Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 34.5s

Alternatives: 8

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(2.0 * N[(Pi * n), $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/ 99.6%

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

   2. *-un-lft-identity 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

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. Final simplification 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 6.2e-45)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (sqrt (/ (pow (* 2.0 (* PI n)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 6.2e-45) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt((pow((2.0 * (((double) M_PI) * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 6.2e-45) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.sqrt((Math.pow((2.0 * (Math.PI * n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 6.2e-45:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.sqrt((math.pow((2.0 * (math.pi * n)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 6.2e-45)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt(Float64((Float64(2.0 * Float64(pi * n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 6.2e-45)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = sqrt((((2.0 * (pi * n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 6.2e-45], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 6.2000000000000002e-45

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 98.9%

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

      2. sqrt-unprod 76.5%

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

      3. *-commutative 76.5%

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

      4. associate-*r* 76.5%

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

      5. div-sub 76.5%

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

      6. metadata-eval 76.5%

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

      7. div-inv 76.6%

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

      8. *-commutative 76.6%

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

   4. Applied egg-rr 76.6%

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

   6. Simplified 76.9%

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

   7. Taylor expanded in k around 0 76.9%

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

      1. pow1/2 76.9%

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

      2. associate-/l* 76.8%

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

      3. associate-*r* 76.8%

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

      4. *-commutative 76.8%

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

      5. metadata-eval 76.8%

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

      6. associate-*l* 76.8%

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

      7. unpow-prod-down 99.5%

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

      8. metadata-eval 99.5%

         \[\leadsto {n}^{\color{blue}{0.5}} \cdot {\left(2 \cdot \frac{\pi}{k}\right)}^{\left(0.25 \cdot 2\right)} \]

      9. pow1/2 99.5%

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

      10. metadata-eval 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. unpow1/2 99.5%

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

   11. Simplified 99.5%

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

   if 6.2000000000000002e-45 < k

   1. Initial program 99.7%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*r* 99.7%

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

      5. div-sub 99.7%

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

      6. metadata-eval 99.7%

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

      7. div-inv 99.7%

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

      8. *-commutative 99.7%

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

   4. Applied egg-rr 99.7%

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

   6. Simplified 99.7%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6.2 \cdot 10^{-45}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{+186}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8.2e+186)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (pow (* (pow (* n (/ PI k)) 2.0) 4.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 8.2e+186) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow((pow((n * (((double) M_PI) / k)), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 8.2e+186) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow((Math.pow((n * (Math.PI / k)), 2.0) * 4.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 8.2e+186:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow((math.pow((n * (math.pi / k)), 2.0) * 4.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 8.2e+186)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64((Float64(n * Float64(pi / k)) ^ 2.0) * 4.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 8.2e+186)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = (((n * (pi / k)) ^ 2.0) * 4.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 8.2e+186], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[Power[N[(n * N[(Pi / k), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] * 4.0), $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 8.2e186

   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. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 86.8%

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

      3. *-commutative 86.8%

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

      4. associate-*r* 86.8%

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

      5. div-sub 86.8%

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

      6. metadata-eval 86.8%

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

      7. div-inv 86.8%

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

      8. *-commutative 86.8%

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

   4. Applied egg-rr 86.9%

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

   6. Simplified 87.0%

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

   7. Taylor expanded in k around 0 51.8%

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

      1. pow1/2 51.8%

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

      2. associate-/l* 51.8%

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

      3. associate-*r* 51.8%

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

      4. *-commutative 51.8%

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

      5. metadata-eval 51.8%

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

      6. associate-*l* 51.8%

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

      7. unpow-prod-down 64.2%

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

      8. metadata-eval 64.2%

         \[\leadsto {n}^{\color{blue}{0.5}} \cdot {\left(2 \cdot \frac{\pi}{k}\right)}^{\left(0.25 \cdot 2\right)} \]

