 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. Final simplification 99.5%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(n * N[(Pi * 2.0), $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. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 8: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-/r/ 99.4%

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

   2. associate-*l* 99.4%

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

   3. div-sub 99.4%

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

   4. metadata-eval 99.4%

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

   5. sub-neg 99.4%

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

   6. div-inv 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in k around 0 47.9%

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

   1. associate-/r/ 47.9%

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

   2. pow1/2 47.9%

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

   3. pow-flip 47.9%

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

   4. metadata-eval 47.9%

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

   5. sqrt-unprod 48.0%

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

   6. *-commutative 48.0%

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

   7. *-commutative 48.0%

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

7. Applied egg-rr 48.0%

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

8. Final simplification 48.0%

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

Alternative 9: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. pow-sub 87.4%

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

   2. pow1 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

11. Simplified 35.7%

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

    1. associate-*r* 35.7%

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

    2. sqrt-prod 48.0%

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

13. Applied egg-rr 48.0%

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

    1. *-commutative 48.0%

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

15. Simplified 48.0%

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

16. Final simplification 48.0%

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

Alternative 10: 49.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi * N[(n * 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. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r/ 99.4%

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

   2. associate-*l* 99.4%

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

   3. div-sub 99.4%

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

   4. metadata-eval 99.4%

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

   5. sub-neg 99.4%

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

   6. div-inv 99.4%

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

   7. metadata-eval 99.4%

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

   8. distribute-rgt-neg-in 99.4%

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

   9. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in k around 0 47.9%

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

   1. clear-num 47.9%

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

   2. add-sqr-sqrt 47.8%

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

   3. *-un-lft-identity 47.8%

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

   4. times-frac 47.8%

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

   5. sqrt-unprod 47.9%

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

   6. *-commutative 47.9%

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

   7. associate-*r* 47.9%

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

   8. *-commutative 47.9%

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

   9. pow1/2 47.9%

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

   10. sqrt-pow1 47.9%

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

   11. *-commutative 47.9%

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

   12. associate-*r* 47.9%

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

   13. *-commutative 47.9%

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

   14. metadata-eval 47.9%

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

7. Applied egg-rr 47.9%

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

   1. /-rgt-identity 47.9%

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

   2. associate-*r/ 47.8%

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

   3. pow-sqr 48.0%

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

   4. metadata-eval 48.0%

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

   5. unpow1/2 48.0%

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

   6. *-commutative 48.0%

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

   7. associate-*r* 48.0%

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

9. Simplified 48.0%

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

10. Final simplification 48.0%

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

Alternative 11: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. clear-num 87.1%

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

   2. sqrt-div 87.8%

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

   3. metadata-eval 87.8%

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

8. Applied egg-rr 87.8%

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

   1. inv-pow 87.8%

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

   2. sqrt-pow2 87.9%

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

   3. associate-*r* 87.9%

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

   4. *-commutative 87.9%

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

   5. *-commutative 87.9%

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

   6. sub-neg 87.9%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 35.9%

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

   9. sqr-neg 35.9%

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

   10. sqrt-unprod 35.9%

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

   11. add-sqr-sqrt 35.9%

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

   12. metadata-eval 35.9%

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

10. Applied egg-rr 35.9%

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

    1. associate-*r* 35.9%

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

    2. *-commutative 35.9%

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

    3. +-commutative 35.9%

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

12. Simplified 35.9%

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

13. Taylor expanded in k around 0 36.4%

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

14. Final simplification 36.4%

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

Alternative 12: 39.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. clear-num 87.1%

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

   2. sqrt-div 87.8%

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

   3. metadata-eval 87.8%

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

8. Applied egg-rr 87.8%

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

   1. inv-pow 87.8%

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

   2. sqrt-pow2 87.9%

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

   3. associate-*r* 87.9%

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

   4. *-commutative 87.9%

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

   5. *-commutative 87.9%

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

   6. sub-neg 87.9%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 35.9%

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

   9. sqr-neg 35.9%

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

   10. sqrt-unprod 35.9%

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

   11. add-sqr-sqrt 35.9%

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

   12. metadata-eval 35.9%

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

10. Applied egg-rr 35.9%

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

    1. associate-*r* 35.9%

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

    2. *-commutative 35.9%

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

    3. +-commutative 35.9%

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

12. Simplified 35.9%

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

13. Taylor expanded in k around 0 36.4%

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

    1. *-commutative 36.4%

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

    2. associate-/r* 36.5%

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

15. Simplified 36.5%

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

16. Final simplification 36.5%

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

Alternative 13: 38.5% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. pow-sub 87.4%

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

   2. pow1 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

11. Simplified 35.7%

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

12. Final simplification 35.7%

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

Alternative 14: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. pow-sub 87.4%

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

   2. pow1 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-*r/ 35.7%

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

    2. associate-*r* 35.7%

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

    3. associate-*l/ 35.7%

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

    4. associate-/l* 35.7%

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

    5. associate-*l* 35.7%

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

11. Simplified 35.7%

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

12. Final simplification 35.7%

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

Alternative 15: 38.6% accurate, 2.0× 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. Add Preprocessing
3. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. pow-sub 87.4%

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

   2. pow1 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

    1. associate-/l* 35.7%

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

11. Simplified 35.7%

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

    1. clear-num 35.7%

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

    2. un-div-inv 35.7%

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

13. Applied egg-rr 35.7%

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

14. Final simplification 35.7%

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

Alternative 16: 38.6% accurate, 2.0× 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 99.5%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 87.0%

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

   3. *-commutative 87.0%

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

   4. div-sub 87.0%

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

   5. metadata-eval 87.0%

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

   6. div-inv 87.1%

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

   7. *-commutative 87.1%

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

   8. div-sub 87.1%

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

   9. metadata-eval 87.1%

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

   10. div-inv 87.1%

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

4. Applied egg-rr 87.2%

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

   1. *-commutative 87.2%

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

   2. associate-*l* 87.2%

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

   3. distribute-lft-in 87.2%

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

   4. metadata-eval 87.2%

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

   5. *-commutative 87.2%

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

   6. associate-*r* 87.2%

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

   7. metadata-eval 87.2%

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

   8. mul-1-neg 87.2%

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

   9. sub-neg 87.2%

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

6. Simplified 87.2%

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

   1. pow-sub 87.4%

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

   2. pow1 87.4%

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

8. Applied egg-rr 87.4%

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

9. Taylor expanded in k around 0 35.7%

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

10. Final simplification 35.7%

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

Reproduce

herbie shell --seed 2024039 
(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))))