Migdal et al, Equation (51)

Average Accuracy: 99.2% → 98.8%

Time: 12.4s

Precision: binary64

Cost: 19908

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

FPCore

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

↓

(FPCore (k n)
 :precision binary64
 (if (<= k 4.5e-47)
   (* (sqrt (* 2.0 (/ PI k))) (sqrt n))
   (sqrt (/ (pow (* PI (+ n n)) (- 1.0 k)) k))))

C

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

↓

double code(double k, double n) {
	double tmp;
	if (k <= 4.5e-47) {
		tmp = sqrt((2.0 * (((double) M_PI) / k))) * sqrt(n);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (n + n)), (1.0 - k)) / k));
	}
	return tmp;
}

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));
}

↓

public static double code(double k, double n) {
	double tmp;
	if (k <= 4.5e-47) {
		tmp = Math.sqrt((2.0 * (Math.PI / k))) * Math.sqrt(n);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (n + n)), (1.0 - k)) / k));
	}
	return tmp;
}

Python

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

↓

def code(k, n):
	tmp = 0
	if k <= 4.5e-47:
		tmp = math.sqrt((2.0 * (math.pi / k))) * math.sqrt(n)
	else:
		tmp = math.sqrt((math.pow((math.pi * (n + n)), (1.0 - k)) / k))
	return tmp

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

↓

function code(k, n)
	tmp = 0.0
	if (k <= 4.5e-47)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi / k))) * sqrt(n));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(n + n)) ^ Float64(1.0 - k)) / k));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(k, n)
	tmp = 0.0;
	if (k <= 4.5e-47)
		tmp = sqrt((2.0 * (pi / k))) * sqrt(n);
	else
		tmp = sqrt((((pi * (n + n)) ^ (1.0 - k)) / k));
	end
	tmp_2 = tmp;
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]

↓

code[k_, n_] := If[LessEqual[k, 4.5e-47], N[(N[Sqrt[N[(2.0 * N[(Pi / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[n], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(n + n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.5e-47

   1. Initial program 99.2%

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

   2. Simplified 99.3%

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

      [Start] 99.2	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      *-lft-identity [=>] 99.3	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      *-commutative [=>] 99.3	\[ \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      associate-*l* [=>] 99.3	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]

   3. Applied egg-rr 70.6%

      \[\leadsto \color{blue}{0 + \sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}} \]
      Proof

      [Start] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      add-log-exp [=>] 4.4	\[ \color{blue}{\log \left(e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)} \]
      *-un-lft-identity [=>] 4.4	\[ \log \color{blue}{\left(1 \cdot e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)} \]
      log-prod [=>] 4.4	\[ \color{blue}{\log 1 + \log \left(e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)} \]
      metadata-eval [=>] 4.4	\[ \color{blue}{0} + \log \left(e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right) \]
      add-log-exp [<=] 99.3	\[ 0 + \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      add-sqr-sqrt [=>] 98.9	\[ 0 + \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}} \]
      sqrt-unprod [=>] 70.4	\[ 0 + \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}} \]
      frac-times [=>] 70.3	\[ 0 + \sqrt{\color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k} \cdot \sqrt{k}}}} \]

   4. Simplified 70.6%

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

      [Start] 70.6	\[ 0 + \sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}} \]
      +-lft-identity [=>] 70.6	\[ \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}} \]

   5. Taylor expanded in k around 0 70.6%

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

   6. Applied egg-rr 69.9%

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

      [Start] 70.6	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k}} \]
      pow1/2 [=>] 70.6	\[ \color{blue}{{\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.5}} \]
      add-cube-cbrt [=>] 69.9	\[ {\color{blue}{\left(\left(\sqrt[3]{2 \cdot \frac{n \cdot \pi}{k}} \cdot \sqrt[3]{2 \cdot \frac{n \cdot \pi}{k}}\right) \cdot \sqrt[3]{2 \cdot \frac{n \cdot \pi}{k}}\right)}}^{0.5} \]
      pow3 [=>] 69.9	\[ {\color{blue}{\left({\left(\sqrt[3]{2 \cdot \frac{n \cdot \pi}{k}}\right)}^{3}\right)}}^{0.5} \]
      metadata-eval [<=] 69.9	\[ {\left({\left(\sqrt[3]{2 \cdot \frac{n \cdot \pi}{k}}\right)}^{\color{blue}{\left(1 + 2\right)}}\right)}^{0.5} \]
      pow-pow [=>] 69.9	\[ \color{blue}{{\left(\sqrt[3]{2 \cdot \frac{n \cdot \pi}{k}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)}} \]
      clear-num [=>] 69.9	\[ {\left(\sqrt[3]{2 \cdot \color{blue}{\frac{1}{\frac{k}{n \cdot \pi}}}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      un-div-inv [=>] 69.9	\[ {\left(\sqrt[3]{\color{blue}{\frac{2}{\frac{k}{n \cdot \pi}}}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      *-commutative [=>] 69.9	\[ {\left(\sqrt[3]{\frac{2}{\frac{k}{\color{blue}{\pi \cdot n}}}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      associate-/r* [=>] 69.9	\[ {\left(\sqrt[3]{\frac{2}{\color{blue}{\frac{\frac{k}{\pi}}{n}}}}\right)}^{\left(\left(1 + 2\right) \cdot 0.5\right)} \]
      metadata-eval [=>] 69.9	\[ {\left(\sqrt[3]{\frac{2}{\frac{\frac{k}{\pi}}{n}}}\right)}^{\left(\color{blue}{3} \cdot 0.5\right)} \]
      metadata-eval [=>] 69.9	\[ {\left(\sqrt[3]{\frac{2}{\frac{\frac{k}{\pi}}{n}}}\right)}^{\color{blue}{1.5}} \]

