Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 19.3s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*r* 99.4%

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

   2. div-sub 99.4%

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

   3. metadata-eval 99.4%

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

   4. pow-div 99.6%

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

   5. pow1/2 99.6%

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

   6. associate-*r/ 99.6%

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

   7. pow1/2 99.6%

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

   8. pow-flip 99.6%

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

   9. metadata-eval 99.6%

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

   10. div-inv 99.6%

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

   11. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/l* 99.6%

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

   3. associate-*r* 99.6%

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

   4. associate-*r* 99.6%

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

6. Simplified 99.6%

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

Alternative 2: 99.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-un-lft-identity 99.4%

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

   3. associate-*r* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

   6. pow-div 99.6%

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

   7. pow1/2 99.6%

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

   8. associate-/l/ 99.6%

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

   9. div-inv 99.6%

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

   10. metadata-eval 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-*r* 99.6%

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

   2. associate-*r* 99.6%

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

6. Simplified 99.6%

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

7. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.08 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.08e-23)
   (* (sqrt (* PI (/ 2.0 k))) (sqrt n))
   (pow (/ k (pow (* (* 2.0 PI) n) (- 1.0 k))) -0.5)))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.08e-23) {
		tmp = sqrt((((double) M_PI) * (2.0 / k))) * sqrt(n);
	} else {
		tmp = pow((k / pow(((2.0 * ((double) M_PI)) * n), (1.0 - k))), -0.5);
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.08e-23) {
		tmp = Math.sqrt((Math.PI * (2.0 / k))) * Math.sqrt(n);
	} else {
		tmp = Math.pow((k / Math.pow(((2.0 * Math.PI) * n), (1.0 - k))), -0.5);
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1.08e-23:
		tmp = math.sqrt((math.pi * (2.0 / k))) * math.sqrt(n)
	else:
		tmp = math.pow((k / math.pow(((2.0 * math.pi) * n), (1.0 - k))), -0.5)
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.08e-23)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 / k))) * sqrt(n));
	else
		tmp = Float64(k / (Float64(Float64(2.0 * pi) * n) ^ Float64(1.0 - k))) ^ -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1.08e-23)
		tmp = sqrt((pi * (2.0 / k))) * sqrt(n);
	else
		tmp = (k / (((2.0 * pi) * n) ^ (1.0 - k))) ^ -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.08e-23], N[(N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Power[N[(k / N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.08000000000000003e-23

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 78.8%

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

      1. associate-/l* 78.8%

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

   5. Simplified 78.8%

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

      1. pow1 78.8%

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

      2. *-commutative 78.8%

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

      3. sqrt-unprod 78.9%

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

      4. associate-*r/ 78.9%

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

      5. *-commutative 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. unpow1 78.9%

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

      2. *-commutative 78.9%

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

      3. associate-/l* 78.9%

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

   9. Simplified 78.9%

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

       1. add-cbrt-cube 62.1%

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

       2. pow1/3 57.8%

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

       3. add-sqr-sqrt 57.8%

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

       4. pow1 57.8%

          \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\right)}^{0.3333333333333333} \]

       5. pow1/2 57.8%

          \[\leadsto {\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

       6. pow-prod-up 57.8%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

       7. associate-*r/ 57.8%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       8. *-commutative 57.8%

          \[\leadsto {\left({\left(2 \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       9. associate-/l* 57.8%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       10. metadata-eval 57.8%

           \[\leadsto {\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 57.8%

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

       1. unpow1/3 62.1%

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

       2. *-commutative 62.1%

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

       3. associate-*r/ 62.0%

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

       4. associate-*l/ 62.1%

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

       5. *-commutative 62.1%

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

       6. associate-*l* 62.1%

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

   13. Simplified 62.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}}} \]
   14. Step-by-step derivation

       1. pow1/3 57.8%

          \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 78.9%

          \[\leadsto \color{blue}{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 78.9%

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

       4. pow1/2 78.9%

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

       5. *-commutative 78.9%

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

       6. sqrt-prod 99.5%

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

       7. *-commutative 99.5%

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

       8. clear-num 99.4%

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

       9. un-div-inv 99.4%

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

   15. Applied egg-rr 99.4%

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

       1. associate-/r/ 99.5%

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

   17. Simplified 99.5%

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

   if 1.08000000000000003e-23 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.5%

