Migdal et al, Equation (51)

Average Error: 0.5 → 1.0

Time: 11.0s

Precision: binary64

Cost: 19908

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \frac{n}{0.5}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

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-60)
   (/ (sqrt (* PI (/ n 0.5))) (sqrt k))
   (sqrt (/ (pow (* n (* PI 2.0)) (- 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-60) {
		tmp = sqrt((((double) M_PI) * (n / 0.5))) / sqrt(k);
	} else {
		tmp = sqrt((pow((n * (((double) M_PI) * 2.0)), (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-60) {
		tmp = Math.sqrt((Math.PI * (n / 0.5))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((n * (Math.PI * 2.0)), (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-60:
		tmp = math.sqrt((math.pi * (n / 0.5))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((n * (math.pi * 2.0)), (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-60)
		tmp = Float64(sqrt(Float64(pi * Float64(n / 0.5))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(n * Float64(pi * 2.0)) ^ 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-60)
		tmp = sqrt((pi * (n / 0.5))) / sqrt(k);
	else
		tmp = sqrt((((n * (pi * 2.0)) ^ (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-60], N[(N[Sqrt[N[(Pi * N[(n / 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 4.50000000000000001e-60

   1. Initial program 0.6

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

   2. Simplified 0.5

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

      [Start] 0.6	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      associate-*l/ [=>] 0.5	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      *-lft-identity [=>] 0.5	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [=>] 0.7	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [<=] 0.5	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      *-commutative [=>] 0.5	\[ \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      associate-*l* [=>] 0.5	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      div-sub [=>] 0.5	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]
      metadata-eval [=>] 0.5	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   3. Applied egg-rr 1.4

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

   4. Taylor expanded in k around 0 22.5

      \[\leadsto {\color{blue}{\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.16666666666666666}\right)}}^{3} \]

   5. Simplified 22.5

      \[\leadsto {\color{blue}{\left({\left(\frac{n}{k} \cdot \left(2 \cdot \pi\right)\right)}^{0.16666666666666666}\right)}}^{3} \]
      Proof

      [Start] 22.5	\[ {\left({\left(2 \cdot \frac{n \cdot \pi}{k}\right)}^{0.16666666666666666}\right)}^{3} \]
      associate-*r/ [=>] 22.5	\[ {\left({\color{blue}{\left(\frac{2 \cdot \left(n \cdot \pi\right)}{k}\right)}}^{0.16666666666666666}\right)}^{3} \]
      associate-*r* [=>] 22.5	\[ {\left({\left(\frac{\color{blue}{\left(2 \cdot n\right) \cdot \pi}}{k}\right)}^{0.16666666666666666}\right)}^{3} \]
      *-commutative [=>] 22.5	\[ {\left({\left(\frac{\color{blue}{\left(n \cdot 2\right)} \cdot \pi}{k}\right)}^{0.16666666666666666}\right)}^{3} \]
      associate-*r* [<=] 22.5	\[ {\left({\left(\frac{\color{blue}{n \cdot \left(2 \cdot \pi\right)}}{k}\right)}^{0.16666666666666666}\right)}^{3} \]
      associate-/l* [=>] 22.5	\[ {\left({\color{blue}{\left(\frac{n}{\frac{k}{2 \cdot \pi}}\right)}}^{0.16666666666666666}\right)}^{3} \]
      associate-/r/ [=>] 22.5	\[ {\left({\color{blue}{\left(\frac{n}{k} \cdot \left(2 \cdot \pi\right)\right)}}^{0.16666666666666666}\right)}^{3} \]

   6. Applied egg-rr 0.5

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

   7. Simplified 0.5

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

      [Start] 0.5	\[ \frac{\sqrt{\left(n \cdot 2\right) \cdot \pi}}{\sqrt{k}} \]
      remove-double-neg [<=] 0.5	\[ \frac{\sqrt{\color{blue}{-\left(-\left(n \cdot 2\right) \cdot \pi\right)}}}{\sqrt{k}} \]
      associate-*l* [=>] 0.5	\[ \frac{\sqrt{-\left(-\color{blue}{n \cdot \left(2 \cdot \pi\right)}\right)}}{\sqrt{k}} \]
      distribute-lft-neg-out [<=] 0.5	\[ \frac{\sqrt{-\color{blue}{\left(-n\right) \cdot \left(2 \cdot \pi\right)}}}{\sqrt{k}} \]
      /-rgt-identity [<=] 0.5	\[ \frac{\sqrt{-\left(-n\right) \cdot \color{blue}{\frac{2 \cdot \pi}{1}}}}{\sqrt{k}} \]
      associate-*r/ [=>] 0.5	\[ \frac{\sqrt{-\color{blue}{\frac{\left(-n\right) \cdot \left(2 \cdot \pi\right)}{1}}}}{\sqrt{k}} \]
      associate-/l* [=>] 0.5	\[ \frac{\sqrt{-\color{blue}{\frac{-n}{\frac{1}{2 \cdot \pi}}}}}{\sqrt{k}} \]
      associate-/r* [=>] 0.5	\[ \frac{\sqrt{-\frac{-n}{\color{blue}{\frac{\frac{1}{2}}{\pi}}}}}{\sqrt{k}} \]
      metadata-eval [=>] 0.5	\[ \frac{\sqrt{-\frac{-n}{\frac{\color{blue}{0.5}}{\pi}}}}{\sqrt{k}} \]
      distribute-frac-neg [<=] 0.5	\[ \frac{\sqrt{\color{blue}{\frac{-\left(-n\right)}{\frac{0.5}{\pi}}}}}{\sqrt{k}} \]
      remove-double-neg [=>] 0.5	\[ \frac{\sqrt{\frac{\color{blue}{n}}{\frac{0.5}{\pi}}}}{\sqrt{k}} \]
      associate-/r/ [=>] 0.5	\[ \frac{\sqrt{\color{blue}{\frac{n}{0.5} \cdot \pi}}}{\sqrt{k}} \]

