Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.0%

Time: 15.6s

Alternatives: 14

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.45e-62)
   (* (sqrt (/ PI k)) (sqrt (* n 2.0)))
   (/ 1.0 (sqrt (/ k (pow (* PI (* n 2.0)) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.45e-62) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n * 2.0));
	} else {
		tmp = 1.0 / sqrt((k / pow((((double) M_PI) * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.45e-62) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((n * 2.0));
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow((Math.PI * (n * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 2.45e-62:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((n * 2.0))
	else:
		tmp = 1.0 / math.sqrt((k / math.pow((math.pi * (n * 2.0)), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.45e-62)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n * 2.0)));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 2.45e-62)
		tmp = sqrt((pi / k)) * sqrt((n * 2.0));
	else
		tmp = 1.0 / sqrt((k / ((pi * (n * 2.0)) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.45e-62], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.4500000000000002e-62

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.1%

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

      2. sqrt-unprod 69.0%

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

      3. *-commutative 69.0%

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

      4. div-inv 69.0%

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

      5. *-commutative 69.0%

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

      6. div-inv 69.0%

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

      7. frac-times 69.0%

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

   3. Applied egg-rr 69.1%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in k around 0 69.3%

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

      1. associate-*r/ 69.3%

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

      2. associate-*r* 69.3%

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

      3. *-commutative 69.3%

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

      4. associate-/l* 69.3%

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

   8. Simplified 69.3%

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

      1. associate-/r/ 69.3%

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

      2. sqrt-prod 99.5%

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

      3. *-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   if 2.4500000000000002e-62 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. div-inv 99.8%

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

      5. *-commutative 99.8%

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

      6. div-inv 99.8%

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

      7. frac-times 99.8%

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

   3. Applied egg-rr 99.8%

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

   5. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. sqrt-div 99.9%

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

      3. metadata-eval 99.9%

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

   7. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.45 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 2: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative 99.6%

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

   2. associate-*r* 99.6%

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

   3. div-sub 99.6%

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

   4. metadata-eval 99.6%

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

   5. div-inv 99.6%

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

   6. pow-sub 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%

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

3. Applied egg-rr 99.7%

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

   1. *-commutative 99.7%

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

   2. associate-*r* 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

5. Simplified 99.7%

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

   1. div-inv 99.6%

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

   2. *-commutative 99.6%

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

   3. pow1/2 99.6%

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

   4. pow-unpow 99.6%

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

   5. pow-prod-down 99.6%

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

   6. *-commutative 99.6%

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

7. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-*l/ 99.7%

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

   3. *-lft-identity 99.7%

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

   4. *-commutative 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*l* 99.7%

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

   7. unpow1/2 99.7%

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

   8. *-commutative 99.7%

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

   9. *-commutative 99.7%

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

   10. associate-*l* 99.7%

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

9. Simplified 99.7%

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

10. Final simplification 99.7%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. sqr-pow 99.4%

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

   4. pow-sqr 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. associate-/l* 99.6%

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

   9. metadata-eval 99.6%

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

   10. /-rgt-identity 99.6%

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

   11. div-sub 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

   1. div-inv 99.6%

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

   2. div-inv 99.6%

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

   3. metadata-eval 99.6%

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

   4. pow1/2 99.6%

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

   5. pow-flip 99.6%

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

   6. metadata-eval 99.6%

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

5. Applied egg-rr 99.6%

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

6. Final simplification 99.6%

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

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.55e-65)
   (* (sqrt (/ PI k)) (sqrt (* n 2.0)))
   (sqrt (/ (pow (* PI (* n 2.0)) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.55e-65) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n * 2.0));
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.55e-65) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((n * 2.0));
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n * 2.0)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.55e-65:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((n * 2.0))
	else:
		tmp = math.sqrt((math.pow((math.pi * (n * 2.0)), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.55e-65)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n * 2.0)));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n * 2.0)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.55e-65)
		tmp = sqrt((pi / k)) * sqrt((n * 2.0));
	else
		tmp = sqrt((((pi * (n * 2.0)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.55e-65], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.55000000000000008e-65