      9. pow1/2 64.2%

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

      10. metadata-eval 64.2%

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

   9. Applied egg-rr 64.2%

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

       1. unpow1/2 64.3%

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

   11. Simplified 64.3%

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

   if 8.2e186 < k

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. div-sub 100.0%

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

      6. metadata-eval 100.0%

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

      7. div-inv 100.0%

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

      8. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in k around 0 2.7%

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

      1. pow1/2 2.7%

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

      2. associate-/l* 2.7%

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

      3. associate-*r* 2.7%

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

      4. *-commutative 2.7%

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

      5. metadata-eval 2.7%

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

      6. pow-sqr 2.7%

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

      7. pow-prod-down 15.0%

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

   9. Applied egg-rr 15.0%

      \[\leadsto \color{blue}{{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.2 \cdot 10^{+186}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \frac{\pi}{k}\right)}^{2} \cdot 4\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(2 \cdot \frac{\pi \cdot n}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.7e+150)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (pow (pow (* 2.0 (/ (* PI n) k)) 3.0) 0.16666666666666666)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.7e+150) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow(pow((2.0 * ((((double) M_PI) * n) / k)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.7e+150) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow(Math.pow((2.0 * ((Math.PI * n) / k)), 3.0), 0.16666666666666666);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.7e+150:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow(math.pow((2.0 * ((math.pi * n) / k)), 3.0), 0.16666666666666666)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.7e+150)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = (Float64(2.0 * Float64(Float64(pi * n) / k)) ^ 3.0) ^ 0.16666666666666666;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.7e+150)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = ((2.0 * ((pi * n) / k)) ^ 3.0) ^ 0.16666666666666666;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.7e+150], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(N[(Pi * n), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision], 0.16666666666666666], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.70000000000000004e150

   1. Initial program 99.3%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 85.9%

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

      3. *-commutative 85.9%

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

      4. associate-*r* 85.9%

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

      5. div-sub 85.9%

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

      6. metadata-eval 85.9%

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

      7. div-inv 86.0%

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

      8. *-commutative 86.0%

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

   4. Applied egg-rr 86.0%

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

   6. Simplified 86.2%

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

   7. Taylor expanded in k around 0 55.1%

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

      1. pow1/2 55.1%

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

      2. associate-/l* 55.0%

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

      3. associate-*r* 55.0%

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

      4. *-commutative 55.0%

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

      5. metadata-eval 55.0%

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

      6. associate-*l* 55.0%

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

      7. unpow-prod-down 68.3%

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

      8. metadata-eval 68.3%

         \[\leadsto {n}^{\color{blue}{0.5}} \cdot {\left(2 \cdot \frac{\pi}{k}\right)}^{\left(0.25 \cdot 2\right)} \]

      9. pow1/2 68.3%

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

      10. metadata-eval 68.3%

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

   9. Applied egg-rr 68.3%

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

       1. unpow1/2 68.3%

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

   11. Simplified 68.3%

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

   if 4.70000000000000004e150 < k

   1. Initial program 100.0%

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

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-*r* 100.0%

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

      5. div-sub 100.0%

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

      6. metadata-eval 100.0%

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

      7. div-inv 100.0%

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

      8. *-commutative 100.0%

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

   4. Applied egg-rr 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in k around 0 2.7%

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

      1. associate-*r/ 2.7%

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

      2. associate-*r* 2.7%

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

      3. *-commutative 2.7%

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

   9. Simplified 2.7%

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

       1. pow1/2 2.7%

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

       2. associate-*r* 2.7%

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

       3. metadata-eval 2.7%

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

       4. pow-pow 6.5%

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

       5. sqr-pow 6.5%

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

       6. pow-prod-down 20.7%

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

       7. pow-prod-up 20.7%

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

       8. metadata-eval 20.7%

          \[\leadsto {\left({\left(n \cdot \left(2 \cdot \frac{\pi}{k}\right)\right)}^{\color{blue}{3}}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       9. associate-*r/ 20.7%

          \[\leadsto {\left({\left(n \cdot \color{blue}{\frac{2 \cdot \pi}{k}}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       10. *-commutative 20.7%

           \[\leadsto {\left({\left(n \cdot \frac{\color{blue}{\pi \cdot 2}}{k}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       11. associate-/l* 20.7%

           \[\leadsto {\left({\left(n \cdot \color{blue}{\left(\pi \cdot \frac{2}{k}\right)}\right)}^{3}\right)}^{\left(\frac{0.3333333333333333}{2}\right)} \]

