Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.9s

Alternatives: 10

Speedup: 1.0×

• could not determine a ground truth

  1. k = 2.00000000000000007e154
  2. n = 0.0

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \sqrt{\frac{1}{k}} \cdot \frac{\sqrt{t_0}}{{t_0}^{\left(k \cdot 0.5\right)}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 PI) n)))
   (* (sqrt (/ 1.0 k)) (/ (sqrt t_0) (pow t_0 (* k 0.5))))))

C

double code(double k, double n) {
	double t_0 = (2.0 * ((double) M_PI)) * n;
	return sqrt((1.0 / k)) * (sqrt(t_0) / 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((1.0 / k)) * (Math.sqrt(t_0) / Math.pow(t_0, (k * 0.5)));
}

Python

def code(k, n):
	t_0 = (2.0 * math.pi) * n
	return math.sqrt((1.0 / k)) * (math.sqrt(t_0) / math.pow(t_0, (k * 0.5)))

Julia

function code(k, n)
	t_0 = Float64(Float64(2.0 * pi) * n)
	return Float64(sqrt(Float64(1.0 / k)) * Float64(sqrt(t_0) / (t_0 ^ Float64(k * 0.5))))
end

MATLAB

function tmp = code(k, n)
	t_0 = (2.0 * pi) * n;
	tmp = sqrt((1.0 / k)) * (sqrt(t_0) / (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[N[(1.0 / k), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Power[t$95$0, N[(k * 0.5), $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. div-sub 99.6%

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

   2. metadata-eval 99.6%

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

   3. pow-sub 99.7%

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

   4. pow1/2 99.7%

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

   5. associate-*l* 99.7%

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

   6. associate-*l* 99.7%

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

   7. div-inv 99.7%

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

   8. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. associate-*r* 99.7%

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

   2. associate-*r* 99.7%

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

5. Simplified 99.7%

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

   1. add-sqr-sqrt 99.5%

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

   2. sqrt-unprod 99.7%

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

   3. frac-times 99.7%

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

   4. metadata-eval 99.7%

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

   5. add-sqr-sqrt 99.7%

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

7. Applied egg-rr 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.2e-72)
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))
   (sqrt (/ (pow (* (* 2.0 PI) n) (- 1.0 k)) k))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 3.2e-72) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	} 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 <= 3.2e-72) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	} 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 <= 3.2e-72:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	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 <= 3.2e-72)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	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 <= 3.2e-72)
		tmp = sqrt((2.0 * (pi * n))) / sqrt(k);
	else
		tmp = sqrt(((((2.0 * pi) * n) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 3.2e-72], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $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 < 3.19999999999999999e-72

   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. *-commutative 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-*r* 99.4%

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

      4. div-inv 99.6%

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

      5. expm1-log1p-u 92.6%

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

      6. expm1-udef 78.2%

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

   3. Applied egg-rr 48.1%

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

      1. expm1-def 62.6%

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

      2. expm1-log1p 65.9%

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

      3. associate-*r* 65.9%

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

   5. Simplified 65.9%

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

   6. Taylor expanded in k around 0 65.9%

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

      1. *-commutative 65.9%

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

      2. *-commutative 65.9%

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

      3. associate-*l* 65.9%

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

   8. Simplified 65.9%

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

      1. sqrt-div 99.6%

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

      2. associate-*r* 99.6%

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

      3. *-commutative 99.6%

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

   10. Applied egg-rr 99.6%

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

   if 3.19999999999999999e-72 < k

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

      2. *-commutative 99.7%

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

      3. associate-*r* 99.7%

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

      4. div-inv 99.7%

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

      5. expm1-log1p-u 99.2%

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

      6. expm1-udef 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. expm1-def 98.6%

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

      2. expm1-log1p 99.1%

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

      3. associate-*r* 99.1%

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

   5. Simplified 99.1%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(N[(2.0 * Pi), $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision] * N[Power[n, N[(0.5 - N[(k * 0.5), $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. 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. *-un-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. unpow-prod-down 71.3%

