Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 15.5s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. rem-square-sqrt 99.7%

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

   3. rem-sqrt-square 99.7%

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

   4. fabs-neg 99.7%

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

   5. neg-mul-1 99.7%

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

   6. rem-square-sqrt 0.0%

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

   7. unpow1/2 0.0%

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

   8. metadata-eval 0.0%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {k}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right|} \]

   9. pow-sqr 0.0%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({k}^{0.25} \cdot {k}^{0.25}\right)}\right|} \]

   10. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}\right|} \]

   11. fabs-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}} \]

   12. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)}} \]

   13. rem-square-sqrt 20.9%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{-1} \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)} \]

   14. pow-sqr 20.9%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{-1 \cdot \color{blue}{{k}^{\left(2 \cdot 0.25\right)}}} \]

   15. metadata-eval 20.9%

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

   16. unpow1/2 20.9%

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

   17. neg-mul-1 20.9%

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

   18. associate-/r* 20.9%

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

6. Simplified 99.7%

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

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*r* 99.7%

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

   6. pow1/2 99.7%

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

   7. pow-unpow 99.7%

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

   8. pow-prod-down 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*r* 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. unpow1/2 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

10. Simplified 99.7%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.7e-20)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (pow (* k (pow (* PI (* n 2.0)) (+ k -1.0))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.7e-20) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = pow((k * pow((((double) M_PI) * (n * 2.0)), (k + -1.0))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.7e-20) {
		tmp = Math.sqrt(n) * Math.sqrt((2.0 * (Math.PI / k)));
	} else {
		tmp = Math.pow((k * Math.pow((Math.PI * (n * 2.0)), (k + -1.0))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.7e-20:
		tmp = math.sqrt(n) * math.sqrt((2.0 * (math.pi / k)))
	else:
		tmp = math.pow((k * math.pow((math.pi * (n * 2.0)), (k + -1.0))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.7e-20)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = Float64(k * (Float64(pi * Float64(n * 2.0)) ^ Float64(k + -1.0))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.7e-20)
		tmp = sqrt(n) * sqrt((2.0 * (pi / k)));
	else
		tmp = (k * ((pi * (n * 2.0)) ^ (k + -1.0))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.7e-20], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[(k * N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(k + -1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.7e-20

   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 69.1%

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

      1. *-commutative 69.1%

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

      2. associate-/l* 69.1%

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

   5. Simplified 69.1%

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

      1. sqrt-unprod 69.3%

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

      2. associate-*r/ 69.3%

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

   7. Applied egg-rr 69.3%

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

      1. *-commutative 69.3%

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

      2. associate-/l* 69.3%

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

   9. Applied egg-rr 69.3%

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

       1. pow1/2 69.3%

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

       2. associate-*r/ 69.3%

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

       3. associate-*r/ 69.3%

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

       4. *-commutative 69.3%

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

       5. associate-*l* 69.3%

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

       6. *-commutative 69.3%

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

       7. associate-*r/ 69.3%

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

       8. associate-*l* 69.3%

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

       9. unpow-prod-down 99.6%

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

       10. pow1/2 99.6%

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

   11. Applied egg-rr 99.6%

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

       1. unpow1/2 99.6%

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

   13. Simplified 99.6%

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

   if 2.7e-20 < 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. associate-*l/ 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. associate-*r* 99.5%

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

      4. div-sub 99.5%

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

      5. metadata-eval 99.5%

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

      6. pow-div 100.0%

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

      7. pow1/2 100.0%

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

      8. associate-/l/ 100.0%

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

      9. div-inv 100.0%

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

      10. metadata-eval 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. associate-/l/ 100.0%

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

      2. rem-square-sqrt 100.0%

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

      3. rem-sqrt-square 100.0%

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

      4. fabs-neg 100.0%

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

      5. neg-mul-1 100.0%

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

      6. rem-square-sqrt 0.0%

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

      7. unpow1/2 0.0%

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

      8. metadata-eval 0.0%

         \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {k}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right|} \]