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

      8. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. *-commutative 99.5%

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

      5. associate-*l* 99.5%

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

   8. Applied egg-rr 99.5%

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

      1. *-commutative 99.5%

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

      2. *-commutative 99.5%

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

      3. associate-*l* 99.5%

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

   10. Simplified 99.5%

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

       1. inv-pow 99.5%

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

       2. sqrt-pow2 99.5%

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

       3. metadata-eval 99.5%

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

   12. Applied egg-rr 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.08 \cdot 10^{-23}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{k}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 10^{-23}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1e-23)
   (* (sqrt (* PI (/ 2.0 k))) (sqrt n))
   (sqrt (/ (pow (* (* 2.0 PI) n) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1e-23) {
		tmp = sqrt((((double) M_PI) * (2.0 / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow(((2.0 * ((double) M_PI)) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1e-23) {
		tmp = Math.sqrt((Math.PI * (2.0 / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow(((2.0 * Math.PI) * n), (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 1e-23:
		tmp = math.sqrt((math.pi * (2.0 / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow(((2.0 * math.pi) * n), (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1e-23)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(Float64(2.0 * pi) * n) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 1e-23)
		tmp = sqrt((pi * (2.0 / k))) * sqrt(n);
	else
		tmp = sqrt(((((2.0 * pi) * n) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1e-23], N[(N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(N[(2.0 * Pi), $MachinePrecision] * n), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 9.9999999999999996e-24

   1. Initial program 99.2%

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

   3. Taylor expanded in k around 0 78.8%

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

      1. associate-/l* 78.8%

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

   5. Simplified 78.8%

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

      1. pow1 78.8%

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

      2. *-commutative 78.8%

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

      3. sqrt-unprod 78.9%

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

      4. associate-*r/ 78.9%

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

      5. *-commutative 78.9%

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

   7. Applied egg-rr 78.9%

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

      1. unpow1 78.9%

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

      2. *-commutative 78.9%

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

      3. associate-/l* 78.9%

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

   9. Simplified 78.9%

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

       1. add-cbrt-cube 62.1%

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

       2. pow1/3 57.8%

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

       3. add-sqr-sqrt 57.8%

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

       4. pow1 57.8%

          \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\right)}^{0.3333333333333333} \]

       5. pow1/2 57.8%

          \[\leadsto {\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

       6. pow-prod-up 57.8%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

       7. associate-*r/ 57.8%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       8. *-commutative 57.8%

          \[\leadsto {\left({\left(2 \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       9. associate-/l* 57.8%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       10. metadata-eval 57.8%

           \[\leadsto {\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 57.8%

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

       1. unpow1/3 62.1%

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

       2. *-commutative 62.1%

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

       3. associate-*r/ 62.0%

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

       4. associate-*l/ 62.1%

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

       5. *-commutative 62.1%

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

       6. associate-*l* 62.1%

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

   13. Simplified 62.1%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}}} \]
   14. Step-by-step derivation

       1. pow1/3 57.8%

          \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 78.9%

          \[\leadsto \color{blue}{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 78.9%

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

       4. pow1/2 78.9%

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

       5. *-commutative 78.9%

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

       6. sqrt-prod 99.5%

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

       7. *-commutative 99.5%

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

       8. clear-num 99.4%

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

       9. un-div-inv 99.4%

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

   15. Applied egg-rr 99.4%

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

       1. associate-/r/ 99.5%

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

   17. Simplified 99.5%

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

   if 9.9999999999999996e-24 < k

   1. Initial program 99.5%

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

      1. add-sqr-sqrt 99.5%

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

      2. sqrt-unprod 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. div-sub 99.5%

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

      6. metadata-eval 99.5%

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

      7. div-inv 99.5%

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

      8. *-commutative 99.5%

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

   4. Applied egg-rr 99.5%

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

   6. Simplified 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 10^{-23}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.02e+79)
   (* (sqrt (* PI (/ 2.0 k))) (sqrt n))
   (sqrt (* 2.0 (+ -1.0 (fma n (/ PI k) 1.0))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.02e+79) {
		tmp = sqrt((((double) M_PI) * (2.0 / k))) * sqrt(n);
	} else {
		tmp = sqrt((2.0 * (-1.0 + fma(n, (((double) M_PI) / k), 1.0))));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.02e+79)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 / k))) * sqrt(n));
	else
		tmp = sqrt(Float64(2.0 * Float64(-1.0 + fma(n, Float64(pi / k), 1.0))));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.02e+79], N[(N[Sqrt[N[(Pi * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(2.0 * N[(-1.0 + N[(n * N[(Pi / k), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.02000000000000006e79

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0 63.5%

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

      1. associate-/l* 63.5%

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

   5. Simplified 63.5%

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

      1. pow1 63.5%

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

      2. *-commutative 63.5%

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

      3. sqrt-unprod 63.6%

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

      4. associate-*r/ 63.5%

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

      5. *-commutative 63.5%

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

   7. Applied egg-rr 63.5%

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

      1. unpow1 63.5%

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

      2. *-commutative 63.5%

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

      3. associate-/l* 63.6%

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

   9. Simplified 63.6%

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

       1. add-cbrt-cube 50.9%

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

       2. pow1/3 47.7%

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

       3. add-sqr-sqrt 47.7%

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

       4. pow1 47.7%

          \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\right)}^{0.3333333333333333} \]

