Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.6%

Time: 24.9s

Alternatives: 14

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: 99.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.3%

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

   2. sqrt-unprod 86.7%

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

   3. *-commutative 86.7%

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

   4. div-sub 86.7%

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

   5. metadata-eval 86.7%

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

   6. div-inv 86.7%

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

   7. *-commutative 86.7%

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

   8. div-sub 86.7%

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

   9. metadata-eval 86.7%

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

   10. div-inv 86.7%

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

4. Applied egg-rr 86.8%

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

   1. sqr-pow 86.6%

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

   2. pow-sqr 86.8%

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

   3. associate-*r* 86.8%

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

   4. associate-/l* 86.8%

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

   5. metadata-eval 86.8%

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

   6. associate-*r* 86.8%

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

   7. *-commutative 86.8%

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

   8. associate-*r* 86.8%

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

   9. metadata-eval 86.8%

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

   10. associate-*r* 86.8%

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

   11. metadata-eval 86.8%

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

   12. *-lft-identity 86.8%

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

   13. *-commutative 86.8%

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

   14. associate-*l* 86.8%

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

6. Simplified 86.8%

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

   1. sqrt-div 99.5%

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

   2. pow1/2 99.5%

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

   3. pow-unpow 99.5%

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

   4. *-commutative 99.5%

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

   5. pow-unpow 99.5%

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

   6. *-commutative 99.5%

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

   7. associate-*r* 99.5%

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

   8. pow1/2 99.5%

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

   9. pow-div 99.7%

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

   10. pow1 99.7%

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

   11. pow1/2 99.7%

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

   12. pow-unpow 99.7%

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

   13. associate-/r* 99.7%

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

8. Applied egg-rr 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{-62}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.36e-62)
   (* (pow k -0.5) (sqrt (* n (* PI 2.0))))
   (pow (/ k (pow (* PI (* n 2.0)) (- 1.0 k))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.36e-62) {
		tmp = pow(k, -0.5) * sqrt((n * (((double) M_PI) * 2.0)));
	} else {
		tmp = pow((k / pow((((double) M_PI) * (n * 2.0)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.36e-62) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (Math.PI * 2.0)));
	} else {
		tmp = Math.pow((k / Math.pow((Math.PI * (n * 2.0)), (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.36e-62:
		tmp = math.pow(k, -0.5) * math.sqrt((n * (math.pi * 2.0)))
	else:
		tmp = math.pow((k / math.pow((math.pi * (n * 2.0)), (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.36e-62)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(pi * 2.0))));
	else
		tmp = Float64(k / (Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.36e-62)
		tmp = (k ^ -0.5) * sqrt((n * (pi * 2.0)));
	else
		tmp = (k / ((pi * (n * 2.0)) ^ (1.0 - k))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.36e-62], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.35999999999999999e-62

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 99.3%

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

      1. pow1 99.3%

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

      2. associate-*l/ 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. sqrt-unprod 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. sqrt-undiv 67.5%

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

   5. Applied egg-rr 67.5%

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

      1. unpow1 67.5%

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

      2. associate-/l* 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-/l* 67.5%

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

   7. Simplified 67.5%

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

      1. sqrt-prod 67.1%

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

      2. associate-*r/ 67.1%

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

      3. *-commutative 67.1%

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

      4. sqrt-div 99.0%

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

      5. associate-/l* 99.1%

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

      6. sqrt-prod 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. associate-*l/ 99.4%

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

      9. pow1/2 99.4%

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

      10. pow-flip 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-*l* 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 1.35999999999999999e-62 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. div-sub 99.5%

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

      5. metadata-eval 99.5%

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

      6. div-inv 99.5%

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

      7. *-commutative 99.5%

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

      8. div-sub 99.5%

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

      9. metadata-eval 99.5%

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

      10. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. sqr-pow 99.5%

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

      2. pow-sqr 99.5%

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

      3. associate-*r* 99.5%

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

      4. associate-/l* 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*r* 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r* 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-lft-identity 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-*l* 99.5%

