Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 25.5s

Alternatives: 11

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(2 \cdot \pi\right)\\ \mathbf{if}\;k \leq 2.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {t\_0}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* 2.0 PI))))
   (if (<= k 2.4e-26)
     (/ (sqrt t_0) (sqrt k))
     (pow (* k (pow t_0 (+ k -1.0))) -0.5))))

C

double code(double k, double n) {
	double t_0 = n * (2.0 * ((double) M_PI));
	double tmp;
	if (k <= 2.4e-26) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = pow((k * pow(t_0, (k + -1.0))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (2.0 * Math.PI);
	double tmp;
	if (k <= 2.4e-26) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.pow((k * Math.pow(t_0, (k + -1.0))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (2.0 * math.pi)
	tmp = 0
	if k <= 2.4e-26:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.pow((k * math.pow(t_0, (k + -1.0))), -0.5)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(2.0 * pi))
	tmp = 0.0
	if (k <= 2.4e-26)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = Float64(k * (t_0 ^ Float64(k + -1.0))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (2.0 * pi);
	tmp = 0.0;
	if (k <= 2.4e-26)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = (k * (t_0 ^ (k + -1.0))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.4e-26], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[(k * N[Power[t$95$0, N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4000000000000001e-26

   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 99.0%

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

      2. sqrt-unprod 77.3%

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

      3. *-commutative 77.3%

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

      4. associate-*r* 77.3%

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

      5. div-sub 77.3%

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

      6. metadata-eval 77.3%

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

      7. div-inv 77.4%

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

      8. *-commutative 77.4%

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

   4. Applied egg-rr 77.5%

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

   6. Simplified 77.6%

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

      1. pow-sub 77.6%

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

      2. pow1 77.6%

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

      3. *-commutative 77.6%

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

      4. clear-num 77.6%

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

      5. *-commutative 77.6%

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

      6. associate-*r* 77.6%

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

      7. *-commutative 77.6%

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

      8. associate-*r* 77.6%

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

      9. associate-*r* 77.6%

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

      10. *-commutative 77.6%

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

      11. associate-*r* 77.6%

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

   8. Applied egg-rr 77.6%

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

   9. Taylor expanded in k around 0 77.6%

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

       1. sqrt-div 99.5%

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

       2. associate-/r/ 99.5%

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

       3. metadata-eval 99.5%

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

       4. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. associate-*r* 99.5%

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

       2. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 2.4000000000000001e-26 < k

   1. Initial program 99.6%

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

      1. add-sqr-sqrt 99.6%

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

      2. sqrt-unprod 99.6%

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

      3. frac-times 99.6%

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

      4. metadata-eval 99.6%

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

      5. add-sqr-sqrt 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. *-commutative 99.6%

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

      2. sqrt-pow1 99.0%

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

      3. sqrt-prod 99.0%

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

      4. div-inv 99.0%

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

      5. clear-num 99.0%

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

      6. sqrt-div 99.0%

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

      7. metadata-eval 99.0%

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

      8. inv-pow 99.0%

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

      9. sqrt-pow2 99.0%

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

   6. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. *-commutative 99.3%

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

      3. associate-*r* 99.3%

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

      4. *-commutative 99.3%

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

      5. sub-neg 99.3%

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

      6. metadata-eval 99.3%

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

   8. Simplified 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.3e-44)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (sqrt (/ (pow (* (* 2.0 n) PI) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.3e-44) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = sqrt((pow(((2.0 * n) * ((double) M_PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.3e-44) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(((2.0 * n) * Math.PI), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 4.3e-44:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(((2.0 * n) * math.pi), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.3e-44)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * n) * pi) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.3e-44)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = sqrt(((((2.0 * n) * pi) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.3e-44], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * n), $MachinePrecision] * Pi), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.30000000000000013e-44

   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 99.0%

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

      2. sqrt-unprod 75.4%

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

      3. *-commutative 75.4%

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

      4. associate-*r* 75.4%

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

      5. div-sub 75.4%

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

      6. metadata-eval 75.4%

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

      7. div-inv 75.5%

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

      8. *-commutative 75.5%

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

   4. Applied egg-rr 75.6%

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

   6. Simplified 75.7%

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

      1. pow-sub 75.7%

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

      2. pow1 75.7%

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

      3. *-commutative 75.7%

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

      4. clear-num 75.6%

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

      5. *-commutative 75.6%

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

      6. associate-*r* 75.6%

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

      7. *-commutative 75.6%

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

      8. associate-*r* 75.6%

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

      9. associate-*r* 75.6%

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

      10. *-commutative 75.6%

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

      11. associate-*r* 75.6%

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

   8. Applied egg-rr 75.6%

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

   9. Taylor expanded in k around 0 75.6%

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

       1. sqrt-div 99.5%

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

       2. associate-/r/ 99.5%

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

       3. metadata-eval 99.5%

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

       4. *-commutative 99.5%

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

   11. Applied egg-rr 99.5%

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

       1. associate-*r* 99.5%

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

       2. *-commutative 99.5%

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

   13. Simplified 99.5%

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

   if 4.30000000000000013e-44 < k

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. 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 99.0%