       12. metadata-eval 20.7%

           \[\leadsto {\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{3}\right)}^{\color{blue}{0.16666666666666666}} \]

   11. Applied egg-rr 20.7%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}} \]
   12. Step-by-step derivation

       1. associate-*r* 20.7%

          \[\leadsto {\left({\color{blue}{\left(\left(n \cdot \pi\right) \cdot \frac{2}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       2. associate-*r/ 20.7%

          \[\leadsto {\left({\color{blue}{\left(\frac{\left(n \cdot \pi\right) \cdot 2}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

       3. associate-*l/ 20.7%

          \[\leadsto {\left({\color{blue}{\left(\frac{n \cdot \pi}{k} \cdot 2\right)}}^{3}\right)}^{0.16666666666666666} \]

       4. *-commutative 20.7%

          \[\leadsto {\left({\color{blue}{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}}^{3}\right)}^{0.16666666666666666} \]

   13. Simplified 20.7%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{3}\right)}^{0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.7 \cdot 10^{+150}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(2 \cdot \frac{\pi \cdot n}{k}\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. associate-*l* 99.6%

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

   4. div-sub 99.6%

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

   5. metadata-eval 99.6%

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

3. Simplified 99.6%

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

5. Final simplification 99.6%

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

Alternative 6: 49.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * 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 90.0%

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

   3. *-commutative 90.0%

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

   4. associate-*r* 90.0%

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

   5. div-sub 90.0%

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

   6. metadata-eval 90.0%

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

   7. div-inv 90.0%

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

   8. *-commutative 90.0%

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

4. Applied egg-rr 90.1%

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

6. Simplified 90.2%

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

7. Taylor expanded in k around 0 39.9%

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

   1. pow1/2 39.9%

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

   2. associate-/l* 39.9%

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

   3. associate-*r* 39.9%

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

   4. *-commutative 39.9%

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

   5. metadata-eval 39.9%

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

   6. associate-*l* 39.9%

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

   7. unpow-prod-down 49.4%

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

   8. metadata-eval 49.4%

      \[\leadsto {n}^{\color{blue}{0.5}} \cdot {\left(2 \cdot \frac{\pi}{k}\right)}^{\left(0.25 \cdot 2\right)} \]

   9. pow1/2 49.4%

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

   10. metadata-eval 49.4%

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

9. Applied egg-rr 49.4%

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

    1. unpow1/2 49.4%

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

11. Simplified 49.4%

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

12. Final simplification 49.4%

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

Alternative 7: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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 90.0%

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

   3. *-commutative 90.0%

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

   4. associate-*r* 90.0%

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

   5. div-sub 90.0%

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

   6. metadata-eval 90.0%

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

   7. div-inv 90.0%

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

   8. *-commutative 90.0%

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

4. Applied egg-rr 90.1%

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

6. Simplified 90.2%

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

7. Taylor expanded in k around 0 39.9%

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

   1. associate-*r/ 39.9%

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

   2. associate-*r* 39.9%

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

   3. *-commutative 39.9%

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

9. Simplified 39.9%

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

    1. associate-*r* 39.9%

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

    2. associate-*r/ 39.9%

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

    3. *-commutative 39.9%

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

    4. associate-*r/ 39.9%

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

    5. associate-*r* 39.9%

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

    6. sqrt-div 49.4%

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

    7. clear-num 49.3%

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

    8. sqrt-undiv 40.6%

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

    9. associate-*r* 40.6%

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

11. Applied egg-rr 40.6%

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

12. Final simplification 40.6%

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

Alternative 8: 38.3% 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 90.0%

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

   3. *-commutative 90.0%

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

   4. associate-*r* 90.0%

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

   5. div-sub 90.0%

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

   6. metadata-eval 90.0%

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

   7. div-inv 90.0%

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

   8. *-commutative 90.0%

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

4. Applied egg-rr 90.1%

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

6. Simplified 90.2%

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

7. Taylor expanded in k around 0 39.9%

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

8. Final simplification 39.9%

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

Reproduce

herbie shell --seed 2024053 
(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))))