      9. pow-sqr 0.0%

         \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({k}^{0.25} \cdot {k}^{0.25}\right)}\right|} \]

      10. unswap-sqr 0.0%

          \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}\right|} \]

      11. fabs-sqr 0.0%

          \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}} \]

      12. unswap-sqr 0.0%

          \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)}} \]

      13. rem-square-sqrt 42.4%

          \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{-1} \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)} \]

      14. pow-sqr 42.4%

          \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{-1 \cdot \color{blue}{{k}^{\left(2 \cdot 0.25\right)}}} \]

      15. metadata-eval 42.4%

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

      16. unpow1/2 42.4%

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

      17. neg-mul-1 42.4%

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

      18. associate-/r* 42.4%

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

   6. Simplified 100.0%

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

      1. div-inv 99.9%

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

      2. *-commutative 99.9%

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

      3. associate-*r* 99.9%

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

      4. *-commutative 99.9%

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

      5. associate-*r* 99.9%

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

      6. pow1/2 99.9%

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

      7. pow-unpow 99.9%

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

      8. pow-prod-down 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*r* 99.9%

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

      11. *-commutative 99.9%

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

      12. associate-*r* 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. associate-*r/ 100.0%

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

      2. *-rgt-identity 100.0%

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

      3. *-commutative 100.0%

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

      4. *-commutative 100.0%

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

      5. unpow1/2 100.0%

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

      6. *-commutative 100.0%

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

      7. *-commutative 100.0%

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

   10. Simplified 100.0%

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

       1. clear-num 100.0%

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

       2. inv-pow 100.0%

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

       3. sqrt-undiv 99.9%

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

   12. Applied egg-rr 99.9%

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

       1. unpow-1 99.9%

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

       2. associate-/l* 99.9%

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

   14. Simplified 99.9%

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

       1. inv-pow 99.9%

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

       2. sqrt-pow2 100.0%

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

       3. pow1 100.0%

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

       4. pow-div 99.5%

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

       5. metadata-eval 99.5%

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

   16. Applied egg-rr 99.5%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.7 \cdot 10^{-20}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;{\left(k \cdot {\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(k + -1\right)}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 + \mathsf{fma}\left(n, \frac{2}{\frac{k}{\pi}}, -1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.5e+19)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (sqrt (+ 1.0 (fma n (/ 2.0 (/ k PI)) -1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.5e+19) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt((1.0 + fma(n, (2.0 / (k / ((double) M_PI))), -1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.5e+19)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt(Float64(1.0 + fma(n, Float64(2.0 / Float64(k / pi)), -1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.5e+19], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(1.0 + N[(n * N[(2.0 / N[(k / Pi), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5e19

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-/l* 66.3%

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

   5. Simplified 66.3%

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

      1. sqrt-unprod 66.5%

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

      2. associate-*r/ 66.5%

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

   7. Applied egg-rr 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-/l* 66.5%

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

   9. Applied egg-rr 66.5%

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

       1. pow1/2 66.5%

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

       2. associate-*r/ 66.5%

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

       3. associate-*r/ 66.5%

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

       4. *-commutative 66.5%

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

       5. associate-*l* 66.5%

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

       6. *-commutative 66.5%

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

       7. associate-*r/ 66.5%

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

       8. associate-*l* 66.5%

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

       9. unpow-prod-down 93.5%

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

       10. pow1/2 93.5%

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

   11. Applied egg-rr 93.5%

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

       1. unpow1/2 93.5%

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

   13. Simplified 93.5%

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

   if 4.5e19 < 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 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. sqrt-unprod 2.6%