   7. Simplified 69.8%

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

      [Start] 69.9	\[ {\left(\sqrt[3]{\frac{2}{\frac{\frac{k}{\pi}}{n}}}\right)}^{1.5} \]
      associate-/r/ [=>] 69.8	\[ {\left(\sqrt[3]{\color{blue}{\frac{2}{\frac{k}{\pi}} \cdot n}}\right)}^{1.5} \]
      *-commutative [=>] 69.8	\[ {\left(\sqrt[3]{\color{blue}{n \cdot \frac{2}{\frac{k}{\pi}}}}\right)}^{1.5} \]
      associate-/r/ [=>] 69.8	\[ {\left(\sqrt[3]{n \cdot \color{blue}{\left(\frac{2}{k} \cdot \pi\right)}}\right)}^{1.5} \]

   8. Applied egg-rr 99.3%

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

      [Start] 69.8	\[ {\left(\sqrt[3]{n \cdot \left(\frac{2}{k} \cdot \pi\right)}\right)}^{1.5} \]
      pow1/3 [=>] 64.8	\[ {\color{blue}{\left({\left(n \cdot \left(\frac{2}{k} \cdot \pi\right)\right)}^{0.3333333333333333}\right)}}^{1.5} \]
      pow-pow [=>] 70.5	\[ \color{blue}{{\left(n \cdot \left(\frac{2}{k} \cdot \pi\right)\right)}^{\left(0.3333333333333333 \cdot 1.5\right)}} \]
      metadata-eval [=>] 70.5	\[ {\left(n \cdot \left(\frac{2}{k} \cdot \pi\right)\right)}^{\color{blue}{0.5}} \]
      pow1/2 [<=] 70.5	\[ \color{blue}{\sqrt{n \cdot \left(\frac{2}{k} \cdot \pi\right)}} \]
      *-commutative [=>] 70.5	\[ \sqrt{\color{blue}{\left(\frac{2}{k} \cdot \pi\right) \cdot n}} \]
      sqrt-prod [=>] 99.2	\[ \color{blue}{\sqrt{\frac{2}{k} \cdot \pi} \cdot \sqrt{n}} \]
      associate-*l/ [=>] 99.3	\[ \sqrt{\color{blue}{\frac{2 \cdot \pi}{k}}} \cdot \sqrt{n} \]
      *-un-lft-identity [=>] 99.3	\[ \sqrt{\frac{2 \cdot \pi}{\color{blue}{1 \cdot k}}} \cdot \sqrt{n} \]
      times-frac [=>] 99.3	\[ \sqrt{\color{blue}{\frac{2}{1} \cdot \frac{\pi}{k}}} \cdot \sqrt{n} \]
      metadata-eval [=>] 99.3	\[ \sqrt{\color{blue}{2} \cdot \frac{\pi}{k}} \cdot \sqrt{n} \]

   if 4.5e-47 < k

   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. Simplified 99.3%

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

      [Start] 99.3	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      associate-*l/ [=>] 99.3	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      *-lft-identity [=>] 99.3	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      *-commutative [=>] 99.3	\[ \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      associate-*l* [=>] 99.3	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]

   3. Applied egg-rr 98.3%

      \[\leadsto \color{blue}{0 + \sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}} \]
      Proof

      [Start] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      add-log-exp [=>] 78.1	\[ \color{blue}{\log \left(e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)} \]
      *-un-lft-identity [=>] 78.1	\[ \log \color{blue}{\left(1 \cdot e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)} \]
      log-prod [=>] 78.1	\[ \color{blue}{\log 1 + \log \left(e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right)} \]
      metadata-eval [=>] 78.1	\[ \color{blue}{0} + \log \left(e^{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}\right) \]
      add-log-exp [<=] 99.3	\[ 0 + \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      add-sqr-sqrt [=>] 99.3	\[ 0 + \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \cdot \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}} \]
      sqrt-unprod [=>] 98.2	\[ 0 + \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \cdot \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}}} \]
      frac-times [=>] 98.2	\[ 0 + \sqrt{\color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k} \cdot \sqrt{k}}}} \]

   4. Simplified 98.3%

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

      [Start] 98.3	\[ 0 + \sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}} \]
      +-lft-identity [=>] 98.3	\[ \color{blue}{\sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-47}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(n + n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.3%
Cost	19904

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

Alternative 2
Accuracy	72.3%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(2 \cdot \left(\pi \cdot \frac{n}{k}\right)\right)}^{3}\right)}^{0.16666666666666666}\\ \end{array} \]

Alternative 3
Accuracy	72.7%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;k \leq 3.3 \cdot 10^{+141}:\\ \;\;\;\;\sqrt{2 \cdot \frac{\pi}{k}} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{\frac{k}{n + n}}{\pi}\right)}^{3}\right)}^{-0.16666666666666666}\\ \end{array} \]

Alternative 4
Accuracy	65.5%
Cost	19584

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

Alternative 5
Accuracy	49.9%
Cost	13248

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

Alternative 6
Accuracy	48.9%
Cost	13184

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

Alternative 7
Accuracy	48.8%
Cost	13184

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

Alternative 8
Accuracy	48.8%
Cost	13184

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

Reproduce

herbie shell --seed 2023151 
(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))))