Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.0%

Time: 13.8s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.1e-56)
   (* (sqrt (* 2.0 n)) (sqrt (/ PI k)))
   (sqrt (/ (pow (* n (* 2.0 PI)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.1e-56) {
		tmp = sqrt((2.0 * n)) * sqrt((((double) M_PI) / k));
	} else {
		tmp = sqrt((pow((n * (2.0 * ((double) M_PI))), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.1e-56) {
		tmp = Math.sqrt((2.0 * n)) * Math.sqrt((Math.PI / k));
	} else {
		tmp = Math.sqrt((Math.pow((n * (2.0 * Math.PI)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.1e-56:
		tmp = math.sqrt((2.0 * n)) * math.sqrt((math.pi / k))
	else:
		tmp = math.sqrt((math.pow((n * (2.0 * math.pi)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.1e-56)
		tmp = Float64(sqrt(Float64(2.0 * n)) * sqrt(Float64(pi / k)));
	else
		tmp = sqrt(Float64((Float64(n * Float64(2.0 * pi)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.1e-56)
		tmp = sqrt((2.0 * n)) * sqrt((pi / k));
	else
		tmp = sqrt((((n * (2.0 * pi)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.1e-56], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(2.0 * Pi), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000002e-56

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 74.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 74.9%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{k}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 75.1%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 75.1%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(2 \cdot n\right)}{k}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}{k}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{k} \cdot n\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{2}{k}\right) \cdot n\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right)\right)\right) \]

      9. /-lowering-/.f64 75.0%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right)\right)\right) \]

   9. Applied egg-rr 75.0%

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

       1. *-commutative N/A

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

       2. associate-*l/ N/A

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

       3. *-commutative N/A

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

       4. associate-*l/ N/A

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

       5. associate-/l* N/A

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

       6. sqrt-prod N/A

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

       7. pow1/2 N/A

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

       8. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(n \cdot 2\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{\mathsf{PI}\left(\right)}{k}}\right)}\right) \]

       9. pow1/2 N/A

          \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{n \cdot 2}\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k}}}\right)\right) \]

       10. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(n \cdot 2\right)\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k}}}\right)\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, 2\right)\right), \left(\sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right)}}{k}}\right)\right) \]

       12. sqrt-lowering-sqrt.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, 2\right)\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k}\right)\right)\right) \]

       13. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, 2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), k\right)\right)\right) \]

       14. PI-lowering-PI.f64 99.5%

           \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(n, 2\right)\right), \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right)\right)\right) \]

   11. Applied egg-rr 99.5%

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

   if 1.10000000000000002e-56 < k

   1. Initial program 99.8%

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

      1. associate-*r* N/A

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

      2. div-sub N/A

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

      3. metadata-eval N/A

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

      4. *-commutative N/A

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

      5. pow-sub N/A

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

      6. frac-times N/A

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

      7. *-rgt-identity N/A

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

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)}\right) \]

      9. unpow1/2 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

      10. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left({\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot n\right)}\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

      13. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   4. Applied egg-rr 99.9%

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

      1. pow1/2 N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. metadata-eval N/A

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

      5. associate-/l* N/A

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

      6. associate-/r/ N/A

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

      7. *-commutative N/A

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

      8. unpow-prod-down N/A

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

      9. pow1/2 N/A

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

      10. pow-pow N/A

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

      11. associate-*r* N/A

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

      12. *-commutative N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. associate-/r/ N/A

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

   6. Applied egg-rr 99.8%

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

      1. unpow1/2 N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left({\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)} \cdot \frac{1}{k}\right)\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{{\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left({\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}\right), k\right)\right) \]

      5. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), \left(1 - k\right)\right), k\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right), \left(1 - k\right)\right), k\right)\right) \]

      7. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(n \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(1 - k\right)\right), k\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(2 \cdot \mathsf{PI}\left(\right)\right)\right), \left(1 - k\right)\right), k\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \left(1 - k\right)\right), k\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), 2\right)\right), \left(1 - k\right)\right), k\right)\right) \]

      11. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \left(1 - k\right)\right), k\right)\right) \]

      12. --lowering--.f64 99.9%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), 2\right)\right), \mathsf{\_.f64}\left(1, k\right)\right), k\right)\right) \]

   8. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-56}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. associate-*r* N/A