   1. Initial program 99.2%

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

      1. add-sqr-sqrt 99.0%

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

      2. sqrt-unprod 68.4%

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

      3. *-commutative 68.4%

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

      4. div-inv 68.4%

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

      5. *-commutative 68.4%

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

      6. div-inv 68.4%

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

      7. frac-times 68.4%

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

   3. Applied egg-rr 68.5%

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

   5. Simplified 68.7%

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

   6. Taylor expanded in k around 0 68.7%

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

      1. associate-*r/ 68.7%

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

      2. associate-*r* 68.7%

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

      3. *-commutative 68.7%

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

      4. associate-/l* 68.7%

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

   8. Simplified 68.7%

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

      1. associate-/r/ 68.7%

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

      2. sqrt-prod 99.5%

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

      3. *-commutative 99.5%

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

   10. Applied egg-rr 99.5%

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

   if 1.55000000000000008e-65 < k

   1. Initial program 99.8%

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

      1. add-sqr-sqrt 99.8%

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

      2. sqrt-unprod 99.8%

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

      3. *-commutative 99.8%

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

      4. div-inv 99.8%

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

      5. *-commutative 99.8%

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

      6. div-inv 99.8%

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

      7. frac-times 99.8%

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

   3. Applied egg-rr 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.55 \cdot 10^{-65}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 5: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. sqr-pow 99.4%

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

   4. pow-sqr 99.6%

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

   5. associate-*l* 99.6%

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

   6. *-commutative 99.6%

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

   7. associate-*l/ 99.6%

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

   8. associate-/l* 99.6%

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

   9. metadata-eval 99.6%

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

   10. /-rgt-identity 99.6%

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

   11. div-sub 99.6%

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

   12. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 6: 51.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi}{k} \cdot \left(n \cdot 2\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.6e+226)
   (* (sqrt (/ PI k)) (sqrt (* n 2.0)))
   (pow (pow (* (/ PI k) (* n 2.0)) 2.0) 0.25)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.6e+226) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n * 2.0));
	} else {
		tmp = pow(pow(((((double) M_PI) / k) * (n * 2.0)), 2.0), 0.25);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.6e+226) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((n * 2.0));
	} else {
		tmp = Math.pow(Math.pow(((Math.PI / k) * (n * 2.0)), 2.0), 0.25);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.6e+226:
		tmp = math.sqrt((math.pi / k)) * math.sqrt((n * 2.0))
	else:
		tmp = math.pow(math.pow(((math.pi / k) * (n * 2.0)), 2.0), 0.25)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.6e+226)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n * 2.0)));
	else
		tmp = (Float64(Float64(pi / k) * Float64(n * 2.0)) ^ 2.0) ^ 0.25;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.6e+226)
		tmp = sqrt((pi / k)) * sqrt((n * 2.0));
	else
		tmp = (((pi / k) * (n * 2.0)) ^ 2.0) ^ 0.25;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.6e+226], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(N[(Pi / k), $MachinePrecision] * N[(n * 2.0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], 0.25], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.59999999999999989e226

   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. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 84.5%

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

      3. *-commutative 84.5%

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

      4. div-inv 84.5%

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

      5. *-commutative 84.5%

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

      6. div-inv 84.5%

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

      7. frac-times 84.5%

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

   3. Applied egg-rr 84.6%

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

   5. Simplified 84.7%

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

   6. Taylor expanded in k around 0 44.6%

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

      1. associate-*r/ 44.6%

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

      2. associate-*r* 44.6%

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

      3. *-commutative 44.6%

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

      4. associate-/l* 44.6%

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

   8. Simplified 44.6%

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

      1. associate-/r/ 44.6%

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

      2. sqrt-prod 59.6%

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

      3. *-commutative 59.6%

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

   10. Applied egg-rr 59.6%

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

   if 1.59999999999999989e226 < 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. div-inv 100.0%