       5. pow1/2 47.7%

          \[\leadsto {\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

       6. pow-prod-up 47.7%

          \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

       7. associate-*r/ 47.7%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       8. *-commutative 47.7%

          \[\leadsto {\left({\left(2 \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       9. associate-/l* 47.7%

          \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

       10. metadata-eval 47.7%

           \[\leadsto {\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

   11. Applied egg-rr 47.7%

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

       1. unpow1/3 51.0%

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

       2. *-commutative 51.0%

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

       3. associate-*r/ 50.9%

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

       4. associate-*l/ 51.0%

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

       5. *-commutative 51.0%

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

       6. associate-*l* 51.0%

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

   13. Simplified 51.0%

       \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}}} \]
   14. Step-by-step derivation

       1. pow1/3 47.7%

          \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

       2. pow-pow 63.6%

          \[\leadsto \color{blue}{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       3. metadata-eval 63.6%

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

       4. pow1/2 63.6%

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

       5. *-commutative 63.6%

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

       6. sqrt-prod 78.3%

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

       7. *-commutative 78.3%

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

       8. clear-num 78.3%

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

       9. un-div-inv 78.3%

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

   15. Applied egg-rr 78.3%

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

       1. associate-/r/ 78.3%

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

   17. Simplified 78.3%

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

   if 1.02000000000000006e79 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0 2.5%

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

      1. associate-/l* 2.5%

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

   5. Simplified 2.5%

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

      1. pow1 2.5%

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

      2. *-commutative 2.5%

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

      3. sqrt-unprod 2.5%

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

      4. associate-*r/ 2.5%

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

      5. *-commutative 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. unpow1 2.5%

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

      2. *-commutative 2.5%

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

      3. associate-/l* 2.5%

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

   9. Simplified 2.5%

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

       1. associate-*r/ 2.5%

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

       2. expm1-log1p-u 2.5%

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

       3. expm1-undefine 20.2%

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

       4. *-commutative 20.2%

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

       5. associate-/l* 20.2%

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

   11. Applied egg-rr 20.2%

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

       1. sub-neg 20.2%

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

       2. metadata-eval 20.2%

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

       3. +-commutative 20.2%

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

       4. log1p-undefine 20.2%

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

       5. rem-exp-log 20.2%

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

       6. +-commutative 20.2%

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

       7. associate-*r/ 20.2%

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

       8. associate-*l/ 20.2%

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

       9. *-commutative 20.2%

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

       10. fma-define 20.2%

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

   13. Simplified 20.2%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.02 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{\pi \cdot \frac{2}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(-1 + \mathsf{fma}\left(n, \frac{\pi}{k}, 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

   1. div-inv 99.4%

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

   2. div-inv 99.4%

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

   3. metadata-eval 99.4%

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

   4. pow1/2 99.4%

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

   5. pow-flip 99.4%

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

   6. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

Alternative 7: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. associate-*l/ 99.4%

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

   2. *-lft-identity 99.4%

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

   3. associate-*l* 99.4%

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

   4. div-sub 99.4%

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

   5. metadata-eval 99.4%

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

3. Simplified 99.4%

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

Alternative 8: 49.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 43.0%

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

   1. associate-/l* 43.0%

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

5. Simplified 43.0%

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

   1. pow1 43.0%

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

   2. *-commutative 43.0%

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

   3. sqrt-unprod 43.0%

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

   4. associate-*r/ 43.0%

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

   5. *-commutative 43.0%

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

7. Applied egg-rr 43.0%

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

   1. unpow1 43.0%

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

   2. *-commutative 43.0%

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

   3. associate-/l* 43.0%

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

9. Simplified 43.0%

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

    1. add-cbrt-cube 35.4%

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

    2. pow1/3 33.2%

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

    3. add-sqr-sqrt 33.2%

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

    4. pow1 33.2%

       \[\leadsto {\left(\color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1}} \cdot \sqrt{2 \cdot \left(n \cdot \frac{\pi}{k}\right)}\right)}^{0.3333333333333333} \]

    5. pow1/2 33.2%

       \[\leadsto {\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{0.5}}\right)}^{0.3333333333333333} \]

    6. pow-prod-up 33.2%

       \[\leadsto {\color{blue}{\left({\left(2 \cdot \left(n \cdot \frac{\pi}{k}\right)\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

    7. associate-*r/ 33.2%

       \[\leadsto {\left({\left(2 \cdot \color{blue}{\frac{n \cdot \pi}{k}}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