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

      2. associate-*r/ 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-/l* 2.6%

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

   9. Applied egg-rr 2.6%

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

       1. associate-*r/ 2.6%

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

       2. *-un-lft-identity 2.6%

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

       3. associate-*l/ 2.6%

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

       4. expm1-log1p-u 2.6%

          \[\leadsto \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(\frac{1}{k} \cdot \left(\pi \cdot n\right)\right)\right)\right)}} \]

       5. expm1-undefine 26.7%

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

       6. *-commutative 26.7%

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

       7. associate-*l/ 26.7%

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

       8. *-un-lft-identity 26.7%

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

       9. associate-*r/ 26.7%

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

       10. associate-*l* 26.7%

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

   11. Applied egg-rr 26.7%

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

       1. log1p-undefine 26.7%

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

       2. rem-exp-log 26.7%

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

       3. associate-+r- 26.7%

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

       4. *-commutative 26.7%

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

       5. associate-*l/ 26.7%

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

       6. associate-/r/ 26.7%

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

       7. associate-/l* 26.7%

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

       8. fma-neg 26.7%

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

       9. metadata-eval 26.7%

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

   13. Simplified 26.7%

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

Alternative 4: 60.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{+19}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{2 \cdot \frac{\pi}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-1 + \mathsf{fma}\left(\pi, \frac{n \cdot 2}{k}, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 4.5e+19)
   (* (sqrt n) (sqrt (* 2.0 (/ PI k))))
   (sqrt (+ -1.0 (fma PI (/ (* n 2.0) k) 1.0)))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 4.5e+19) {
		tmp = sqrt(n) * sqrt((2.0 * (((double) M_PI) / k)));
	} else {
		tmp = sqrt((-1.0 + fma(((double) M_PI), ((n * 2.0) / k), 1.0)));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 4.5e+19)
		tmp = Float64(sqrt(n) * sqrt(Float64(2.0 * Float64(pi / k))));
	else
		tmp = sqrt(Float64(-1.0 + fma(pi, Float64(Float64(n * 2.0) / k), 1.0)));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 4.5e+19], N[(N[Sqrt[n], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(-1.0 + N[(Pi * N[(N[(n * 2.0), $MachinePrecision] / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5e19

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-/l* 66.3%

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

   5. Simplified 66.3%

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

      1. sqrt-unprod 66.5%

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

      2. associate-*r/ 66.5%

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

   7. Applied egg-rr 66.5%

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

      1. *-commutative 66.5%

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

      2. associate-/l* 66.5%

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

   9. Applied egg-rr 66.5%

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

       1. pow1/2 66.5%

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

       2. associate-*r/ 66.5%

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

       3. associate-*r/ 66.5%

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

       4. *-commutative 66.5%

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

       5. associate-*l* 66.5%

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

       6. *-commutative 66.5%

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

       7. associate-*r/ 66.5%

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

       8. associate-*l* 66.5%

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

       9. unpow-prod-down 93.5%

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

       10. pow1/2 93.5%

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

   11. Applied egg-rr 93.5%

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

       1. unpow1/2 93.5%

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

   13. Simplified 93.5%

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

   if 4.5e19 < 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 2.6%

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

      1. *-commutative 2.6%

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

      2. associate-/l* 2.6%

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

   5. Simplified 2.6%

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

      1. sqrt-unprod 2.6%

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

      2. associate-*r/ 2.6%

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

   7. Applied egg-rr 2.6%

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

      1. expm1-log1p-u 2.6%

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

      2. expm1-undefine 26.7%

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

      3. associate-/l* 26.7%

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

      4. associate-*r* 26.7%

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

      5. *-commutative 26.7%

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

   9. Applied egg-rr 26.7%

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

       1. sub-neg 26.7%

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

       2. metadata-eval 26.7%

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

       3. +-commutative 26.7%

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

       4. log1p-undefine 26.7%

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

       5. rem-exp-log 26.7%

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

       6. +-commutative 26.7%

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

       7. associate-*r/ 26.7%

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

       8. associate-*l/ 26.7%

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

       9. associate-*r/ 26.7%

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

       10. *-commutative 26.7%

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

       11. fma-define 26.7%

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

       12. associate-*r/ 26.7%

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

   11. Simplified 26.7%

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

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   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. associate-*l* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