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

   6. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. inv-pow 99.5%

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

      2. sqrt-pow2 99.5%

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

      3. metadata-eval 99.5%

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

   10. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.36 \cdot 10^{-62}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-62}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.5e-62)
   (* (pow k -0.5) (sqrt (* n (* PI 2.0))))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.5e-62) {
		tmp = pow(k, -0.5) * sqrt((n * (((double) M_PI) * 2.0)));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.5e-62) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (Math.PI * 2.0)));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.5e-62:
		tmp = math.pow(k, -0.5) * math.sqrt((n * (math.pi * 2.0)))
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.5e-62)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(pi * 2.0))));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.5e-62)
		tmp = (k ^ -0.5) * sqrt((n * (pi * 2.0)));
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.5e-62], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.5000000000000001e-62

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 99.3%

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

      1. pow1 99.3%

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

      2. associate-*l/ 99.3%

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

      3. *-un-lft-identity 99.3%

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

      4. sqrt-unprod 99.4%

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

      5. *-commutative 99.4%

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

      6. associate-*r* 99.4%

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

      7. sqrt-undiv 67.5%

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

   5. Applied egg-rr 67.5%

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

      1. unpow1 67.5%

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

      2. associate-/l* 67.5%

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

      3. *-commutative 67.5%

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

      4. associate-/l* 67.5%

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

   7. Simplified 67.5%

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

      1. sqrt-prod 67.1%

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

      2. associate-*r/ 67.1%

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

      3. *-commutative 67.1%

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

      4. sqrt-div 99.0%

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

      5. associate-/l* 99.1%

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

      6. sqrt-prod 99.4%

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

      7. *-un-lft-identity 99.4%

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

      8. associate-*l/ 99.4%

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

      9. pow1/2 99.4%

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

      10. pow-flip 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-commutative 99.5%

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

      13. associate-*l* 99.5%

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

   9. Applied egg-rr 99.5%

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

   if 1.5000000000000001e-62 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. div-sub 99.5%

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

      5. metadata-eval 99.5%

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

      6. div-inv 99.5%

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

      7. *-commutative 99.5%

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

      8. div-sub 99.5%

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

      9. metadata-eval 99.5%

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

      10. div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. sqr-pow 99.5%

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

      2. pow-sqr 99.5%

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

      3. associate-*r* 99.5%

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

      4. associate-/l* 99.5%

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

      5. metadata-eval 99.5%

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

      6. associate-*r* 99.5%

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

      7. *-commutative 99.5%

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

      8. associate-*r* 99.5%

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

      9. metadata-eval 99.5%

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

      10. associate-*r* 99.5%

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

      11. metadata-eval 99.5%

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

      12. *-lft-identity 99.5%

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

      13. *-commutative 99.5%

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

      14. associate-*l* 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.5 \cdot 10^{-62}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, 2 \cdot \frac{\pi}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.4e+49)
   (* (pow k -0.5) (sqrt (* n (* PI 2.0))))
   (sqrt (+ -1.0 (fma n (* 2.0 (/ PI k)) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.4e+49) {
		tmp = pow(k, -0.5) * sqrt((n * (((double) M_PI) * 2.0)));
	} else {
		tmp = sqrt((-1.0 + fma(n, (2.0 * (((double) M_PI) / k)), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.4e+49)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(pi * 2.0))));
	else
		tmp = sqrt(Float64(-1.0 + fma(n, Float64(2.0 * Float64(pi / k)), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.4e+49], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(n * N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.3999999999999999e49

   1. Initial program 99.0%

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

   3. Taylor expanded in k around 0 84.4%

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

      1. pow1 84.4%

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

      2. associate-*l/ 84.4%

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

      3. *-un-lft-identity 84.4%

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

      4. sqrt-unprod 84.5%

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

      5. *-commutative 84.5%

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

      6. associate-*r* 84.5%

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

      7. sqrt-undiv 62.7%

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

   5. Applied egg-rr 62.7%

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

      1. unpow1 62.7%

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

      2. associate-/l* 62.7%

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

      3. *-commutative 62.7%

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

      4. associate-/l* 62.7%

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

   7. Simplified 62.7%

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

      1. sqrt-prod 62.4%

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

      2. associate-*r/ 62.4%

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

      3. *-commutative 62.4%

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

      4. sqrt-div 84.2%

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

      5. associate-/l* 84.3%

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

      6. sqrt-prod 84.5%

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

      7. *-un-lft-identity 84.5%

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

      8. associate-*l/ 84.5%

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

      9. pow1/2 84.5%

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

      10. pow-flip 84.6%

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

      11. metadata-eval 84.6%

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

      12. *-commutative 84.6%

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

      13. associate-*l* 84.6%

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

   9. Applied egg-rr 84.6%

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

   if 1.3999999999999999e49 < 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 2.8%