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

   4. associate-/l* 71.3%

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

   5. div-sub 71.3%

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

   6. metadata-eval 71.3%

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

   7. div-inv 71.3%

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

   8. metadata-eval 71.3%

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

   9. div-sub 71.3%

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

   10. metadata-eval 71.3%

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

   11. div-inv 71.3%

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

   12. metadata-eval 71.3%

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

3. Applied egg-rr 71.3%

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

4. Taylor expanded in k around 0 98.3%

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

   1. associate-/r/ 98.3%

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

   2. sqrt-unprod 98.2%

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

   3. sqrt-undiv 98.4%

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

6. Applied egg-rr 98.4%

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

7. Final simplification 98.4%

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

Alternative 4: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - k), $MachinePrecision] / 2.0), $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. *-commutative 99.6%

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

   4. associate-*l* 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(2.0 * N[(Pi * n), $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. *-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. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

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

   6. expm1-udef 88.8%

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

3. Applied egg-rr 77.3%

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

   1. expm1-def 85.0%

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

   2. expm1-log1p 86.5%

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

   3. associate-*r* 86.5%

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

5. Simplified 86.5%

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

6. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. *-commutative 33.7%

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

   3. associate-*l* 33.7%

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

8. Simplified 33.7%

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

   1. sqrt-div 46.5%

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

   2. associate-*r* 46.5%

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

   3. *-commutative 46.5%

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

10. Applied egg-rr 46.5%

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

11. Final simplification 46.5%

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

Alternative 6: 38.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(k / N[(2.0 * N[(Pi * n), $MachinePrecision]), $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. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

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

   6. expm1-udef 88.8%

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

3. Applied egg-rr 77.3%

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

   1. expm1-def 85.0%

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

   2. expm1-log1p 86.5%

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

   3. associate-*r* 86.5%

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

5. Simplified 86.5%

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

6. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. *-commutative 33.7%

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

   3. associate-*l* 33.7%

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

8. Simplified 33.7%

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

   1. pow1/2 33.7%

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

   2. div-inv 33.7%

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

   3. associate-*r* 33.7%

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

   4. *-commutative 33.7%

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

   5. unpow-prod-down 46.4%

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

   6. pow1/2 46.4%

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

   7. inv-pow 46.4%

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

   8. pow-pow 46.5%

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

   9. metadata-eval 46.5%

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

10. Applied egg-rr 46.5%

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

    1. *-commutative 46.5%

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

    2. *-commutative 46.5%

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

12. Simplified 46.5%

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

    1. *-commutative 46.5%

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

    2. *-commutative 46.5%

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

    3. add-cube-cbrt 46.1%

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

    4. unpow3 46.1%

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

    5. add-sqr-sqrt 46.0%

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

    6. sqrt-unprod 46.1%

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

    7. pow-prod-up 46.1%

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

    8. metadata-eval 46.1%

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

    9. inv-pow 46.1%

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

    10. sqrt-prod 33.4%

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

    11. div-inv 33.4%

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

    12. clear-num 33.3%

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

    13. sqrt-div 33.7%

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

    14. metadata-eval 33.7%

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

    15. unpow3 33.8%

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

14. Applied egg-rr 34.0%

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

15. Final simplification 34.0%

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

Alternative 7: 38.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(N[(k / Pi), $MachinePrecision] / N[(2.0 * n), $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. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