    8. *-commutative 33.2%

       \[\leadsto {\left({\left(2 \cdot \frac{\color{blue}{\pi \cdot n}}{k}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

    9. associate-/l* 33.2%

       \[\leadsto {\left({\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{\left(1 + 0.5\right)}\right)}^{0.3333333333333333} \]

    10. metadata-eval 33.2%

        \[\leadsto {\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{\color{blue}{1.5}}\right)}^{0.3333333333333333} \]

11. Applied egg-rr 33.2%

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

    1. unpow1/3 35.4%

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

    2. *-commutative 35.4%

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

    3. associate-*r/ 35.4%

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

    4. associate-*l/ 35.4%

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

    5. *-commutative 35.4%

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

    6. associate-*l* 35.4%

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

13. Simplified 35.4%

    \[\leadsto \color{blue}{\sqrt[3]{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}}} \]
14. Step-by-step derivation

    1. pow1/3 33.2%

       \[\leadsto \color{blue}{{\left({\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]

    2. pow-pow 43.0%

       \[\leadsto \color{blue}{{\left(n \cdot \left(\frac{\pi}{k} \cdot 2\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    3. metadata-eval 43.0%

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

    4. pow1/2 43.0%

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

    5. *-commutative 43.0%

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

    6. sqrt-prod 52.9%

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

    7. *-commutative 52.9%

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

    8. clear-num 52.9%

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

    9. un-div-inv 52.9%

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

15. Applied egg-rr 52.9%

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

    1. associate-/r/ 52.9%

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

17. Simplified 52.9%

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

18. Final simplification 52.9%

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

Alternative 9: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 43.0%

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

   1. associate-/l* 43.0%

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

5. Simplified 43.0%

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

   1. pow1 43.0%

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

   2. *-commutative 43.0%

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

   3. sqrt-unprod 43.0%

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

   4. associate-*r/ 43.0%

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

   5. *-commutative 43.0%

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

7. Applied egg-rr 43.0%

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

   1. unpow1 43.0%

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

   2. *-commutative 43.0%

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

   3. associate-/l* 43.0%

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

9. Simplified 43.0%

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

    1. pow1/2 43.0%

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

    2. associate-*r* 43.0%

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

    3. unpow-prod-down 52.8%

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

    4. pow1/2 52.8%

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

11. Applied egg-rr 52.8%

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

    1. unpow1/2 52.8%

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

    2. *-commutative 52.8%

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

13. Simplified 52.8%

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

14. Final simplification 52.8%

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

Alternative 10: 38.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 43.0%

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

   1. associate-/l* 43.0%

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

5. Simplified 43.0%

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

   1. pow1 43.0%

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

   2. *-commutative 43.0%

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

   3. sqrt-unprod 43.0%

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

   4. associate-*r/ 43.0%

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

   5. *-commutative 43.0%

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

7. Applied egg-rr 43.0%

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

   1. unpow1 43.0%

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

   2. *-commutative 43.0%

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

   3. associate-/l* 43.0%

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

9. Simplified 43.0%

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

    1. associate-*r/ 43.0%

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

    2. associate-*r/ 43.0%

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

    3. *-commutative 43.0%

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

    4. sqrt-undiv 52.8%

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

    5. clear-num 52.7%

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

    6. frac-2neg 52.7%

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

    7. metadata-eval 52.7%

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

    8. div-inv 52.7%

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

    9. sqrt-undiv 43.9%

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

11. Applied egg-rr 43.9%

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

    1. mul-1-neg 43.9%

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

    2. distribute-frac-neg2 43.9%

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

    3. remove-double-neg 43.9%

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

    4. associate-/r* 43.9%

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

    5. *-commutative 43.9%

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

13. Simplified 43.9%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{\frac{k}{2}}{n \cdot \pi}}}} \]
14. Step-by-step derivation

    1. pow1/2 43.9%

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

    2. pow-flip 44.0%

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

    3. div-inv 44.0%

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

    4. metadata-eval 44.0%

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

    5. *-commutative 44.0%

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

    6. times-frac 44.0%

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

    7. metadata-eval 44.0%

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

15. Applied egg-rr 44.0%

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

Alternative 11: 38.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in k around 0 43.0%

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

   1. associate-/l* 43.0%

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

5. Simplified 43.0%

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

   1. pow1 43.0%

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

   2. *-commutative 43.0%

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

   3. sqrt-unprod 43.0%

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

   4. associate-*r/ 43.0%

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

   5. *-commutative 43.0%

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

7. Applied egg-rr 43.0%

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

   1. unpow1 43.0%

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

   2. *-commutative 43.0%

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

   3. associate-/l* 43.0%

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

9. Simplified 43.0%

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

Reproduce

herbie shell --seed 2024093 
(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))))