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

      1. pow1 2.8%

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

      2. associate-*l/ 2.8%

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

      3. *-un-lft-identity 2.8%

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

      4. sqrt-unprod 2.8%

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

      5. *-commutative 2.8%

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

      6. associate-*r* 2.8%

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

      7. sqrt-undiv 2.7%

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

   5. Applied egg-rr 2.7%

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

      1. unpow1 2.7%

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

      2. associate-/l* 2.7%

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

      3. *-commutative 2.7%

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

      4. associate-/l* 2.7%

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

   7. Simplified 2.7%

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

      1. expm1-log1p-u 2.7%

         \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)\right)}} \]

      2. expm1-undefine 32.3%

         \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(\pi \cdot \left(2 \cdot \frac{n}{k}\right)\right)} - 1}} \]

      3. *-commutative 32.3%

         \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{\left(2 \cdot \frac{n}{k}\right) \cdot \pi}\right)} - 1} \]

      4. associate-*l* 32.3%

         \[\leadsto \sqrt{e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(\frac{n}{k} \cdot \pi\right)}\right)} - 1} \]

   9. Applied egg-rr 32.3%

      \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)} - 1}} \]
   10. Step-by-step derivation

       1. sub-neg 32.3%

          \[\leadsto \sqrt{\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)} + \left(-1\right)}} \]

       2. metadata-eval 32.3%

          \[\leadsto \sqrt{e^{\mathsf{log1p}\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)} + \color{blue}{-1}} \]

       3. +-commutative 32.3%

          \[\leadsto \sqrt{\color{blue}{-1 + e^{\mathsf{log1p}\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}}} \]

       4. log1p-undefine 32.3%

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

       5. rem-exp-log 32.3%

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

       6. +-commutative 32.3%

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

       7. *-commutative 32.3%

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

       8. associate-*l/ 32.3%

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

       9. associate-/l* 32.3%

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

       10. associate-*l* 32.3%

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

       11. fma-define 32.3%

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

   11. Simplified 32.3%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.4 \cdot 10^{+49}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(n, 2 \cdot \frac{\pi}{k}, 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+234}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.4e+234)
   (/ (sqrt (* PI (* n 2.0))) (sqrt k))
   (cbrt (pow (* 2.0 (* PI (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.4e+234) {
		tmp = sqrt((((double) M_PI) * (n * 2.0))) / sqrt(k);
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.4e+234) {
		tmp = Math.sqrt((Math.PI * (n * 2.0))) / Math.sqrt(k);
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.4e+234)
		tmp = Float64(sqrt(Float64(pi * Float64(n * 2.0))) / sqrt(k));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.4e+234], N[(N[Sqrt[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.40000000000000015e234

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 57.1%

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

      1. associate-*l/ 57.1%

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

      2. *-un-lft-identity 57.1%

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

      3. sqrt-unprod 57.1%

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

      4. *-commutative 57.1%

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

      5. associate-*r* 57.1%

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

   5. Applied egg-rr 57.1%

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

   if 4.40000000000000015e234 < 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 2.9%

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

      1. pow1 2.9%

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

      2. associate-*l/ 2.9%

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

      3. *-un-lft-identity 2.9%

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

      4. sqrt-unprod 2.9%

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

      5. *-commutative 2.9%

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

      6. associate-*r* 2.9%

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

      7. sqrt-undiv 2.8%

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

   5. Applied egg-rr 2.8%

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

      1. unpow1 2.8%

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

      2. associate-/l* 2.8%

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

      3. *-commutative 2.8%

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

      4. associate-/l* 2.8%

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

   7. Simplified 2.8%

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

      1. add-cbrt-cube 20.7%

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

      2. add-sqr-sqrt 20.7%

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

      3. pow1 20.7%

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

      4. pow1/2 20.7%

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

      5. pow-prod-up 20.7%

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

      6. *-commutative 20.7%

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

      7. associate-*l* 20.7%

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

      8. metadata-eval 20.7%

         \[\leadsto \sqrt[3]{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\color{blue}{1.5}}} \]