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

      5. *-commutative 100.0%

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

      6. div-inv 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 3.0%

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

      1. associate-*r/ 3.0%

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

      2. associate-*r* 3.0%

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

      3. *-commutative 3.0%

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

      4. associate-/l* 3.0%

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

   8. Simplified 3.0%

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

      1. pow1/2 3.0%

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

      2. metadata-eval 3.0%

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

      3. pow-prod-up 3.0%

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

      4. pow-prod-down 20.3%

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

   10. Applied egg-rr 20.3%

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

       1. *-commutative 20.3%

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

       2. associate-*l* 20.3%

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

       3. *-commutative 20.3%

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

       4. associate-*l* 20.3%

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

   12. Simplified 20.3%

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

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.6 \cdot 10^{+226}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\pi}{k} \cdot \left(n \cdot 2\right)\right)}^{2}\right)}^{0.25}\\ \end{array} \]

Alternative 7: 50.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.4e+228)
   (* (sqrt (/ PI k)) (sqrt (* n 2.0)))
   (cbrt (pow (* 2.0 (* PI (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 2.4e+228) {
		tmp = sqrt((((double) M_PI) / k)) * sqrt((n * 2.0));
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 2.4e+228) {
		tmp = Math.sqrt((Math.PI / k)) * Math.sqrt((n * 2.0));
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 2.4e+228)
		tmp = Float64(sqrt(Float64(pi / k)) * sqrt(Float64(n * 2.0)));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 2.4e+228], N[(N[Sqrt[N[(Pi / k), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(n * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 2.39999999999999989e228

   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. add-sqr-sqrt 99.4%

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

      2. sqrt-unprod 84.5%

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

      3. *-commutative 84.5%

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

      4. div-inv 84.5%

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

      5. *-commutative 84.5%

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

      6. div-inv 84.5%

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

      7. frac-times 84.5%

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

   3. Applied egg-rr 84.6%

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

   5. Simplified 84.7%

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

   6. Taylor expanded in k around 0 44.6%

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

      1. associate-*r/ 44.6%

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

      2. associate-*r* 44.6%

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

      3. *-commutative 44.6%

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

      4. associate-/l* 44.6%

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

   8. Simplified 44.6%

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

      1. associate-/r/ 44.6%

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

      2. sqrt-prod 59.6%

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

      3. *-commutative 59.6%

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

   10. Applied egg-rr 59.6%

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

   if 2.39999999999999989e228 < 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. Step-by-step derivation

      1. add-sqr-sqrt 100.0%

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

      2. sqrt-unprod 100.0%

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

      3. *-commutative 100.0%

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

      4. div-inv 100.0%

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

      5. *-commutative 100.0%

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

      6. div-inv 100.0%

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

      7. frac-times 100.0%

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

   3. Applied egg-rr 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in k around 0 3.0%

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

      1. associate-*r/ 3.0%

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

      2. associate-*r* 3.0%

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

      3. *-commutative 3.0%

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

      4. associate-/l* 3.0%

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

   8. Simplified 3.0%

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

      1. pow1/2 3.0%

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

      2. metadata-eval 3.0%

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

      3. pow-pow 3.0%

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

      4. add-cbrt-cube 13.8%

         \[\leadsto \color{blue}{\sqrt[3]{\left({\left({\left(\frac{\pi}{\frac{k}{2 \cdot n}}\right)}^{0.25}\right)}^{2} \cdot {\left({\left(\frac{\pi}{\frac{k}{2 \cdot n}}\right)}^{0.25}\right)}^{2}\right) \cdot {\left({\left(\frac{\pi}{\frac{k}{2 \cdot n}}\right)}^{0.25}\right)}^{2}}} \]

      5. pow1/3 13.8%

         \[\leadsto \color{blue}{{\left(\left({\left({\left(\frac{\pi}{\frac{k}{2 \cdot n}}\right)}^{0.25}\right)}^{2} \cdot {\left({\left(\frac{\pi}{\frac{k}{2 \cdot n}}\right)}^{0.25}\right)}^{2}\right) \cdot {\left({\left(\frac{\pi}{\frac{k}{2 \cdot n}}\right)}^{0.25}\right)}^{2}\right)}^{0.3333333333333333}} \]

