Migdal et al, Equation (51)

Average Error: 0.5 → 0.5

Time: 12.9s

Precision: binary64

Cost: 20036

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

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 3.7e-17)
   (/ (sqrt (* 2.0 n)) (sqrt (/ k PI)))
   (/ 1.0 (sqrt (/ k (pow (* n (* PI 2.0)) (- 1.0 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 <= 3.7e-17) {
		tmp = sqrt((2.0 * n)) / sqrt((k / ((double) M_PI)));
	} 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) {
	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 <= 3.7e-17) {
		tmp = Math.sqrt((2.0 * n)) / Math.sqrt((k / Math.PI));
	} else {
		tmp = 1.0 / Math.sqrt((k / Math.pow((n * (Math.PI * 2.0)), (1.0 - 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 <= 3.7e-17:
		tmp = math.sqrt((2.0 * n)) / math.sqrt((k / math.pi))
	else:
		tmp = 1.0 / math.sqrt((k / math.pow((n * (math.pi * 2.0)), (1.0 - 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 <= 3.7e-17)
		tmp = Float64(sqrt(Float64(2.0 * n)) / sqrt(Float64(k / pi)));
	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 = 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 <= 3.7e-17)
		tmp = sqrt((2.0 * n)) / sqrt((k / pi));
	else
		tmp = 1.0 / sqrt((k / ((n * (pi * 2.0)) ^ (1.0 - 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, 3.7e-17], N[(N[Sqrt[N[(2.0 * n), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(k / Pi), $MachinePrecision]], $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 < 3.6999999999999997e-17

   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 [=>] 0.8	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}{\sqrt{k}} \]
      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 17.5

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

   4. Simplified 17.5

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

      [Start] 17.5	\[ \sqrt{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      associate-*r* [=>] 17.5	\[ \sqrt{\frac{{\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [=>] 17.5	\[ \sqrt{\frac{{\color{blue}{\left(n \cdot \left(\pi \cdot 2\right)\right)}}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      *-commutative [<=] 17.5	\[ \sqrt{\frac{{\left(n \cdot \color{blue}{\left(2 \cdot \pi\right)}\right)}^{\left(2 \cdot \left(0.5 + k \cdot -0.5\right)\right)}}{k}} \]
      distribute-rgt-in [=>] 17.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{\left(0.5 \cdot 2 + \left(k \cdot -0.5\right) \cdot 2\right)}}}{k}} \]
      metadata-eval [=>] 17.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\color{blue}{1} + \left(k \cdot -0.5\right) \cdot 2\right)}}{k}} \]
      associate-*l* [=>] 17.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + \color{blue}{k \cdot \left(-0.5 \cdot 2\right)}\right)}}{k}} \]
      metadata-eval [=>] 17.5	\[ \sqrt{\frac{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + k \cdot \color{blue}{-1}\right)}}{k}} \]

   5. Taylor expanded in k around 0 17.5

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

   6. Simplified 17.5

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

      [Start] 17.5	\[ \sqrt{2 \cdot \frac{n \cdot \pi}{k}} \]
      associate-/l* [=>] 17.5	\[ \sqrt{2 \cdot \color{blue}{\frac{n}{\frac{k}{\pi}}}} \]

   7. Applied egg-rr 0.5

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

   8. Simplified 0.5

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

      [Start] 0.5	\[ \frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}} \]
      *-commutative [=>] 0.5	\[ \frac{\sqrt{\color{blue}{n \cdot 2}}}{\sqrt{\frac{k}{\pi}}} \]

   if 3.6999999999999997e-17 < 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. Applied egg-rr 0.4