   9. Applied egg-rr 20.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{+234}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 50.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+236}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7.2e+236)
   (* (pow k -0.5) (sqrt (* n (* PI 2.0))))
   (cbrt (pow (* 2.0 (* PI (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 7.2e+236) {
		tmp = pow(k, -0.5) * sqrt((n * (((double) M_PI) * 2.0)));
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 7.2e+236) {
		tmp = Math.pow(k, -0.5) * Math.sqrt((n * (Math.PI * 2.0)));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 7.2e+236)
		tmp = Float64((k ^ -0.5) * sqrt(Float64(n * Float64(pi * 2.0))));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 7.2e+236], N[(N[Power[k, -0.5], $MachinePrecision] * N[Sqrt[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 7.1999999999999999e236

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0 57.1%

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

      1. pow1 57.1%

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

      2. associate-*l/ 57.1%

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

      3. *-un-lft-identity 57.1%

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

      4. sqrt-unprod 57.1%

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

      5. *-commutative 57.1%

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

      6. associate-*r* 57.1%

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

      7. sqrt-undiv 42.6%

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

   5. Applied egg-rr 42.6%

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

      1. unpow1 42.6%

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

      2. associate-/l* 42.6%

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

      3. *-commutative 42.6%

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

      4. associate-/l* 42.6%

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

   7. Simplified 42.6%

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

      1. sqrt-prod 42.4%

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

      2. associate-*r/ 42.4%

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

      3. *-commutative 42.4%

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

      4. sqrt-div 57.0%

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

      5. associate-/l* 57.0%

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

      6. sqrt-prod 57.1%

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

      7. *-un-lft-identity 57.1%

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

      8. associate-*l/ 57.1%

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

      9. pow1/2 57.1%

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

      10. pow-flip 57.2%

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

      11. metadata-eval 57.2%

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

      12. *-commutative 57.2%

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

      13. associate-*l* 57.2%

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

   9. Applied egg-rr 57.2%

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

   if 7.1999999999999999e236 < 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 2.9%

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

      1. pow1 2.9%

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

      2. associate-*l/ 2.9%

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

      3. *-un-lft-identity 2.9%

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

      4. sqrt-unprod 2.9%

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

      5. *-commutative 2.9%

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

      6. associate-*r* 2.9%

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

      7. sqrt-undiv 2.8%

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

   5. Applied egg-rr 2.8%

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

      1. unpow1 2.8%

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

      2. associate-/l* 2.8%

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

      3. *-commutative 2.8%

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

      4. associate-/l* 2.8%

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

   7. Simplified 2.8%

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

      1. add-cbrt-cube 20.7%

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

      2. add-sqr-sqrt 20.7%

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

      3. pow1 20.7%

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

      4. pow1/2 20.7%

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

      5. pow-prod-up 20.7%

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

      6. *-commutative 20.7%

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

      7. associate-*l* 20.7%

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

      8. metadata-eval 20.7%

         \[\leadsto \sqrt[3]{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\color{blue}{1.5}}} \]

   9. Applied egg-rr 20.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1.5}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 7.2 \cdot 10^{+236}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{n \cdot \left(\pi \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