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

   2. div-sub N/A

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

   3. metadata-eval N/A

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

   4. *-commutative N/A

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

   5. pow-sub N/A

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

   6. frac-times N/A

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

   7. *-rgt-identity N/A

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

   8. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)}\right) \]

   9. unpow1/2 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

   10. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}^{\left(\frac{k}{2}\right)}} \cdot \sqrt{k}\right)\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left({\color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)}}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot n\right)}\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

   13. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right)\right), \left({\left(2 \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot n\right)\right)}^{\left(\frac{k}{2}\right)} \cdot \sqrt{k}\right)\right) \]

4. Applied egg-rr 99.7%

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

5. Final simplification 99.7%

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

Alternative 3: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{k}{\pi \cdot n}\\ \mathbf{if}\;k \leq 5.5 \cdot 10^{+183}:\\ \;\;\;\;{\left(\frac{k}{\pi}\right)}^{-0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_0 \cdot t\_0}{4}\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ k (* PI n))))
   (if (<= k 5.5e+183)
     (* (pow (/ k PI) -0.5) (pow (* 2.0 n) 0.5))
     (pow (/ (* t_0 t_0) 4.0) -0.25))))

C

double code(double k, double n) {
	double t_0 = k / (((double) M_PI) * n);
	double tmp;
	if (k <= 5.5e+183) {
		tmp = pow((k / ((double) M_PI)), -0.5) * pow((2.0 * n), 0.5);
	} else {
		tmp = pow(((t_0 * t_0) / 4.0), -0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = k / (Math.PI * n);
	double tmp;
	if (k <= 5.5e+183) {
		tmp = Math.pow((k / Math.PI), -0.5) * Math.pow((2.0 * n), 0.5);
	} else {
		tmp = Math.pow(((t_0 * t_0) / 4.0), -0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = k / (math.pi * n)
	tmp = 0
	if k <= 5.5e+183:
		tmp = math.pow((k / math.pi), -0.5) * math.pow((2.0 * n), 0.5)
	else:
		tmp = math.pow(((t_0 * t_0) / 4.0), -0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(k / Float64(pi * n))
	tmp = 0.0
	if (k <= 5.5e+183)
		tmp = Float64((Float64(k / pi) ^ -0.5) * (Float64(2.0 * n) ^ 0.5));
	else
		tmp = Float64(Float64(t_0 * t_0) / 4.0) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = k / (pi * n);
	tmp = 0.0;
	if (k <= 5.5e+183)
		tmp = ((k / pi) ^ -0.5) * ((2.0 * n) ^ 0.5);
	else
		tmp = ((t_0 * t_0) / 4.0) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 5.5e+183], N[(N[Power[N[(k / Pi), $MachinePrecision], -0.5], $MachinePrecision] * N[Power[N[(2.0 * n), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision], -0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 5.5e183

   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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 46.2%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{k}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 46.3%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 46.3%

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

      1. sqrt-div N/A

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

      2. sqrt-prod N/A

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

      3. pow1/2 N/A

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

      4. associate-*l/ N/A

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

      5. sqrt-div N/A

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

      6. pow1/2 N/A

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

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{\mathsf{PI}\left(\right)}{k}\right)}^{\frac{1}{2}}\right), \color{blue}{\left({\left(n \cdot 2\right)}^{\frac{1}{2}}\right)}\right) \]

      8. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{1}{\frac{k}{\mathsf{PI}\left(\right)}}\right)}^{\frac{1}{2}}\right), \left({\left(\color{blue}{n} \cdot 2\right)}^{\frac{1}{2}}\right)\right) \]

      9. inv-pow N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left({\left(\frac{k}{\mathsf{PI}\left(\right)}\right)}^{-1}\right)}^{\frac{1}{2}}\right), \left({\left(\color{blue}{n} \cdot 2\right)}^{\frac{1}{2}}\right)\right) \]

      10. pow-pow N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{k}{\mathsf{PI}\left(\right)}\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\right), \left({\color{blue}{\left(n \cdot 2\right)}}^{\frac{1}{2}}\right)\right) \]

      11. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\left({\left(\frac{k}{\mathsf{PI}\left(\right)}\right)}^{\frac{-1}{2}}\right), \left({\left(n \cdot \color{blue}{2}\right)}^{\frac{1}{2}}\right)\right) \]