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

      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. associate-*r* 99.0%

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

      5. div-sub 99.0%

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

      6. metadata-eval 99.0%

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

      7. div-inv 99.0%

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

      8. *-commutative 99.0%

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

   4. Applied egg-rr 99.0%

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

   6. Simplified 99.0%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.3 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \pi\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.5e+204)
   (/ (sqrt (* n (* 2.0 PI))) (sqrt k))
   (pow (pow (* n (* PI (/ 2.0 k))) 2.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.5e+204) {
		tmp = sqrt((n * (2.0 * ((double) M_PI)))) / sqrt(k);
	} else {
		tmp = pow(pow((n * (((double) M_PI) * (2.0 / k))), 2.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.5e+204) {
		tmp = Math.sqrt((n * (2.0 * Math.PI))) / Math.sqrt(k);
	} else {
		tmp = Math.pow(Math.pow((n * (Math.PI * (2.0 / k))), 2.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.5e+204:
		tmp = math.sqrt((n * (2.0 * math.pi))) / math.sqrt(k)
	else:
		tmp = math.pow(math.pow((n * (math.pi * (2.0 / k))), 2.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.5e+204)
		tmp = Float64(sqrt(Float64(n * Float64(2.0 * pi))) / sqrt(k));
	else
		tmp = (Float64(n * Float64(pi * Float64(2.0 / k))) ^ 2.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.5e+204)
		tmp = sqrt((n * (2.0 * pi))) / sqrt(k);
	else
		tmp = ((n * (pi * (2.0 / k))) ^ 2.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.5e+204], N[(N[Sqrt[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(n * N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.50000000000000004e204

   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 99.2%

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

      2. sqrt-unprod 87.2%

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

      3. *-commutative 87.2%

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

      4. associate-*r* 87.2%

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

      5. div-sub 87.2%

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

      6. metadata-eval 87.2%

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

      7. div-inv 87.2%

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

      8. *-commutative 87.2%

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

   4. Applied egg-rr 87.3%

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

   6. Simplified 87.3%

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

      1. pow-sub 87.5%

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

      2. pow1 87.5%

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

      3. *-commutative 87.5%

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

      4. clear-num 87.5%

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

      5. *-commutative 87.5%

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

      6. associate-*r* 87.5%

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

      7. *-commutative 87.5%

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

      8. associate-*r* 87.5%

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

      9. associate-*r* 87.5%

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

      10. *-commutative 87.5%

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

      11. associate-*r* 87.5%

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

   8. Applied egg-rr 87.5%

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

   9. Taylor expanded in k around 0 45.6%

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

       1. sqrt-div 57.3%

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

       2. associate-/r/ 57.3%

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

       3. metadata-eval 57.3%

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

       4. *-commutative 57.3%

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

   11. Applied egg-rr 57.3%

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

       1. associate-*r* 57.3%

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

       2. *-commutative 57.3%

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

   13. Simplified 57.3%

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

   if 2.50000000000000004e204 < 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%

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

      1. pow-sub 100.0%

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

      2. pow1 100.0%

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

      3. *-commutative 100.0%

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

      4. clear-num 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-*r* 100.0%

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

      7. *-commutative 100.0%

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

      8. associate-*r* 100.0%

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

      9. associate-*r* 100.0%

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

      10. *-commutative 100.0%

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

      11. associate-*r* 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in k around 0 2.5%

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

       1. pow1/2 2.5%

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

       2. associate-/r/ 2.5%

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

       3. metadata-eval 2.5%

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

       4. *-commutative 2.5%

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

       5. associate-*r* 2.5%

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

       6. *-commutative 2.5%

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

       7. metadata-eval 2.5%

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

       8. pow-prod-up 2.5%

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

       9. pow-prod-down 14.4%

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

       10. pow2 14.4%

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

       11. *-commutative 14.4%

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

       12. associate-*r* 14.4%

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

       13. *-commutative 14.4%

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

       14. *-un-lft-identity 14.4%

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

       15. times-frac 14.4%

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

       16. metadata-eval 14.4%

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

       17. *-commutative 14.4%

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

   11. Applied egg-rr 14.4%

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

       1. *-commutative 14.4%

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

       2. *-commutative 14.4%

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

       3. associate-/l* 14.4%

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

       4. associate-*l* 14.4%

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

       5. associate-*l/ 14.4%

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

       6. associate-/l* 14.4%

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

   13. Simplified 14.4%

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

4. Final simplification 49.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.5 \cdot 10^{+204}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   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%