   1. div-inv 99.4%

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

   2. metadata-eval 99.4%

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

   3. div-sub 99.4%

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

   4. unpow-prod-down 75.8%

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

   5. unpow-prod-down 99.4%

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

   6. associate-*l* 99.4%

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

   7. div-inv 99.4%

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

   8. metadata-eval 99.4%

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

   9. pow1/2 99.4%

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

   10. pow-flip 99.5%

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

   11. metadata-eval 99.5%

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

6. Applied egg-rr 99.5%

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

7. Final simplification 99.5%

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

Alternative 8: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.5%

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

   2. *-lft-identity 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

3. Simplified 99.5%

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

5. Final simplification 99.5%

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

Alternative 9: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 50.3%

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

   1. associate-*l/ 50.3%

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

   2. *-un-lft-identity 50.3%

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

   3. sqrt-unprod 50.4%

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

   4. *-commutative 50.4%

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

   5. associate-*r* 50.4%

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

5. Applied egg-rr 50.4%

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

6. Final simplification 50.4%

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

Alternative 10: 38.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 50.3%

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

   1. pow1 50.3%

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

   2. associate-*l/ 50.3%

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

   3. *-un-lft-identity 50.3%

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

   4. sqrt-unprod 50.4%

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

   5. *-commutative 50.4%

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

   6. associate-*r* 50.4%

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

   7. sqrt-undiv 37.6%

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

5. Applied egg-rr 37.6%

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

   1. unpow1 37.6%

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

   2. associate-/l* 37.6%

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

   3. *-commutative 37.6%

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

   4. associate-/l* 37.6%

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

7. Simplified 37.6%

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

   1. sqrt-prod 37.5%

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

   2. associate-*r/ 37.5%

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

   3. *-commutative 37.5%

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

   4. sqrt-div 50.2%

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

   5. associate-/l* 50.2%

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

   6. sqrt-prod 50.4%

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

   7. clear-num 50.3%

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

   8. inv-pow 50.3%

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

   9. sqrt-undiv 38.4%

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

   10. sqrt-pow2 38.5%

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

   11. *-commutative 38.5%

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

   12. associate-*l* 38.5%

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

   13. metadata-eval 38.5%

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

9. Applied egg-rr 38.5%

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

10. Final simplification 38.5%

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

Alternative 11: 37.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 50.3%

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

   1. pow1 50.3%

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

   2. associate-*l/ 50.3%

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

   3. *-un-lft-identity 50.3%

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

   4. sqrt-unprod 50.4%

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

   5. *-commutative 50.4%

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

   6. associate-*r* 50.4%

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

   7. sqrt-undiv 37.6%

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

5. Applied egg-rr 37.6%

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

   1. unpow1 37.6%

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

   2. associate-/l* 37.6%

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

   3. *-commutative 37.6%

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

   4. associate-/l* 37.6%

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

7. Simplified 37.6%

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

8. Taylor expanded in n around 0 37.6%

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

   1. associate-*r/ 37.6%

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

   2. associate-*r* 37.6%

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

   3. *-commutative 37.6%

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

   4. associate-*r* 37.6%

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

   5. associate-/l* 37.6%

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

10. Simplified 37.6%

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

11. Final simplification 37.6%

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

Alternative 12: 37.7% accurate, 2.0× speedup

Math

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

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(pi * Float64(2.0 * Float64(n / k))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 50.3%

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

   1. pow1 50.3%

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

   2. associate-*l/ 50.3%

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

   3. *-un-lft-identity 50.3%

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

   4. sqrt-unprod 50.4%

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

   5. *-commutative 50.4%

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

   6. associate-*r* 50.4%

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

   7. sqrt-undiv 37.6%

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

5. Applied egg-rr 37.6%

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

   1. unpow1 37.6%

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

   2. associate-/l* 37.6%

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

   3. *-commutative 37.6%

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

   4. associate-/l* 37.6%

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

7. Simplified 37.6%

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

8. Final simplification 37.6%

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

Alternative 13: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 50.3%

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

   1. pow1 50.3%

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

   2. associate-*l/ 50.3%

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

   3. *-un-lft-identity 50.3%

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

   4. sqrt-unprod 50.4%

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

   5. *-commutative 50.4%

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

   6. associate-*r* 50.4%

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

   7. sqrt-undiv 37.6%

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

5. Applied egg-rr 37.6%

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

   1. unpow1 37.6%

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

   2. associate-/l* 37.6%

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

   3. *-commutative 37.6%

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

   4. associate-/l* 37.6%

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

7. Simplified 37.6%

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

   1. associate-*r* 37.6%

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

   2. clear-num 37.6%

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

   3. un-div-inv 37.6%

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

   4. *-commutative 37.6%

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

9. Applied egg-rr 37.6%

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

10. Final simplification 37.6%

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

Alternative 14: 37.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 50.3%

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

   1. associate-*l/ 50.3%

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

   2. *-un-lft-identity 50.3%

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

   3. sqrt-unprod 50.4%

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

   4. *-commutative 50.4%

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

   5. associate-*r* 50.4%

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

   6. sqrt-undiv 37.6%

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

5. Applied egg-rr 37.6%

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

6. Final simplification 37.6%

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

Reproduce

herbie shell --seed 2024043 
(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))))