Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 13.2s

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: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. associate-*r* 99.4%

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

   3. div-sub 99.4%

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

   4. metadata-eval 99.4%

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

   5. div-inv 99.4%

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

   6. pow-sub 99.7%

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

   7. pow1/2 99.7%

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

   8. associate-/l/ 99.7%

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

   9. associate-*r* 99.7%

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

   10. *-commutative 99.7%

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

   11. associate-*l* 99.7%

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

   12. div-inv 99.7%

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

   13. metadata-eval 99.7%

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

3. Applied egg-rr 99.7%

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

   1. expm1-log1p-u 96.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}\right)\right)} \]

   2. expm1-udef 83.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k} \cdot {\left({\left(\pi \cdot \left(2 \cdot n\right)\right)}^{k}\right)}^{0.5}}\right)} - 1} \]

5. Applied egg-rr 83.4%

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

   1. expm1-def 96.5%

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

   2. expm1-log1p 99.7%

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

   3. *-commutative 99.7%

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

   4. associate-*r* 99.7%

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

   5. *-commutative 99.7%

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

   6. associate-*r* 99.7%

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

7. Simplified 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \mathbf{if}\;k \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* n (* PI 2.0))))
   (if (<= k 2e-30) (/ (sqrt t_0) (sqrt k)) (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = n * (((double) M_PI) * 2.0);
	double tmp;
	if (k <= 2e-30) {
		tmp = sqrt(t_0) / sqrt(k);
	} else {
		tmp = sqrt((pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double t_0 = n * (Math.PI * 2.0);
	double tmp;
	if (k <= 2e-30) {
		tmp = Math.sqrt(t_0) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow(t_0, (1.0 - k)) / k));
	}
	return tmp;
}

Python

def code(k, n):
	t_0 = n * (math.pi * 2.0)
	tmp = 0
	if k <= 2e-30:
		tmp = math.sqrt(t_0) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow(t_0, (1.0 - k)) / k))
	return tmp

Julia

function code(k, n)
	t_0 = Float64(n * Float64(pi * 2.0))
	tmp = 0.0
	if (k <= 2e-30)
		tmp = Float64(sqrt(t_0) / sqrt(k));
	else
		tmp = sqrt(Float64((t_0 ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	t_0 = n * (pi * 2.0);
	tmp = 0.0;
	if (k <= 2e-30)
		tmp = sqrt(t_0) / sqrt(k);
	else
		tmp = sqrt(((t_0 ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := Block[{t$95$0 = N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2e-30], N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[t$95$0, N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if k < 2e-30

   1. Initial program 99.3%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. sqr-pow 99.1%

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

      4. pow-sqr 99.5%

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

      5. associate-*l* 99.5%

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

      6. *-commutative 99.5%

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

      7. associate-*l/ 99.5%

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

      8. associate-/l* 99.5%

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

      9. metadata-eval 99.5%

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

      10. /-rgt-identity 99.5%

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

      11. div-sub 99.5%

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

      12. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in k around inf 92.3%

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

   5. Taylor expanded in k around 0 99.2%

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

      1. expm1-log1p-u 95.7%

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

      2. expm1-udef 49.7%

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

      3. *-commutative 49.7%

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

      4. sqrt-unprod 49.7%

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

      5. *-commutative 49.7%

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

   7. Applied egg-rr 49.7%

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

      1. expm1-def 95.8%

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

      2. expm1-log1p 99.5%

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

      3. *-commutative 99.5%

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

      4. associate-*r* 99.5%

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

      5. *-commutative 99.5%

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

      6. associate-*r* 99.5%

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

   9. Simplified 99.5%

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

   if 2e-30 < k

   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. add-sqr-sqrt 99.3%

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

      2. sqrt-unprod 99.4%

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

      3. *-commutative 99.4%

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

      4. div-inv 99.4%

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

      5. *-commutative 99.4%

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

      6. div-inv 99.4%

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

      7. frac-times 99.4%

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

   3. Applied egg-rr 99.4%

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

   5. Simplified 99.4%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2 \cdot 10^{-30}:\\ \;\;\;\;\frac{\sqrt{n \cdot \left(\pi \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 3: 99.4% 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. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

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. Final simplification 99.4%

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

Alternative 4: 48.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(Pi * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / 2.0), $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. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

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. Taylor expanded in k around inf 95.7%

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

5. Taylor expanded in k around 0 53.3%

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

   1. associate-/l* 53.4%

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

   2. div-inv 53.3%

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

   3. *-commutative 53.3%

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

   4. sqrt-undiv 53.4%

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

7. Applied egg-rr 53.4%

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

   1. associate-*r/ 53.4%

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

   2. *-rgt-identity 53.4%

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

   3. *-commutative 53.4%

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

9. Simplified 53.4%

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

10. Final simplification 53.4%

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

Alternative 5: 48.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(N[Sqrt[N[(n * N[(Pi * 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. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

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. Taylor expanded in k around inf 95.7%