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\frac{k}{\mathsf{PI}\left(\right)}\right), \frac{-1}{2}\right), \left({\color{blue}{\left(n \cdot 2\right)}}^{\frac{1}{2}}\right)\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{PI}\left(\right)\right), \frac{-1}{2}\right), \left({\left(\color{blue}{n} \cdot 2\right)}^{\frac{1}{2}}\right)\right) \]

      14. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{2}\right), \left({\left(n \cdot 2\right)}^{\frac{1}{2}}\right)\right) \]

      15. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(\left(n \cdot 2\right), \color{blue}{\frac{1}{2}}\right)\right) \]

      16. *-lowering-*.f64 57.1%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{PI.f64}\left(\right)\right), \frac{-1}{2}\right), \mathsf{pow.f64}\left(\mathsf{*.f64}\left(n, 2\right), \frac{1}{2}\right)\right) \]

   9. Applied egg-rr 57.1%

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

   if 5.5e183 < 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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.7%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{k}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 2.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(2 \cdot n\right)}{k}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}{k}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{k} \cdot n\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{2}{k}\right) \cdot n\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right)\right)\right) \]

      9. /-lowering-/.f64 2.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right)\right)\right) \]

   9. Applied egg-rr 2.7%

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

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. inv-pow N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

       9. clear-num N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. sqr-pow N/A

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

       13. pow-prod-down N/A

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

       14. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)} \cdot \frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{2}\right)}\right) \]

   11. Applied egg-rr 16.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{+183}:\\ \;\;\;\;{\left(\frac{k}{\pi}\right)}^{-0.5} \cdot {\left(2 \cdot n\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{k}{\pi \cdot n} \cdot \frac{k}{\pi \cdot n}}{4}\right)}^{-0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right), \color{blue}{\left(\sqrt{k}\right)}\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{\color{blue}{k}}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} - \frac{k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\left(\frac{1}{2}\right), \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   13. sqrt-lowering-sqrt.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

3. Simplified 99.7%

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

5. Taylor expanded in n around -inf

   \[\leadsto \mathsf{/.f64}\left(\color{blue}{\left(e^{\left(\log \left(-2 \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{-1}{n}\right)\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)}\right)}, \mathsf{sqrt.f64}\left(k\right)\right) \]
6. Step-by-step derivation

   1. exp-prod N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(e^{\log \left(-2 \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{-1}{n}\right)}\right)}^{\left(\frac{1}{2} - \frac{1}{2} \cdot k\right)}\right), \mathsf{sqrt.f64}\left(\color{blue}{k}\right)\right) \]

   2. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(e^{\log \left(-2 \cdot \mathsf{PI}\left(\right)\right) + -1 \cdot \log \left(\frac{-1}{n}\right)}\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(\color{blue}{k}\right)\right) \]

   3. mul-1-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(e^{\log \left(-2 \cdot \mathsf{PI}\left(\right)\right) + \left(\mathsf{neg}\left(\log \left(\frac{-1}{n}\right)\right)\right)}\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   4. unsub-neg N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(e^{\log \left(-2 \cdot \mathsf{PI}\left(\right)\right) - \log \left(\frac{-1}{n}\right)}\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   5. exp-diff N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\frac{e^{\log \left(-2 \cdot \mathsf{PI}\left(\right)\right)}}{e^{\log \left(\frac{-1}{n}\right)}}\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(e^{\log \left(-2 \cdot \mathsf{PI}\left(\right)\right)}\right), \left(e^{\log \left(\frac{-1}{n}\right)}\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   7. rem-exp-log N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(-2 \cdot \mathsf{PI}\left(\right)\right), \left(e^{\log \left(\frac{-1}{n}\right)}\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot -2\right), \left(e^{\log \left(\frac{-1}{n}\right)}\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), -2\right), \left(e^{\log \left(\frac{-1}{n}\right)}\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), -2\right), \left(e^{\log \left(\frac{-1}{n}\right)}\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   11. rem-exp-log N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), -2\right), \left(\frac{-1}{n}\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), -2\right), \mathsf{/.f64}\left(-1, n\right)\right), \left(\frac{1}{2} - \frac{1}{2} \cdot k\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   13. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), -2\right), \mathsf{/.f64}\left(-1, n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{1}{2} \cdot k\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), -2\right), \mathsf{/.f64}\left(-1, n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(k \cdot \frac{1}{2}\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

   15. *-lowering-*.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), -2\right), \mathsf{/.f64}\left(-1, n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(k, \frac{1}{2}\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