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

Alternative 6: 48.9% 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 99.4%

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

   2. sqrt-unprod 89.6%

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

   3. *-commutative 89.6%

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

   4. associate-*r* 89.6%

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

   5. div-sub 89.6%

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

   6. metadata-eval 89.6%

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

   7. div-inv 89.6%

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

   8. *-commutative 89.6%

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

4. Applied egg-rr 89.7%

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

6. Simplified 89.7%

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

   1. pow-sub 89.9%

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

   2. pow1 89.9%

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

   3. *-commutative 89.9%

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

   4. clear-num 89.9%

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

   5. *-commutative 89.9%

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

   6. associate-*r* 89.9%

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

   7. *-commutative 89.9%

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

   8. associate-*r* 89.9%

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

   9. associate-*r* 89.9%

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

   10. *-commutative 89.9%

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

   11. associate-*r* 89.9%

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

8. Applied egg-rr 89.9%

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

9. Taylor expanded in k around 0 37.5%

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

    1. pow1/2 37.5%

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

    2. associate-/r/ 37.5%

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

    3. metadata-eval 37.5%

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

    4. *-commutative 37.5%

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

    5. associate-*r* 37.5%

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

    6. *-commutative 37.5%

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

    7. associate-/l* 37.5%

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

    8. unpow-prod-down 47.0%

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

    9. pow1/2 47.0%

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

    10. *-un-lft-identity 47.0%

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

    11. times-frac 47.0%

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

    12. metadata-eval 47.0%

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

11. Applied egg-rr 47.0%

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

    1. unpow1/2 47.0%

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

13. Simplified 47.0%

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

14. Final simplification 47.0%

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

Alternative 7: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / 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 99.4%

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

   2. sqrt-unprod 89.6%

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

   3. *-commutative 89.6%

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

   4. associate-*r* 89.6%

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

   5. div-sub 89.6%

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

   6. metadata-eval 89.6%

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

   7. div-inv 89.6%

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

   8. *-commutative 89.6%

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

4. Applied egg-rr 89.7%

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

6. Simplified 89.7%

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

   1. pow-sub 89.9%

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

   2. pow1 89.9%

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

   3. *-commutative 89.9%

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

   4. clear-num 89.9%

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

   5. *-commutative 89.9%

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

   6. associate-*r* 89.9%

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

   7. *-commutative 89.9%

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

   8. associate-*r* 89.9%

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

   9. associate-*r* 89.9%

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

   10. *-commutative 89.9%

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

   11. associate-*r* 89.9%

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

8. Applied egg-rr 89.9%

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

9. Taylor expanded in k around 0 37.5%

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

    1. pow1/2 37.5%

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

    2. associate-/r/ 37.5%

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

    3. metadata-eval 37.5%

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

    4. *-commutative 37.5%

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

    5. associate-*r* 37.5%

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

    6. *-commutative 37.5%

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

    7. associate-/l* 37.5%

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

    8. unpow-prod-down 47.0%

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

    9. pow1/2 47.0%

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

    10. *-un-lft-identity 47.0%

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

    11. times-frac 47.0%

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

    12. metadata-eval 47.0%

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

11. Applied egg-rr 47.0%

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

    1. unpow1/2 47.0%

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

    2. associate-*r/ 47.0%

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

    3. *-commutative 47.0%

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

    4. associate-/l* 47.0%

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

13. Simplified 47.0%

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

14. Final simplification 47.0%

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

Alternative 8: 48.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. add-sqr-sqrt 99.4%

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

   2. sqrt-unprod 89.6%

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

   3. *-commutative 89.6%

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

   4. associate-*r* 89.6%

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

   5. div-sub 89.6%

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

   6. metadata-eval 89.6%

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

   7. div-inv 89.6%

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

   8. *-commutative 89.6%

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

4. Applied egg-rr 89.7%

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

6. Simplified 89.7%

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

   1. pow-sub 89.9%

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

   2. pow1 89.9%

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

   3. *-commutative 89.9%

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

   4. clear-num 89.9%

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

   5. *-commutative 89.9%

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

   6. associate-*r* 89.9%

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

   7. *-commutative 89.9%

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

   8. associate-*r* 89.9%

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

   9. associate-*r* 89.9%

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

   10. *-commutative 89.9%

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

   11. associate-*r* 89.9%

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

8. Applied egg-rr 89.9%

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

9. Taylor expanded in k around 0 37.5%

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

    1. sqrt-div 47.0%

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

    2. associate-/r/ 47.0%

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

    3. metadata-eval 47.0%

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

    4. *-commutative 47.0%

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

11. Applied egg-rr 47.0%

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

    1. associate-*r* 47.0%

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

    2. *-commutative 47.0%

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

13. Simplified 47.0%

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

14. Final simplification 47.0%

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

Alternative 9: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(k * N[(0.5 / N[(n * 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 99.4%