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

   3. Applied egg-rr 0.5

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

   4. Simplified 0.5

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

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

   5. Taylor expanded in n around 0 0.7

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

   6. Simplified 0.5

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

      [Start] 0.7	\[ \frac{1}{\sqrt{\frac{k}{e^{\left(1 + -1 \cdot k\right) \cdot \left(\log n + \log \left(2 \cdot \pi\right)\right)}}}} \]
      *-commutative [=>] 0.7	\[ \frac{1}{\sqrt{\frac{k}{e^{\color{blue}{\left(\log n + \log \left(2 \cdot \pi\right)\right) \cdot \left(1 + -1 \cdot k\right)}}}}} \]
      exp-prod [=>] 0.6	\[ \frac{1}{\sqrt{\frac{k}{\color{blue}{{\left(e^{\log n + \log \left(2 \cdot \pi\right)}\right)}^{\left(1 + -1 \cdot k\right)}}}}} \]
      +-commutative [=>] 0.6	\[ \frac{1}{\sqrt{\frac{k}{{\left(e^{\color{blue}{\log \left(2 \cdot \pi\right) + \log n}}\right)}^{\left(1 + -1 \cdot k\right)}}}} \]
      log-prod [=>] 0.6	\[ \frac{1}{\sqrt{\frac{k}{{\left(e^{\color{blue}{\left(\log 2 + \log \pi\right)} + \log n}\right)}^{\left(1 + -1 \cdot k\right)}}}} \]
      associate-+l+ [=>] 0.6	\[ \frac{1}{\sqrt{\frac{k}{{\left(e^{\color{blue}{\log 2 + \left(\log \pi + \log n\right)}}\right)}^{\left(1 + -1 \cdot k\right)}}}} \]
      log-prod [<=] 0.6	\[ \frac{1}{\sqrt{\frac{k}{{\left(e^{\log 2 + \color{blue}{\log \left(\pi \cdot n\right)}}\right)}^{\left(1 + -1 \cdot k\right)}}}} \]
      log-prod [<=] 0.6	\[ \frac{1}{\sqrt{\frac{k}{{\left(e^{\color{blue}{\log \left(2 \cdot \left(\pi \cdot n\right)\right)}}\right)}^{\left(1 + -1 \cdot k\right)}}}} \]
      *-commutative [=>] 0.6	\[ \frac{1}{\sqrt{\frac{k}{{\left(e^{\log \color{blue}{\left(\left(\pi \cdot n\right) \cdot 2\right)}}\right)}^{\left(1 + -1 \cdot k\right)}}}} \]
      associate-*r* [<=] 0.6	\[ \frac{1}{\sqrt{\frac{k}{{\left(e^{\log \color{blue}{\left(\pi \cdot \left(n \cdot 2\right)\right)}}\right)}^{\left(1 + -1 \cdot k\right)}}}} \]
      rem-exp-log [=>] 0.5	\[ \frac{1}{\sqrt{\frac{k}{{\color{blue}{\left(\pi \cdot \left(n \cdot 2\right)\right)}}^{\left(1 + -1 \cdot k\right)}}}} \]
      *-commutative [=>] 0.5	\[ \frac{1}{\sqrt{\frac{k}{{\color{blue}{\left(\left(n \cdot 2\right) \cdot \pi\right)}}^{\left(1 + -1 \cdot k\right)}}}} \]
      associate-*l* [=>] 0.5	\[ \frac{1}{\sqrt{\frac{k}{{\color{blue}{\left(n \cdot \left(2 \cdot \pi\right)\right)}}^{\left(1 + -1 \cdot k\right)}}}} \]
      mul-1-neg [=>] 0.5	\[ \frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(1 + \color{blue}{\left(-k\right)}\right)}}}} \]
      sub-neg [<=] 0.5	\[ \frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\color{blue}{\left(1 - k\right)}}}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.7 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 4.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 2
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 3
Error	21.1
Cost	19780

\[\begin{array}{l} \mathbf{if}\;k \leq 10^{+212}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\left(\pi \cdot 2\right) \cdot \frac{n}{k}\right)}^{1.5}}\\ \end{array} \]

Alternative 4
Error	32.8
Cost	19584

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

Alternative 5
Error	22.0
Cost	19584

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

Alternative 6
Error	22.0
Cost	19584

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

Alternative 7
Error	32.8
Cost	13184

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

Alternative 8
Error	32.8
Cost	13184

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

Reproduce

herbie shell --seed 2023059 
(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))))