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

   6. expm1-udef 88.8%

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

3. Applied egg-rr 77.3%

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

   1. expm1-def 85.0%

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

   2. expm1-log1p 86.5%

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

   3. associate-*r* 86.5%

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

5. Simplified 86.5%

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

6. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. *-commutative 33.7%

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

   3. associate-*l* 33.7%

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

8. Simplified 33.7%

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

   1. pow1/2 33.7%

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

   2. div-inv 33.7%

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

   3. associate-*r* 33.7%

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

   4. *-commutative 33.7%

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

   5. unpow-prod-down 46.4%

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

   6. pow1/2 46.4%

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

   7. inv-pow 46.4%

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

   8. pow-pow 46.5%

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

   9. metadata-eval 46.5%

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

10. Applied egg-rr 46.5%

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

    1. *-commutative 46.5%

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

    2. *-commutative 46.5%

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

12. Simplified 46.5%

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

    1. *-commutative 46.5%

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

    2. *-commutative 46.5%

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

    3. add-cube-cbrt 46.1%

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

    4. unpow3 46.1%

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

    5. add-sqr-sqrt 46.0%

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

    6. sqrt-unprod 46.1%

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

    7. pow-prod-up 46.1%

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

    8. metadata-eval 46.1%

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

    9. inv-pow 46.1%

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

    10. sqrt-prod 33.4%

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

    11. div-inv 33.4%

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

    12. clear-num 33.3%

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

    13. sqrt-div 33.7%

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

    14. metadata-eval 33.7%

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

    15. unpow3 33.8%

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

14. Applied egg-rr 34.0%

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

    1. *-commutative 34.0%

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

    2. associate-*l* 34.0%

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

    3. *-commutative 34.0%

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

    4. associate-/r* 34.0%

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

16. Simplified 34.0%

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

17. Final simplification 34.0%

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

Alternative 8: 38.1% 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. *-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. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

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

   6. expm1-udef 88.8%

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

3. Applied egg-rr 77.3%

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

   1. expm1-def 85.0%

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

   2. expm1-log1p 86.5%

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

   3. associate-*r* 86.5%

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

5. Simplified 86.5%

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

6. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. *-commutative 33.7%

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

   3. associate-*l* 33.7%

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

8. Simplified 33.7%

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

9. Taylor expanded in n around 0 33.7%

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

    1. associate-/l* 33.7%

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

    2. associate-/r/ 33.7%

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

11. Simplified 33.7%

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

12. Final simplification 33.7%

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

Alternative 9: 38.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 * N[(n / N[(k / Pi), $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. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

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

   6. expm1-udef 88.8%

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

3. Applied egg-rr 77.3%

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

   1. expm1-def 85.0%

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

   2. expm1-log1p 86.5%

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

   3. associate-*r* 86.5%

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

5. Simplified 86.5%

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

6. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. *-commutative 33.7%

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

   3. associate-*l* 33.7%

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

8. Simplified 33.7%

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

9. Taylor expanded in n around 0 33.7%

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

    1. associate-/l* 33.7%

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

11. Simplified 33.7%

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

12. Final simplification 33.7%

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

Alternative 10: 38.1% accurate, 1.5× speedup

Math

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

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(Float64(2.0 * 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[(N[(2.0 * n), $MachinePrecision] * N[(Pi / k), $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. *-commutative 99.6%

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

   3. associate-*r* 99.6%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 96.7%

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

   6. expm1-udef 88.8%

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

3. Applied egg-rr 77.3%

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

   1. expm1-def 85.0%

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

   2. expm1-log1p 86.5%

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

   3. associate-*r* 86.5%

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

5. Simplified 86.5%

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

6. Taylor expanded in k around 0 33.7%

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

   1. *-commutative 33.7%

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

   2. *-commutative 33.7%

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

   3. associate-*l* 33.7%

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

8. Simplified 33.7%

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

   1. expm1-log1p-u 32.2%

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

   2. expm1-udef 35.0%

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

   3. associate-/l* 35.0%

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

   4. *-commutative 35.0%

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

10. Applied egg-rr 35.0%

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

    1. expm1-def 32.1%

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

    2. expm1-log1p 33.7%

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

    3. associate-/r/ 33.7%

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

12. Simplified 33.7%

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

13. Final simplification 33.7%

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

Reproduce

herbie shell --seed 2023223 
(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))))