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

5. Taylor expanded in k around 0 53.3%

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

   1. expm1-log1p-u 51.5%

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

   2. expm1-udef 27.6%

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

   3. *-commutative 27.6%

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

   4. sqrt-unprod 27.6%

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

   5. *-commutative 27.6%

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

7. Applied egg-rr 27.6%

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

   1. expm1-def 51.6%

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

   2. expm1-log1p 53.5%

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

   3. *-commutative 53.5%

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

   4. associate-*r* 53.5%

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

   5. *-commutative 53.5%

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

   6. associate-*r* 53.5%

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

9. Simplified 53.5%

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

10. Final simplification 53.5%

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

Alternative 6: 37.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[(1.0 / N[Sqrt[N[(N[(k / n), $MachinePrecision] * N[(1.0 / N[(Pi * 2.0), $MachinePrecision]), $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. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 84.0%

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

   3. *-commutative 84.0%

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

   4. div-inv 84.0%

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

   5. *-commutative 84.0%

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

   6. div-inv 84.1%

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

   7. frac-times 84.0%

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

3. Applied egg-rr 84.0%

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

5. Simplified 84.1%

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

   1. clear-num 84.1%

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

   2. sqrt-div 85.6%

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

   3. metadata-eval 85.6%

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

   4. *-commutative 85.6%

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

7. Applied egg-rr 85.6%

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

   1. *-commutative 85.6%

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

   2. associate-*r* 85.6%

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

9. Simplified 85.6%

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

10. Taylor expanded in k around 0 39.6%

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

    1. associate-*r* 39.6%

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

    2. *-commutative 39.6%

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

    3. associate-*r* 39.6%

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

12. Simplified 39.6%

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

    1. associate-/r* 39.6%

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

    2. div-inv 39.6%

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

    3. *-commutative 39.6%

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

14. Applied egg-rr 39.6%

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

15. Final simplification 39.6%

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

Alternative 7: 37.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 84.0%

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

   3. *-commutative 84.0%

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

   4. div-inv 84.0%

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

   5. *-commutative 84.0%

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

   6. div-inv 84.1%

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

   7. frac-times 84.0%

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

3. Applied egg-rr 84.0%

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

5. Simplified 84.1%

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

   1. clear-num 84.1%

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

   2. sqrt-div 85.6%

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

   3. metadata-eval 85.6%

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

   4. *-commutative 85.6%

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

7. Applied egg-rr 85.6%

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

   1. *-commutative 85.6%

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

   2. associate-*r* 85.6%

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

9. Simplified 85.6%

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

10. Taylor expanded in k around 0 39.6%

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

    1. associate-*r* 39.6%

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

    2. *-commutative 39.6%

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

    3. associate-*r* 39.6%

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

12. Simplified 39.6%

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

13. Taylor expanded in k around 0 39.6%

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

    1. associate-*r/ 39.6%

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

    2. *-commutative 39.6%

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

    3. associate-*r/ 39.6%

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

15. Simplified 39.6%

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

16. Final simplification 39.6%

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

Alternative 8: 37.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(Pi * n), $MachinePrecision] * N[(2.0 / 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. 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. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

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. Taylor expanded in k around inf 95.7%

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

5. Taylor expanded in k around 0 53.3%

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

   1. expm1-log1p-u 50.4%

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

   2. expm1-udef 45.7%

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

   3. sqrt-unprod 45.7%

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

   4. sqrt-undiv 31.9%

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

   5. associate-*r* 31.9%

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

   6. *-commutative 31.9%

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

   7. *-commutative 31.9%

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

   8. associate-*l* 31.9%

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

7. Applied egg-rr 31.9%

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

   1. expm1-def 36.6%

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

   2. expm1-log1p 38.1%

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

   3. associate-/l* 38.1%

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

   4. *-commutative 38.1%

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

   5. associate-/r/ 38.1%

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

9. Simplified 38.1%

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

10. Final simplification 38.1%

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

Alternative 9: 37.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(2.0 / N[(k / N[(Pi * n), $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. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

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. Taylor expanded in k around inf 95.7%

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

5. Taylor expanded in k around 0 53.3%

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

   1. expm1-log1p-u 50.4%

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

   2. expm1-udef 45.7%

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

   3. sqrt-unprod 45.7%

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

   4. sqrt-undiv 31.9%

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

   5. associate-*r* 31.9%

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

   6. *-commutative 31.9%

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

   7. *-commutative 31.9%

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

   8. associate-*l* 31.9%

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

7. Applied egg-rr 31.9%

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

   1. expm1-def 36.6%

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

   2. expm1-log1p 38.1%

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

   3. associate-/l* 38.1%

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

   4. *-commutative 38.1%

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

9. Simplified 38.1%

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

10. Final simplification 38.1%

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

Alternative 10: 37.3% accurate, 1.5× speedup

Math

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

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. 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. sqr-pow 99.2%

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

   4. pow-sqr 99.4%

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

   5. associate-*l* 99.4%

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

   6. *-commutative 99.4%

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

   7. associate-*l/ 99.4%

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

   8. associate-/l* 99.4%

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

   9. metadata-eval 99.4%

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

   10. /-rgt-identity 99.4%

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

   11. div-sub 99.4%

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

   12. metadata-eval 99.4%

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

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. Taylor expanded in k around inf 95.7%

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

5. Taylor expanded in k around 0 53.3%

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

   1. sqrt-unprod 53.5%

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

   2. sqrt-undiv 38.1%

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

   3. associate-*r* 38.1%

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

   4. *-commutative 38.1%

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

   5. *-commutative 38.1%

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

   6. associate-*l* 38.1%

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

7. Applied egg-rr 38.1%

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

8. Final simplification 38.1%

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

Reproduce

herbie shell --seed 2023292 
(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))))