3. Simplified 99.5%

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

Alternative 6: 49.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. *-commutative 39.6%

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

   2. associate-/l* 39.7%

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

5. Simplified 39.7%

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

   1. sqrt-unprod 39.8%

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

   2. associate-*r/ 39.8%

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

7. Applied egg-rr 39.8%

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

   1. *-commutative 39.8%

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

   2. associate-/l* 39.8%

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

9. Applied egg-rr 39.8%

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

    1. pow1/2 39.8%

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

    2. associate-*r/ 39.8%

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

    3. associate-*r/ 39.8%

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

    4. *-commutative 39.8%

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

    5. associate-*l* 39.8%

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

    6. *-commutative 39.8%

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

    7. associate-*r/ 39.8%

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

    8. associate-*l* 39.8%

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

    9. unpow-prod-down 55.5%

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

    10. pow1/2 55.5%

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

11. Applied egg-rr 55.5%

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

    1. unpow1/2 55.5%

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

13. Simplified 55.5%

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

Alternative 7: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. *-commutative 39.6%

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

   2. associate-/l* 39.7%

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

5. Simplified 39.7%

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

   1. sqrt-unprod 39.8%

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

   2. associate-*r/ 39.8%

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

7. Applied egg-rr 39.8%

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

   1. associate-*r/ 39.8%

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

   2. div-inv 39.7%

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

   3. associate-*r* 39.7%

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

   4. *-commutative 39.7%

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

   5. *-commutative 39.7%

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

   6. div-inv 39.8%

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

   7. clear-num 39.8%

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

   8. sqrt-div 40.7%

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

   9. metadata-eval 40.7%

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

9. Applied egg-rr 40.7%

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

Alternative 8: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. rem-square-sqrt 99.7%

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

   3. rem-sqrt-square 99.7%

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

   4. fabs-neg 99.7%

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

   5. neg-mul-1 99.7%

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

   6. rem-square-sqrt 0.0%

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

   7. unpow1/2 0.0%

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

   8. metadata-eval 0.0%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {k}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right|} \]

   9. pow-sqr 0.0%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({k}^{0.25} \cdot {k}^{0.25}\right)}\right|} \]

   10. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}\right|} \]

   11. fabs-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}} \]

   12. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)}} \]

   13. rem-square-sqrt 20.9%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{-1} \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)} \]

   14. pow-sqr 20.9%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{-1 \cdot \color{blue}{{k}^{\left(2 \cdot 0.25\right)}}} \]

   15. metadata-eval 20.9%

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

   16. unpow1/2 20.9%

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

   17. neg-mul-1 20.9%

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

   18. associate-/r* 20.9%

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

6. Simplified 99.7%

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

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*r* 99.7%

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

   6. pow1/2 99.7%

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

   7. pow-unpow 99.7%

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

   8. pow-prod-down 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*r* 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. unpow1/2 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

10. Simplified 99.7%

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

    1. clear-num 99.7%

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

    2. inv-pow 99.7%

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

    3. sqrt-undiv 85.0%

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

12. Applied egg-rr 85.0%

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

    1. unpow-1 85.0%

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

    2. associate-/l* 85.0%

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

14. Simplified 85.0%

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

15. Taylor expanded in k around 0 40.7%

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

16. Final simplification 40.7%

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

Alternative 9: 39.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.5%

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

   2. *-un-lft-identity 99.5%

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

   3. associate-*r* 99.5%

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

   4. div-sub 99.5%

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

   5. metadata-eval 99.5%

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

   6. pow-div 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. div-inv 99.7%

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

   10. metadata-eval 99.7%

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

4. Applied egg-rr 99.7%

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

   1. associate-/l/ 99.7%

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

   2. rem-square-sqrt 99.7%

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

   3. rem-sqrt-square 99.7%

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

   4. fabs-neg 99.7%

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

   5. neg-mul-1 99.7%

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

   6. rem-square-sqrt 0.0%

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

   7. unpow1/2 0.0%

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

   8. metadata-eval 0.0%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {k}^{\color{blue}{\left(2 \cdot 0.25\right)}}\right|} \]