   10. Applied egg-rr 13.8%

       \[\leadsto \color{blue}{{\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   11. Step-by-step derivation

       1. unpow1/3 13.8%

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

       2. *-commutative 13.8%

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

       3. associate-*r* 13.8%

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

       4. associate-*r/ 13.8%

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

       5. associate-*l/ 13.8%

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

       6. associate-*r/ 13.8%

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

       7. associate-*l* 13.8%

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

   12. Simplified 13.8%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{+228}:\\ \;\;\;\;\sqrt{\frac{\pi}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 8: 49.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.2%

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

   3. *-commutative 87.2%

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

   4. div-inv 87.2%

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

   5. *-commutative 87.2%

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

   6. div-inv 87.2%

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

   7. frac-times 87.2%

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

3. Applied egg-rr 87.2%

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

5. Simplified 87.3%

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

6. Taylor expanded in k around 0 37.5%

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

   1. associate-*r/ 37.5%

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

   2. associate-*r* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-/l* 37.5%

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

8. Simplified 37.5%

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

   1. associate-/r/ 37.5%

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

   2. sqrt-prod 49.9%

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

   3. *-commutative 49.9%

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

10. Applied egg-rr 49.9%

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

11. Final simplification 49.9%

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

Alternative 9: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.2%

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

   3. *-commutative 87.2%

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

   4. div-inv 87.2%

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

   5. *-commutative 87.2%

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

   6. div-inv 87.2%

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

   7. frac-times 87.2%

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

3. Applied egg-rr 87.2%

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

5. Simplified 87.3%

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

6. Taylor expanded in k around 0 37.5%

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

   1. associate-*r/ 37.5%

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

   2. associate-*r* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-/l* 37.5%

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

8. Simplified 37.5%

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

9. Taylor expanded in k around 0 37.5%

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

    1. associate-/l* 37.4%

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

    2. associate-/r/ 37.4%

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

    3. associate-*l* 37.4%

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

    4. *-commutative 37.4%

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

    5. associate-*r* 37.4%

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

    6. *-commutative 37.4%

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

    7. associate-*l* 37.4%

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

11. Simplified 37.4%

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

    1. associate-*r/ 37.5%

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

    2. *-commutative 37.5%

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

    3. metadata-eval 37.5%

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

    4. times-frac 37.5%

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

    5. *-commutative 37.5%

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

    6. associate-/l* 37.5%

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

    7. metadata-eval 37.5%

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

    8. add-sqr-sqrt 37.4%

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

    9. frac-times 37.4%

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

    10. sqrt-unprod 38.0%

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

    11. add-sqr-sqrt 38.0%

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

    12. inv-pow 38.0%

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

    13. sqrt-pow2 38.1%

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

13. Applied egg-rr 38.1%

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

    1. *-commutative 38.1%

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

    2. associate-/r* 38.1%

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

15. Simplified 38.1%

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

16. Final simplification 38.1%

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

Alternative 10: 38.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.2%

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

   3. *-commutative 87.2%

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

   4. div-inv 87.2%

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

   5. *-commutative 87.2%

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

   6. div-inv 87.2%

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

   7. frac-times 87.2%

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

3. Applied egg-rr 87.2%

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

5. Simplified 87.3%

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

6. Taylor expanded in k around 0 37.5%

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

   1. associate-*r/ 37.5%

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

   2. associate-*r* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-/l* 37.5%

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

8. Simplified 37.5%

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

9. Taylor expanded in k around 0 37.5%

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

    1. associate-/l* 37.4%

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

    2. associate-/r/ 37.4%

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

    3. associate-*l* 37.4%

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

    4. *-commutative 37.4%

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

    5. associate-*r* 37.4%

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

    6. *-commutative 37.4%

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

    7. associate-*l* 37.4%

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

11. Simplified 37.4%

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

    1. associate-*r/ 37.5%

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

    2. *-commutative 37.5%

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

    3. metadata-eval 37.5%

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

