Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 18.0s

Alternatives: 9

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

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

FPCore

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

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 * 0.5)));
}

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 * 0.5)));
}

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 * 0.5)))

Julia

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

MATLAB

function tmp = code(k, n)
	t_0 = pi * (2.0 * n);
	tmp = sqrt(t_0) / (sqrt(k) * (t_0 ^ (k * 0.5)));
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[(N[Sqrt[k], $MachinePrecision] * N[Power[t$95$0, N[(k * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

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

   1. unpow-prod-down 68.9%

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

   2. unpow-prod-down 99.5%

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

   3. div-sub 99.5%

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

   4. metadata-eval 99.5%

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

   5. pow-sub 99.7%

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

   6. pow1/2 99.7%

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

   7. frac-times 99.8%

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

   8. *-un-lft-identity 99.8%

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

   9. associate-*l* 99.8%

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

   10. associate-*l* 99.8%

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

   11. div-inv 99.8%

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

   12. metadata-eval 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r* 99.8%

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

   2. *-commutative 99.8%

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

   3. associate-*l* 99.8%

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

   4. associate-*r* 99.8%

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

   5. *-commutative 99.8%

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

   6. associate-*l* 99.8%

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

   7. *-commutative 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

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

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 5.5e-17)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (/ 1.0 (sqrt (/ k (pow (* n (* PI 2.0)) (- 1.0 k)))))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-17) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = 1.0 / sqrt((k / pow((n * (((double) M_PI) * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 5.5e-17) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow((n * (Math.PI * 2.0)), (1.0 - k))));
	}
	return tmp;
}

Python

def code(k, n):
	tmp = 0
	if k <= 5.5e-17:
		tmp = math.sqrt((math.pi * (2.0 * n))) / math.sqrt(k)
	else:
		tmp = 1.0 / math.sqrt((k / math.pow((n * (math.pi * 2.0)), (1.0 - k))))
	return tmp

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 5.5e-17)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = Float64(1.0 / sqrt(Float64(k / (Float64(n * Float64(pi * 2.0)) ^ Float64(1.0 - k)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 5.5e-17)
		tmp = sqrt((pi * (2.0 * n))) / sqrt(k);
	else
		tmp = 1.0 / sqrt((k / ((n * (pi * 2.0)) ^ (1.0 - k))));
	end
	tmp_2 = tmp;
end

Wolfram

code[k_, n_] := If[LessEqual[k, 5.5e-17], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sqrt[N[(k / N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.50000000000000001e-17

   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. Taylor expanded in k around 0 99.1%

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

      1. associate-*l/ 99.3%

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

      2. *-commutative 99.3%

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

      3. pow1/2 99.3%

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

      4. metadata-eval 99.3%

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

      5. pow1/2 99.3%

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

      6. metadata-eval 99.3%

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

      7. pow-prod-down 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*l* 99.5%

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

      10. *-commutative 99.5%

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

      11. metadata-eval 99.5%

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

      12. pow1/2 99.4%

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

      13. *-un-lft-identity 99.4%

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

      14. associate-*l* 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 5.50000000000000001e-17 < k

   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. div-sub 99.6%

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

      3. metadata-eval 99.6%

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

      4. div-inv 99.7%

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

      5. expm1-log1p-u 99.6%

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

      6. expm1-udef 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l* 99.6%

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

   5. Simplified 99.6%

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

      1. clear-num 99.6%

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

      2. sqrt-div 99.6%

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

      3. metadata-eval 99.6%

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

      4. associate-*r* 99.6%

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

   7. Applied egg-rr 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 5.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \pi \cdot \left(2 \cdot n\right)\\ \mathbf{if}\;k \leq 2.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{t_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* PI (* 2.0 n))))
   (if (<= k 2.4e-19)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

C

double code(double k, double n) {
	double t_0 = ((double) M_PI) * (2.0 * n);
	double tmp;
	if (k <= 2.4e-19) {
		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 = Math.PI * (2.0 * n);
	double tmp;
	if (k <= 2.4e-19) {
		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 = math.pi * (2.0 * n)
	tmp = 0
	if k <= 2.4e-19:
		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(pi * Float64(2.0 * n))
	tmp = 0.0
	if (k <= 2.4e-19)
		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 = pi * (2.0 * n);
	tmp = 0.0;
	if (k <= 2.4e-19)
		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[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, 2.4e-19], 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 < 2.40000000000000023e-19

   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. Taylor expanded in k around 0 99.1%