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

   2. sqrt-unprod 89.6%

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

   3. *-commutative 89.6%

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

   4. associate-*r* 89.6%

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

   5. div-sub 89.6%

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

   6. metadata-eval 89.6%

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

   7. div-inv 89.6%

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

   8. *-commutative 89.6%

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

4. Applied egg-rr 89.7%

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

6. Simplified 89.7%

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

   1. pow-sub 89.9%

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

   2. pow1 89.9%

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

   3. *-commutative 89.9%

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

   4. clear-num 89.9%

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

   5. *-commutative 89.9%

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

   6. associate-*r* 89.9%

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

   7. *-commutative 89.9%

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

   8. associate-*r* 89.9%

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

   9. associate-*r* 89.9%

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

   10. *-commutative 89.9%

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

   11. associate-*r* 89.9%

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

8. Applied egg-rr 89.9%

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

9. Taylor expanded in k around 0 37.5%

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

    1. associate-/l/ 37.5%

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

    2. sqrt-div 38.1%

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

    3. metadata-eval 38.1%

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

    4. *-commutative 38.1%

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

11. Applied egg-rr 38.1%

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

12. Final simplification 38.1%

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

Alternative 10: 38.1% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(k * N[(N[(0.5 / n), $MachinePrecision] / 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 99.4%

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

   2. sqrt-unprod 89.6%

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

   3. *-commutative 89.6%

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

   4. associate-*r* 89.6%

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

   5. div-sub 89.6%

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

   6. metadata-eval 89.6%

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

   7. div-inv 89.6%

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

   8. *-commutative 89.6%

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

4. Applied egg-rr 89.7%

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

6. Simplified 89.7%

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

   1. pow-sub 89.9%

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

   2. pow1 89.9%

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

   3. *-commutative 89.9%

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

   4. clear-num 89.9%

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

   5. *-commutative 89.9%

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

   6. associate-*r* 89.9%

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

   7. *-commutative 89.9%

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

   8. associate-*r* 89.9%

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

   9. associate-*r* 89.9%

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

   10. *-commutative 89.9%

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

   11. associate-*r* 89.9%

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

8. Applied egg-rr 89.9%

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

9. Taylor expanded in k around 0 37.5%

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

    1. clear-num 37.5%

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

    2. sqrt-div 38.1%

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

    3. metadata-eval 38.1%

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

    4. *-un-lft-identity 38.1%

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

    5. associate-/r/ 38.1%

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

    6. metadata-eval 38.1%

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

    7. times-frac 38.1%

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

    8. metadata-eval 38.1%

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

    9. *-commutative 38.1%

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

11. Applied egg-rr 38.1%

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

    1. associate-*r/ 38.1%

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

    2. *-commutative 38.1%

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

    3. *-commutative 38.1%

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

    4. associate-*r/ 38.1%

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

    5. associate-/r* 38.1%

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

13. Simplified 38.1%

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

14. Final simplification 38.1%

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

Alternative 11: 37.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(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 99.4%

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

   2. sqrt-unprod 89.6%

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

   3. *-commutative 89.6%

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

   4. associate-*r* 89.6%

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

   5. div-sub 89.6%

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

   6. metadata-eval 89.6%

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

   7. div-inv 89.6%

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

   8. *-commutative 89.6%

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

4. Applied egg-rr 89.7%

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

6. Simplified 89.7%

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

   1. *-commutative 89.7%

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

   2. sub-neg 89.7%

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

   3. unpow-prod-up 89.9%

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

   4. pow1 89.9%

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

   5. associate-*r* 89.9%

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

   6. *-commutative 89.9%

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

   7. associate-*r* 89.9%

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

   8. associate-*r* 89.9%

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

   9. *-commutative 89.9%

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

   10. associate-*r* 89.9%

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

8. Applied egg-rr 89.9%

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

9. Taylor expanded in k around 0 37.5%

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

    1. *-commutative 37.5%

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

    2. associate-/l* 37.5%

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

    3. associate-*l* 37.5%

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

    4. *-commutative 37.5%

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

11. Simplified 37.5%

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

12. Final simplification 37.5%

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

Reproduce

herbie shell --seed 2024044 
(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))))