    4. times-frac 37.5%

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

    5. *-commutative 37.5%

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

    6. associate-/l* 37.5%

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

    7. metadata-eval 37.5%

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

    8. add-sqr-sqrt 37.4%

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

    9. frac-times 37.4%

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

    10. sqrt-unprod 38.0%

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

    11. add-sqr-sqrt 38.0%

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

    12. inv-pow 38.0%

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

    13. sqrt-pow2 38.1%

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

13. Applied egg-rr 38.1%

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

    1. associate-*r/ 38.1%

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

    2. *-commutative 38.1%

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

    3. *-commutative 38.1%

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

15. Simplified 38.1%

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

16. Final simplification 38.1%

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

Alternative 11: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.2%

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

   3. *-commutative 87.2%

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

   4. div-inv 87.2%

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

   5. *-commutative 87.2%

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

   6. div-inv 87.2%

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

   7. frac-times 87.2%

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

3. Applied egg-rr 87.2%

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

5. Simplified 87.3%

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

6. Taylor expanded in k around 0 37.5%

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

   1. associate-*r/ 37.5%

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

   2. associate-*r* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-/l* 37.5%

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

8. Simplified 37.5%

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

9. Taylor expanded in k around 0 37.5%

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

    1. associate-/l* 37.4%

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

    2. associate-/r/ 37.4%

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

    3. associate-*l* 37.4%

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

    4. *-commutative 37.4%

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

    5. associate-*r* 37.4%

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

    6. *-commutative 37.4%

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

    7. associate-*l* 37.4%

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

11. Simplified 37.4%

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

12. Final simplification 37.4%

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

Alternative 12: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.2%

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

   3. *-commutative 87.2%

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

   4. div-inv 87.2%

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

   5. *-commutative 87.2%

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

   6. div-inv 87.2%

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

   7. frac-times 87.2%

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

3. Applied egg-rr 87.2%

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

5. Simplified 87.3%

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

6. Taylor expanded in k around 0 37.5%

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

   1. associate-*r/ 37.5%

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

   2. associate-*r* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-/l* 37.5%

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

8. Simplified 37.5%

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

   1. add-sqr-sqrt 37.2%

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

   2. *-un-lft-identity 37.2%

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

   3. times-frac 37.3%

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

   4. associate-/r* 37.3%

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

   5. div-inv 37.3%

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

   6. metadata-eval 37.3%

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

10. Applied egg-rr 37.3%

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

    1. /-rgt-identity 37.3%

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

    2. associate-*r/ 37.2%

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

    3. rem-square-sqrt 37.5%

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

    4. associate-/l* 37.5%

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

    5. associate-/r/ 37.5%

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

12. Simplified 37.5%

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

13. Final simplification 37.5%

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

Alternative 13: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 87.2%

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

   3. *-commutative 87.2%

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

   4. div-inv 87.2%

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

   5. *-commutative 87.2%

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

   6. div-inv 87.2%

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

   7. frac-times 87.2%

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

3. Applied egg-rr 87.2%

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

5. Simplified 87.3%

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

6. Taylor expanded in k around 0 37.5%

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

   1. associate-*r/ 37.5%

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

   2. associate-*r* 37.5%

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

   3. *-commutative 37.5%

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

   4. associate-/l* 37.5%

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

8. Simplified 37.5%

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

9. Final simplification 37.5%

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

Alternative 14: 37.5% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

2. Taylor expanded in k around 0 49.7%

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

   1. associate-*l/ 49.7%

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

   2. *-un-lft-identity 49.7%

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

   3. sqrt-unprod 49.8%

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

   4. *-commutative 49.8%

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

   5. associate-*l* 49.8%

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

   6. sqrt-undiv 37.5%

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

   7. *-commutative 37.5%

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

4. Applied egg-rr 37.5%

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

5. Final simplification 37.5%

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

Reproduce

herbie shell --seed 2023301 
(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))))