Migdal et al, Equation (51)

Average Error: 0.5 → 0.8

Time: 10.2s

Precision: binary64

Cost: 19908

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

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 6e-56)
   (/ (sqrt (* 2.0 (* PI n))) (sqrt k))
   (sqrt (/ (pow (* PI (* 2.0 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 <= 6e-56) {
		tmp = sqrt((2.0 * (((double) M_PI) * n))) / sqrt(k);
	} else {
		tmp = sqrt((pow((((double) M_PI) * (2.0 * 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 <= 6e-56) {
		tmp = Math.sqrt((2.0 * (Math.PI * n))) / Math.sqrt(k);
	} else {
		tmp = Math.sqrt((Math.pow((Math.PI * (2.0 * 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 <= 6e-56:
		tmp = math.sqrt((2.0 * (math.pi * n))) / math.sqrt(k)
	else:
		tmp = math.sqrt((math.pow((math.pi * (2.0 * 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 <= 6e-56)
		tmp = Float64(sqrt(Float64(2.0 * Float64(pi * n))) / sqrt(k));
	else
		tmp = sqrt(Float64((Float64(pi * Float64(2.0 * 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 <= 6e-56)
		tmp = sqrt((2.0 * (pi * n))) / sqrt(k);
	else
		tmp = sqrt((((pi * (2.0 * 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, 6e-56], N[(N[Sqrt[N[(2.0 * N[(Pi * n), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[N[(Pi * N[(2.0 * n), $MachinePrecision]), $MachinePrecision], N[(1.0 - k), $MachinePrecision]], $MachinePrecision] / k), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < 5.99999999999999979e-56

   1. Initial program 0.5

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

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

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

   4. Simplified 0.5

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

      [Start] 0.5	\[ \frac{\sqrt{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 - \frac{k}{2}\right)\right)}}}{\sqrt{k}} \]
      associate-*r* [=>] 0.5	\[ \frac{\sqrt{{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(2 \cdot \left(0.5 - \frac{k}{2}\right)\right)}}}{\sqrt{k}} \]
      *-commutative [<=] 0.5	\[ \frac{\sqrt{{\left(\color{blue}{\left(2 \cdot \pi\right)} \cdot n\right)}^{\left(2 \cdot \left(0.5 - \frac{k}{2}\right)\right)}}}{\sqrt{k}} \]

   5. Taylor expanded in k around 0 0.5

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

   if 5.99999999999999979e-56 < k

   1. Initial program 0.5

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

   2. Applied egg-rr 0.5

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

   3. Simplified 0.5

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

      [Start] 0.5	\[ \left({\left({k}^{0.25}\right)}^{-1} \cdot {\left({k}^{0.25}\right)}^{-1}\right) \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      pow-sqr [=>] 0.5	\[ \color{blue}{{\left({k}^{0.25}\right)}^{\left(2 \cdot -1\right)}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
      metadata-eval [=>] 0.5	\[ {\left({k}^{0.25}\right)}^{\color{blue}{-2}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]

   4. Applied egg-rr 8.0

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)} - 1} \]

   5. Simplified 1.1

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

      [Start] 8.0	\[ e^{\mathsf{log1p}\left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)} - 1 \]
      expm1-def [=>] 1.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\right)\right)} \]
      expm1-log1p [=>] 0.5	\[ \color{blue}{{\left({k}^{0.25}\right)}^{-2} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}} \]
      *-commutative [=>] 0.5	\[ \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{-2}} \]
      sqr-pow [=>] 0.5	\[ \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}\right)} \cdot {\left({k}^{0.25}\right)}^{-2} \]
      pow-sqr [=>] 0.5	\[ \color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \frac{\frac{1 - k}{2}}{2}\right)}} \cdot {\left({k}^{0.25}\right)}^{-2} \]
      associate-/l/ [=>] 0.5	\[ {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \color{blue}{\frac{1 - k}{2 \cdot 2}}\right)} \cdot {\left({k}^{0.25}\right)}^{-2} \]
      metadata-eval [=>] 0.5	\[ {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \frac{1 - k}{\color{blue}{4}}\right)} \cdot {\left({k}^{0.25}\right)}^{-2} \]
      div-sub [=>] 0.5	\[ {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \color{blue}{\left(\frac{1}{4} - \frac{k}{4}\right)}\right)} \cdot {\left({k}^{0.25}\right)}^{-2} \]
      metadata-eval [=>] 0.5	\[ {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(2 \cdot \left(\color{blue}{0.25} - \frac{k}{4}\right)\right)} \cdot {\left({k}^{0.25}\right)}^{-2} \]
      pow-sqr [<=] 0.5	\[ \color{blue}{\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)}\right)} \cdot {\left({k}^{0.25}\right)}^{-2} \]
      rem-exp-log [<=] 1.0	\[ \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)}\right) \cdot \color{blue}{e^{\log \left({\left({k}^{0.25}\right)}^{-2}\right)}} \]
      sqr-pow [=>] 1.0	\[ \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)}\right) \cdot e^{\log \color{blue}{\left({\left({k}^{0.25}\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}\right)}} \]
      log-prod [=>] 1.0	\[ \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)}\right) \cdot e^{\color{blue}{\log \left({\left({k}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}\right) + \log \left({\left({k}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}\right)}} \]
      log-prod [<=] 1.0	\[ \left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(0.25 - \frac{k}{4}\right)}\right) \cdot e^{\color{blue}{\log \left({\left({k}^{0.25}\right)}^{\left(\frac{-2}{2}\right)} \cdot {\left({k}^{0.25}\right)}^{\left(\frac{-2}{2}\right)}\right)}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 6 \cdot 10^{-56}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

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	21.1
Cost	19780

\[\begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \left(\pi \cdot \frac{2}{k}\right)\right)}^{1.5}}\\ \end{array} \]

Alternative 3
Error	21.8
Cost	19584

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

Alternative 4
Error	21.9
Cost	19584

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

Alternative 5
Error	21.8
Cost	19584

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

Alternative 6
Error	31.9
Cost	13312

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

Alternative 7
Error	32.5
Cost	13184

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

Alternative 8
Error	32.5
Cost	13184

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

Alternative 9
Error	32.4
Cost	13184

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

Alternative 10
Error	32.4
Cost	13184

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

Reproduce

herbie shell --seed 2023011 
(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))))