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

      1. associate-*l/ 99.3%

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

      2. *-commutative 99.3%

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

      3. pow1/2 99.3%

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

      4. metadata-eval 99.3%

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

      5. pow1/2 99.3%

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

      6. metadata-eval 99.3%

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

      7. pow-prod-down 99.5%

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

      8. *-commutative 99.5%

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

      9. associate-*l* 99.5%

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

      10. *-commutative 99.5%

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

      11. metadata-eval 99.5%

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

      12. pow1/2 99.4%

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

      13. *-un-lft-identity 99.4%

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

      14. associate-*l* 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 2.40000000000000023e-19 < k

   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. div-sub 99.6%

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

      3. metadata-eval 99.6%

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

      4. div-inv 99.7%

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

      5. expm1-log1p-u 99.6%

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

      6. expm1-udef 97.7%

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

   3. Applied egg-rr 97.7%

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

      1. expm1-def 99.6%

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

      2. expm1-log1p 99.6%

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

      3. associate-*r* 99.6%

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

      4. *-commutative 99.6%

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

      5. associate-*l* 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 2.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. associate-*l/ 99.6%

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

   2. *-lft-identity 99.6%

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

   3. sqr-pow 99.4%

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

   4. pow-sqr 99.6%

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

   5. *-commutative 99.6%

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

   6. associate-*l/ 99.6%

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

   7. associate-/l* 99.6%

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

   8. metadata-eval 99.6%

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

   9. /-rgt-identity 99.6%

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

   10. div-sub 99.6%

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

   11. metadata-eval 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 5: 50.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+236}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.05e+236)
   (/ (sqrt (* PI (* 2.0 n))) (sqrt k))
   (cbrt (pow (* 2.0 (* PI (/ n k))) 1.5))))

C

double code(double k, double n) {
	double tmp;
	if (k <= 1.05e+236) {
		tmp = sqrt((((double) M_PI) * (2.0 * n))) / sqrt(k);
	} else {
		tmp = cbrt(pow((2.0 * (((double) M_PI) * (n / k))), 1.5));
	}
	return tmp;
}

Java

public static double code(double k, double n) {
	double tmp;
	if (k <= 1.05e+236) {
		tmp = Math.sqrt((Math.PI * (2.0 * n))) / Math.sqrt(k);
	} else {
		tmp = Math.cbrt(Math.pow((2.0 * (Math.PI * (n / k))), 1.5));
	}
	return tmp;
}

Julia

function code(k, n)
	tmp = 0.0
	if (k <= 1.05e+236)
		tmp = Float64(sqrt(Float64(pi * Float64(2.0 * n))) / sqrt(k));
	else
		tmp = cbrt((Float64(2.0 * Float64(pi * Float64(n / k))) ^ 1.5));
	end
	return tmp
end

Wolfram

code[k_, n_] := If[LessEqual[k, 1.05e+236], N[(N[Sqrt[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Power[N[Power[N[(2.0 * N[(Pi * N[(n / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.5], $MachinePrecision], 1/3], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 1.05000000000000003e236

   1. Initial program 99.5%

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

   2. Taylor expanded in k around 0 50.6%

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

      1. associate-*l/ 50.7%

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

      2. *-commutative 50.7%

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

      3. pow1/2 50.7%

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

      4. metadata-eval 50.7%

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

      5. pow1/2 50.7%

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

      6. metadata-eval 50.7%

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

      7. pow-prod-down 50.7%

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

      8. *-commutative 50.7%

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

      9. associate-*l* 50.7%

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

      10. *-commutative 50.7%

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

      11. metadata-eval 50.7%

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

      12. pow1/2 50.7%

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

      13. *-un-lft-identity 50.7%

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

      14. associate-*l* 50.7%

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

   4. Applied egg-rr 50.7%

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

   if 1.05000000000000003e236 < k

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. div-sub 100.0%

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

      3. metadata-eval 100.0%

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

      4. div-inv 100.0%

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

      5. expm1-log1p-u 100.0%

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

      6. expm1-udef 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. expm1-def 100.0%

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

      2. expm1-log1p 100.0%

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

      3. associate-*r* 100.0%

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

      4. *-commutative 100.0%

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

      5. associate-*l* 100.0%

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

   5. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. unpow-prod-up 100.0%

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

      3. pow1 100.0%

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

      4. associate-*r* 100.0%

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

      5. associate-*r* 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in k around 0 2.6%

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

      1. associate-/l* 2.6%

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

      2. associate-/r/ 2.6%

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

   10. Simplified 2.6%

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

       1. add-cbrt-cube 15.2%

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

       2. add-sqr-sqrt 15.2%

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

       3. pow1 15.2%

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

       4. pow1/2 15.2%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{1} \cdot \color{blue}{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{0.5}}} \]