7. Simplified 99.7%

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

Alternative 5: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{k}{\pi \cdot n}\\ \mathbf{if}\;k \leq 1.4 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_0 \cdot t\_0}{4}\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ k (* PI n))))
   (if (<= k 1.4e+184)
     (* (sqrt n) (sqrt (* PI (/ 2.0 k))))
     (pow (/ (* t_0 t_0) 4.0) -0.25))))

C

double code(double k, double n) {
	double t_0 = k / (((double) M_PI) * n);
	double tmp;
	if (k <= 1.4e+184) {
		tmp = sqrt(n) * sqrt((((double) M_PI) * (2.0 / k)));
	} else {
		tmp = pow(((t_0 * t_0) / 4.0), -0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = k / (Math.PI * n);
	double tmp;
	if (k <= 1.4e+184) {
		tmp = Math.sqrt(n) * Math.sqrt((Math.PI * (2.0 / k)));
	} else {
		tmp = Math.pow(((t_0 * t_0) / 4.0), -0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = k / (math.pi * n)
	tmp = 0
	if k <= 1.4e+184:
		tmp = math.sqrt(n) * math.sqrt((math.pi * (2.0 / k)))
	else:
		tmp = math.pow(((t_0 * t_0) / 4.0), -0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(k / Float64(pi * n))
	tmp = 0.0
	if (k <= 1.4e+184)
		tmp = Float64(sqrt(n) * sqrt(Float64(pi * Float64(2.0 / k))));
	else
		tmp = Float64(Float64(t_0 * t_0) / 4.0) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = k / (pi * n);
	tmp = 0.0;
	if (k <= 1.4e+184)
		tmp = sqrt(n) * sqrt((pi * (2.0 / k)));
	else
		tmp = ((t_0 * t_0) / 4.0) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.4e+184], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision], -0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.39999999999999995e184

   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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 46.2%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 46.2%

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

      1. sqrt-unprod N/A

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

      2. associate-/l* N/A

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

      3. associate-*l* N/A

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

      4. sqrt-prod N/A

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

      5. pow1/2 N/A

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

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({n}^{\frac{1}{2}}\right), \color{blue}{\left(\sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}\right)}\right) \]

      7. pow1/2 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{n}\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}}\right)\right) \]

      8. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \left(\sqrt{\color{blue}{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}}\right)\right) \]

      9. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k} \cdot 2\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{\mathsf{PI}\left(\right)}{k}\right), 2\right)\right)\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI}\left(\right), k\right), 2\right)\right)\right) \]

      12. PI-lowering-PI.f64 57.0%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{PI.f64}\left(\right), k\right), 2\right)\right)\right) \]

   7. Applied egg-rr 57.0%

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

      1. clear-num N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{1}{\frac{k}{\mathsf{PI}\left(\right)}} \cdot 2\right)\right)\right) \]

      2. associate-*l/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{1 \cdot 2}{\frac{k}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{k}{\mathsf{PI}\left(\right)}}\right)\right)\right) \]

      4. associate-/r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\left(\frac{2}{k} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{2}{k}\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

      7. PI-lowering-PI.f64 57.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(n\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

   9. Applied egg-rr 57.1%

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

   if 1.39999999999999995e184 < 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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.7%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{k}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 2.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(2 \cdot n\right)}{k}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}{k}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{k} \cdot n\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{2}{k}\right) \cdot n\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right)\right)\right) \]

      9. /-lowering-/.f64 2.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right)\right)\right) \]

   9. Applied egg-rr 2.7%

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

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. inv-pow N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

       9. clear-num N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. sqr-pow N/A

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

       13. pow-prod-down N/A

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

       14. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)} \cdot \frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{2}\right)}\right) \]

   11. Applied egg-rr 16.1%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{+184}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\pi \cdot \frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{k}{\pi \cdot n} \cdot \frac{k}{\pi \cdot n}}{4}\right)}^{-0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ N/A

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

   2. *-lft-identity N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right), \color{blue}{\left(\sqrt{k}\right)}\right) \]

   4. pow-lowering-pow.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{\color{blue}{k}}\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \left(\mathsf{PI}\left(\right) \cdot n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   8. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1 - k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   9. div-sub N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \left(\frac{1}{2} - \frac{k}{2}\right)\right), \left(\sqrt{k}\right)\right) \]