   if 4.50000000000000001e-60 < k

   1. Initial program 0.4

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

   2. Simplified 0.4

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

      [Start] 0.4	\[ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      associate-*l/ [=>] 0.4	\[ \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}} \]
      *-lft-identity [=>] 0.4	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [=>] 0.5	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}{\sqrt{k}} \]
      sqr-pow [<=] 0.4	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
      *-commutative [=>] 0.4	\[ \frac{{\left(\color{blue}{\left(\pi \cdot 2\right)} \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      associate-*l* [=>] 0.4	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
      div-sub [=>] 0.4	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]
      metadata-eval [=>] 0.4	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

   3. Applied egg-rr 1.5

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

   4. Simplified 1.5

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

      [Start] 1.5	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      associate-*r* [=>] 1.5	\[ \sqrt{\frac{{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [=>] 1.5	\[ \sqrt{\frac{{\color{blue}{\left(n \cdot \left(\pi \cdot 2\right)\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [=>] 1.5	\[ \sqrt{\frac{{\left(n \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      distribute-rgt-in [=>] 1.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{\left(0.5 \cdot 2 + \left(k \cdot -0.5\right) \cdot 2\right)}}}{k}} \]
      metadata-eval [=>] 1.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\color{blue}{1} + \left(k \cdot -0.5\right) \cdot 2\right)}}{k}} \]
      associate-*l* [=>] 1.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + \color{blue}{k \cdot \left(-0.5 \cdot 2\right)}\right)}}{k}} \]
      metadata-eval [=>] 1.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + k \cdot \color{blue}{-1}\right)}}{k}} \]

   5. Taylor expanded in n around 0 2.6

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

   6. Simplified 1.5

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

      [Start] 2.6	\[ \sqrt{\frac{e^{\left(1 + -1 \cdot k\right) \cdot \left(\log n + \log \left(2 \cdot \pi\right)\right)}}{k}} \]
      distribute-rgt-in [=>] 2.6	\[ \sqrt{\frac{e^{\color{blue}{\log n \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}}{k}} \]
      remove-double-neg [<=] 2.6	\[ \sqrt{\frac{e^{\color{blue}{\left(-\left(-\log n\right)\right)} \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}{k}} \]
      log-rec [<=] 2.6	\[ \sqrt{\frac{e^{\left(-\color{blue}{\log \left(\frac{1}{n}\right)}\right) \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}{k}} \]
      mul-1-neg [<=] 2.6	\[ \sqrt{\frac{e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{n}\right)\right)} \cdot \left(1 + -1 \cdot k\right) + \log \left(2 \cdot \pi\right) \cdot \left(1 + -1 \cdot k\right)}}{k}} \]
      distribute-rgt-in [<=] 2.6	\[ \sqrt{\frac{e^{\color{blue}{\left(1 + -1 \cdot k\right) \cdot \left(-1 \cdot \log \left(\frac{1}{n}\right) + \log \left(2 \cdot \pi\right)\right)}}}{k}} \]
      *-commutative [<=] 2.6	\[ \sqrt{\frac{e^{\color{blue}{\left(-1 \cdot \log \left(\frac{1}{n}\right) + \log \left(2 \cdot \pi\right)\right) \cdot \left(1 + -1 \cdot k\right)}}}{k}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 4.5 \cdot 10^{-60}:\\ \;\;\;\;\frac{\sqrt{\pi \cdot \frac{n}{0.5}}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	19904

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

Alternative 2
Error	22.2
Cost	19584

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

Alternative 3
Error	22.1
Cost	19584

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

Alternative 4
Error	22.2
Cost	19584

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

Alternative 5
Error	32.0
Cost	13312

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

Alternative 6
Error	32.6
Cost	13184

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

Alternative 7
Error	32.6
Cost	13184

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

Alternative 8
Error	32.6
Cost	13184

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

Reproduce

herbie shell --seed 2023045 
(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))))