       5. pow-prod-up 15.2%

          \[\leadsto \sqrt[3]{\color{blue}{{\left(2 \cdot \left(\frac{n}{k} \cdot \pi\right)\right)}^{\left(1 + 0.5\right)}}} \]

       6. *-commutative 15.2%

          \[\leadsto \sqrt[3]{{\left(2 \cdot \color{blue}{\left(\pi \cdot \frac{n}{k}\right)}\right)}^{\left(1 + 0.5\right)}} \]

       7. metadata-eval 15.2%

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

   12. Applied egg-rr 15.2%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.05 \cdot 10^{+236}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \left(2 \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 6: 49.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

2. Taylor expanded in k around 0 43.5%

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

   1. associate-*l/ 43.6%

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

   2. *-commutative 43.6%

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

   3. pow1/2 43.6%

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

   4. metadata-eval 43.6%

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

   5. pow1/2 43.6%

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

   6. metadata-eval 43.6%

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

   7. pow-prod-down 43.6%

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

   8. *-commutative 43.6%

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

   9. associate-*l* 43.6%

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

   10. *-commutative 43.6%

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

   11. metadata-eval 43.6%

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

   12. pow1/2 43.6%

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

   13. *-un-lft-identity 43.6%

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

   14. associate-*l* 43.6%

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

4. Applied egg-rr 43.6%

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

5. Final simplification 43.6%

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

Alternative 7: 38.3% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. div-sub 99.5%

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

   3. metadata-eval 99.5%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 97.1%

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

   6. expm1-udef 87.6%

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

3. Applied egg-rr 79.1%

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

   1. expm1-def 88.6%

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

   2. expm1-log1p 90.1%

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

   3. associate-*r* 90.1%

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

   4. *-commutative 90.1%

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

   5. associate-*l* 90.1%

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

5. Simplified 90.1%

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

   1. clear-num 90.1%

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

   2. sqrt-div 91.5%

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

   3. metadata-eval 91.5%

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

   4. associate-*r* 91.5%

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

7. Applied egg-rr 91.5%

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

8. Taylor expanded in k around 0 35.5%

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

   1. associate-*r* 35.5%

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

   2. *-commutative 35.5%

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

   3. associate-*l* 35.5%

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

10. Simplified 35.5%

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

11. Final simplification 35.5%

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

Alternative 8: 37.7% 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 99.5%

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

   1. *-commutative 99.5%

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

   2. div-sub 99.5%

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

   3. metadata-eval 99.5%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 97.1%

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

   6. expm1-udef 87.6%

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

3. Applied egg-rr 79.1%

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

   1. expm1-def 88.6%

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

   2. expm1-log1p 90.1%

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

   3. associate-*r* 90.1%

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

   4. *-commutative 90.1%

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

   5. associate-*l* 90.1%

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

5. Simplified 90.1%

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

   1. sub-neg 90.1%

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

   2. unpow-prod-up 90.3%

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

   3. pow1 90.3%

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

   4. associate-*r* 90.3%

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

   5. associate-*r* 90.3%

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

7. Applied egg-rr 90.3%

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

8. Taylor expanded in k around 0 34.1%

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

   1. associate-/l* 34.1%

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

   2. associate-/r/ 34.1%

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

10. Simplified 34.1%

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

11. Final simplification 34.1%

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

Alternative 9: 37.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. *-commutative 99.5%

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

   2. div-sub 99.5%

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

   3. metadata-eval 99.5%

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

   4. div-inv 99.6%

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

   5. expm1-log1p-u 97.1%

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

   6. expm1-udef 87.6%

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

3. Applied egg-rr 79.1%

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

   1. expm1-def 88.6%

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

   2. expm1-log1p 90.1%

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

   3. associate-*r* 90.1%

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

   4. *-commutative 90.1%

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

   5. associate-*l* 90.1%

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

5. Simplified 90.1%

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

   1. sub-neg 90.1%

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

   2. unpow-prod-up 90.3%

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

   3. pow1 90.3%

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

   4. associate-*r* 90.3%

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

   5. associate-*r* 90.3%

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

7. Applied egg-rr 90.3%

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

8. Taylor expanded in k around 0 34.1%

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

   1. associate-/l* 34.1%

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

   2. associate-/r/ 34.1%

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

10. Simplified 34.1%

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

11. Taylor expanded in n around 0 34.1%

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

    1. *-commutative 34.1%

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

    2. associate-*r/ 34.1%

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

    3. associate-*r* 34.1%

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

13. Simplified 34.1%

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

14. Final simplification 34.1%

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

Reproduce

herbie shell --seed 2023279 
(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))))