   10. --lowering--.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\left(\frac{1}{2}\right), \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \left(\frac{k}{2}\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   12. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \left(\sqrt{k}\right)\right) \]

   13. sqrt-lowering-sqrt.f64 99.7%

       \[\leadsto \mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), n\right)\right), \mathsf{\_.f64}\left(\frac{1}{2}, \mathsf{/.f64}\left(k, 2\right)\right)\right), \mathsf{sqrt.f64}\left(k\right)\right) \]

3. Simplified 99.7%

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

Alternative 7: 41.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{k}{\pi \cdot n}\\ \mathbf{if}\;k \leq 1.22 \cdot 10^{+182}:\\ \;\;\;\;{\left(\frac{k}{\pi \cdot \left(2 \cdot n\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{t\_0 \cdot t\_0}{4}\right)}^{-0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (/ k (* PI n))))
   (if (<= k 1.22e+182)
     (pow (/ k (* PI (* 2.0 n))) -0.5)
     (pow (/ (* t_0 t_0) 4.0) -0.25))))

C

double code(double k, double n) {
	double t_0 = k / (((double) M_PI) * n);
	double tmp;
	if (k <= 1.22e+182) {
		tmp = pow((k / (((double) M_PI) * (2.0 * n))), -0.5);
	} else {
		tmp = pow(((t_0 * t_0) / 4.0), -0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = k / (Math.PI * n);
	double tmp;
	if (k <= 1.22e+182) {
		tmp = Math.pow((k / (Math.PI * (2.0 * n))), -0.5);
	} else {
		tmp = Math.pow(((t_0 * t_0) / 4.0), -0.25);
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = k / (math.pi * n)
	tmp = 0
	if k <= 1.22e+182:
		tmp = math.pow((k / (math.pi * (2.0 * n))), -0.5)
	else:
		tmp = math.pow(((t_0 * t_0) / 4.0), -0.25)
	return tmp

Julia

function code(k, n)
	t_0 = Float64(k / Float64(pi * n))
	tmp = 0.0
	if (k <= 1.22e+182)
		tmp = Float64(k / Float64(pi * Float64(2.0 * n))) ^ -0.5;
	else
		tmp = Float64(Float64(t_0 * t_0) / 4.0) ^ -0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = k / (pi * n);
	tmp = 0.0;
	if (k <= 1.22e+182)
		tmp = (k / (pi * (2.0 * n))) ^ -0.5;
	else
		tmp = ((t_0 * t_0) / 4.0) ^ -0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(k / N[(Pi * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 1.22e+182], N[Power[N[(k / N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision], N[Power[N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision], -0.25], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 1.22e182

   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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 46.4%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 46.4%

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

      1. *-commutative N/A

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

      2. sqrt-div N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. inv-pow N/A

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

      6. *-commutative N/A

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

      7. sqrt-prod N/A

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

      8. sqrt-undiv N/A

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

      9. sqrt-pow2 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. pow-lowering-pow.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \color{blue}{\left(-1 \cdot \frac{1}{2}\right)}\right) \]

      13. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{1}{2}\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

      15. associate-*l* N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

      16. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

      17. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

      18. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

      19. metadata-eval 46.8%

          \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right)\right), \frac{-1}{2}\right) \]

   7. Applied egg-rr 46.8%

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

   if 1.22e182 < 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. Taylor expanded in k around 0

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      5. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 2.7%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

   5. Simplified 2.7%

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

      1. sqrt-unprod N/A

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

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{k}\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

      8. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      10. PI-lowering-PI.f64 N/A

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

      11. *-lowering-*.f64 2.7%

          \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

   7. Applied egg-rr 2.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(2 \cdot n\right)}{k}\right)\right) \]

      2. associate-*r* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}{k}\right)\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{k} \cdot n\right)\right) \]

      4. associate-/l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{2}{k}\right) \cdot n\right)\right) \]

      5. associate-*l* N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      7. PI-lowering-PI.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right)\right)\right) \]

      9. /-lowering-/.f64 2.7%

         \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right)\right)\right) \]

   9. Applied egg-rr 2.7%

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

       1. pow1/2 N/A

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

       2. metadata-eval N/A

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

       3. pow-pow N/A

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

       4. inv-pow N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. clear-num N/A

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

       8. un-div-inv N/A

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

       9. clear-num N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. sqr-pow N/A

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

       13. pow-prod-down N/A

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

       14. pow-lowering-pow.f64 N/A

           \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)} \cdot \frac{k}{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}\right), \color{blue}{\left(\frac{\frac{-1}{2}}{2}\right)}\right) \]