   9. pow-sqr 0.0%

      \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \color{blue}{\left({k}^{0.25} \cdot {k}^{0.25}\right)}\right|} \]

   10. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\left|\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}\right|} \]

   11. fabs-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot {k}^{0.25}\right) \cdot \left(\sqrt{-1} \cdot {k}^{0.25}\right)}} \]

   12. unswap-sqr 0.0%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)}} \]

   13. rem-square-sqrt 20.9%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{\color{blue}{-1} \cdot \left({k}^{0.25} \cdot {k}^{0.25}\right)} \]

   14. pow-sqr 20.9%

       \[\leadsto \frac{\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot 0.5\right)}}}{-1 \cdot \color{blue}{{k}^{\left(2 \cdot 0.25\right)}}} \]

   15. metadata-eval 20.9%

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

   16. unpow1/2 20.9%

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

   17. neg-mul-1 20.9%

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

   18. associate-/r* 20.9%

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

6. Simplified 99.7%

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

   1. div-inv 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-*r* 99.7%

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

   4. *-commutative 99.7%

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

   5. associate-*r* 99.7%

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

   6. pow1/2 99.7%

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

   7. pow-unpow 99.7%

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

   8. pow-prod-down 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*r* 99.7%

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

   11. *-commutative 99.7%

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

   12. associate-*r* 99.7%

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

8. Applied egg-rr 99.7%

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

   1. associate-*r/ 99.7%

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

   2. *-rgt-identity 99.7%

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

   3. *-commutative 99.7%

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

   4. *-commutative 99.7%

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

   5. unpow1/2 99.7%

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

   6. *-commutative 99.7%

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

   7. *-commutative 99.7%

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

10. Simplified 99.7%

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

    1. clear-num 99.7%

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

    2. inv-pow 99.7%

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

    3. sqrt-undiv 85.0%

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

12. Applied egg-rr 85.0%

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

    1. unpow-1 85.0%

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

    2. associate-/l* 85.0%

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

14. Simplified 85.0%

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

15. Taylor expanded in k around 0 40.7%

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

    1. associate-/r* 40.7%

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

17. Simplified 40.7%

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

Alternative 10: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. *-commutative 39.6%

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

   2. associate-/l* 39.7%

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

5. Simplified 39.7%

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

   1. sqrt-unprod 39.8%

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

   2. associate-*r/ 39.8%

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

7. Applied egg-rr 39.8%

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

   1. *-commutative 39.8%

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

   2. associate-/l* 39.8%

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

9. Applied egg-rr 39.8%

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

    1. associate-*r/ 39.8%

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

    2. associate-*r/ 39.8%

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

    3. *-commutative 39.8%

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

    4. associate-*l* 39.8%

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

    5. *-commutative 39.8%

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

    6. associate-*r/ 39.8%

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

    7. clear-num 39.7%

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

    8. un-div-inv 39.8%

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

11. Applied egg-rr 39.8%

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

Alternative 11: 38.3% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0 39.6%

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

   1. *-commutative 39.6%

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

   2. associate-/l* 39.7%

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

5. Simplified 39.7%

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

   1. sqrt-unprod 39.8%

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

   2. associate-*r/ 39.8%

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

7. Applied egg-rr 39.8%

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

   1. *-commutative 39.8%

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

   2. associate-/l* 39.8%

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

9. Applied egg-rr 39.8%

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

Reproduce

herbie shell --seed 2024121 
(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))))