   11. Applied egg-rr 15.8%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.22 \cdot 10^{+182}:\\ \;\;\;\;{\left(\frac{k}{\pi \cdot \left(2 \cdot n\right)}\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{\frac{k}{\pi \cdot n} \cdot \frac{k}{\pi \cdot n}}{4}\right)}^{-0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 36.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 36.5%

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

   1. *-commutative N/A

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

   2. sqrt-div N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. inv-pow N/A

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

   6. *-commutative N/A

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

   7. sqrt-prod N/A

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

   8. sqrt-undiv N/A

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

   9. sqrt-pow2 N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. pow-lowering-pow.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\left(\frac{k}{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}\right), \color{blue}{\left(-1 \cdot \frac{1}{2}\right)}\right) \]

   13. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right)\right), \left(\color{blue}{-1} \cdot \frac{1}{2}\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   15. associate-*l* N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   16. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   17. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   18. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right)\right), \left(-1 \cdot \frac{1}{2}\right)\right) \]

   19. metadata-eval 36.8%

       \[\leadsto \mathsf{pow.f64}\left(\mathsf{/.f64}\left(k, \mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right)\right), \frac{-1}{2}\right) \]

7. Applied egg-rr 36.8%

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

8. Final simplification 36.8%

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

Alternative 9: 37.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 36.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 36.5%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{k}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   11. *-lowering-*.f64 36.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

7. Applied egg-rr 36.6%

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

8. Final simplification 36.6%

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

Alternative 10: 37.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in k around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}\right), \color{blue}{\left(\sqrt{2}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k}\right)\right), \left(\sqrt{\color{blue}{2}}\right)\right) \]

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(n \cdot \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   5. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \left(\sqrt{2}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 36.5%

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(n, \mathsf{PI.f64}\left(\right)\right), k\right)\right), \mathsf{sqrt.f64}\left(2\right)\right) \]

5. Simplified 36.5%

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

   1. sqrt-unprod N/A

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

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2}{k}\right)\right) \]

   4. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}{k}\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)}{k}\right)\right) \]

   6. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot n\right)\right), k\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\right), k\right)\right) \]

   8. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right), k\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   10. PI-lowering-PI.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(n \cdot 2\right)\right), k\right)\right) \]

   11. *-lowering-*.f64 36.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(n, 2\right)\right), k\right)\right) \]

7. Applied egg-rr 36.6%

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

   1. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot \left(2 \cdot n\right)}{k}\right)\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}{k}\right)\right) \]

   3. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{\mathsf{PI}\left(\right) \cdot 2}{k} \cdot n\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\mathsf{PI}\left(\right) \cdot \frac{2}{k}\right) \cdot n\right)\right) \]

   5. associate-*l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\mathsf{PI}\left(\right) \cdot \left(\frac{2}{k} \cdot n\right)\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

   7. PI-lowering-PI.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \left(\frac{2}{k} \cdot n\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\left(\frac{2}{k}\right), n\right)\right)\right) \]

   9. /-lowering-/.f64 36.6%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{PI.f64}\left(\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(2, k\right), n\right)\right)\right) \]

9. Applied egg-rr 36.6%

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

    1. *-commutative N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(\frac{2}{k} \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

    2. associate-*l/ N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2 \cdot n}{k} \cdot \mathsf{PI}\left(\right)\right)\right) \]

    3. associate-/l* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(\left(2 \cdot \frac{n}{k}\right) \cdot \mathsf{PI}\left(\right)\right)\right) \]

    4. associate-*l* N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \left(\frac{n}{k} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    5. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \left(\frac{n}{k} \cdot \mathsf{PI}\left(\right)\right)\right)\right) \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\left(\frac{n}{k}\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

    7. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(n, k\right), \mathsf{PI}\left(\right)\right)\right)\right) \]

    8. PI-lowering-PI.f64 36.6%

       \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(2, \mathsf{*.f64}\left(\mathsf{/.f64}\left(n, k\right), \mathsf{PI.f64}\left(\right)\right)\right)\right) \]

11. Applied egg-rr 36.6%

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

12. Final simplification 36.6%

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

Reproduce

herbie shell --seed 2024162 
(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))))