Migdal et al, Equation (51)

Average Accuracy: 99.2% → 99.4%

Time: 15.6s

Precision: binary64

Cost: 32896

Math

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

↓

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

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
 (let* ((t_0 (* PI (* 2.0 n))))
   (* (pow t_0 (* -0.5 k)) (/ (sqrt t_0) (sqrt 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 t_0 = ((double) M_PI) * (2.0 * n);
	return pow(t_0, (-0.5 * k)) * (sqrt(t_0) / sqrt(k));
}

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 t_0 = Math.PI * (2.0 * n);
	return Math.pow(t_0, (-0.5 * k)) * (Math.sqrt(t_0) / Math.sqrt(k));
}

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):
	t_0 = math.pi * (2.0 * n)
	return math.pow(t_0, (-0.5 * k)) * (math.sqrt(t_0) / math.sqrt(k))

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

MATLAB

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

↓

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

Derivation

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%

   \[\leadsto \color{blue}{\frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}}} \]
   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	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}{\sqrt{k}} \]
   sqr-pow [=>] 99.0	\[ \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{1 - k}{2}}{2}\right)}}}{\sqrt{k}} \]
   sqr-pow [<=] 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	\[ \frac{{\color{blue}{\left(\pi \cdot \left(2 \cdot n\right)\right)}}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}} \]
   div-sub [=>] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(\frac{1}{2} - \frac{k}{2}\right)}}}{\sqrt{k}} \]
   metadata-eval [=>] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(\color{blue}{0.5} - \frac{k}{2}\right)}}{\sqrt{k}} \]

3. Applied egg-rr 99.4%

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

   [Start] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(0.5 - \frac{k}{2}\right)}}{\sqrt{k}} \]
   sub-neg [=>] 99.3	\[ \frac{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(0.5 + \left(-\frac{k}{2}\right)\right)}}}{\sqrt{k}} \]
   unpow-prod-up [=>] 99.4	\[ \frac{\color{blue}{{\left(\pi \cdot \left(2 \cdot n\right)\right)}^{0.5} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-\frac{k}{2}\right)}}}{\sqrt{k}} \]
   unpow1/2 [=>] 99.4	\[ \frac{\color{blue}{\sqrt{\pi \cdot \left(2 \cdot n\right)}} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-\frac{k}{2}\right)}}{\sqrt{k}} \]
   div-inv [=>] 99.4	\[ \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-\color{blue}{k \cdot \frac{1}{2}}\right)}}{\sqrt{k}} \]
   metadata-eval [=>] 99.4	\[ \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-k \cdot \color{blue}{0.5}\right)}}{\sqrt{k}} \]
   distribute-rgt-neg-in [=>] 99.4	\[ \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(k \cdot \left(-0.5\right)\right)}}}{\sqrt{k}} \]
   metadata-eval [=>] 99.4	\[ \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot \color{blue}{-0.5}\right)}}{\sqrt{k}} \]

4. Simplified 99.4%

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

   [Start] 99.4	\[ \frac{\sqrt{\pi \cdot \left(2 \cdot n\right)} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]
   associate-*r* [=>] 99.4	\[ \frac{\sqrt{\color{blue}{\left(\pi \cdot 2\right) \cdot n}} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]
   *-commutative [=>] 99.4	\[ \frac{\sqrt{\color{blue}{\left(2 \cdot \pi\right)} \cdot n} \cdot {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]
   associate-*r* [=>] 99.4	\[ \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot {\color{blue}{\left(\left(\pi \cdot 2\right) \cdot n\right)}}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]
   *-commutative [=>] 99.4	\[ \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot {\left(\color{blue}{\left(2 \cdot \pi\right)} \cdot n\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]

5. Applied egg-rr 99.3%

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

   [Start] 99.4	\[ \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]
   associate-/l* [=>] 99.4	\[ \color{blue}{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{\frac{\sqrt{k}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot -0.5\right)}}}} \]
   div-inv [=>] 99.3	\[ \color{blue}{\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \frac{1}{\frac{\sqrt{k}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot -0.5\right)}}}} \]
   associate-/l* [<=] 99.3	\[ \sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \color{blue}{\frac{1 \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}}} \]
   *-un-lft-identity [<=] 99.3	\[ \sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \frac{\color{blue}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot -0.5\right)}}}{\sqrt{k}} \]
   associate-*l* [=>] 99.3	\[ \sqrt{\color{blue}{2 \cdot \left(\pi \cdot n\right)}} \cdot \frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]
   associate-*l* [=>] 99.3	\[ \sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot \frac{{\color{blue}{\left(2 \cdot \left(\pi \cdot n\right)\right)}}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]

6. Simplified 99.4%

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

   [Start] 99.3	\[ \sqrt{2 \cdot \left(\pi \cdot n\right)} \cdot \frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \]
   *-commutative [=>] 99.3	\[ \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot -0.5\right)}}{\sqrt{k}} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}} \]
   associate-*l/ [=>] 99.4	\[ \color{blue}{\frac{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}} \]
   associate-*r/ [<=] 99.4	\[ \color{blue}{{\left(2 \cdot \left(\pi \cdot n\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}}} \]
   *-commutative [=>] 99.4	\[ {\color{blue}{\left(\left(\pi \cdot n\right) \cdot 2\right)}}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}} \]
   associate-*l* [=>] 99.4	\[ {\color{blue}{\left(\pi \cdot \left(n \cdot 2\right)\right)}}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}} \]
   *-commutative [<=] 99.4	\[ {\left(\pi \cdot \color{blue}{\left(2 \cdot n\right)}\right)}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}} \]
   *-commutative [=>] 99.4	\[ {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\color{blue}{\left(-0.5 \cdot k\right)}} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{\sqrt{k}} \]
   *-commutative [=>] 99.4	\[ {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-0.5 \cdot k\right)} \cdot \frac{\sqrt{\color{blue}{\left(\pi \cdot n\right) \cdot 2}}}{\sqrt{k}} \]
   associate-*l* [=>] 99.4	\[ {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-0.5 \cdot k\right)} \cdot \frac{\sqrt{\color{blue}{\pi \cdot \left(n \cdot 2\right)}}}{\sqrt{k}} \]
   *-commutative [<=] 99.4	\[ {\left(\pi \cdot \left(2 \cdot n\right)\right)}^{\left(-0.5 \cdot k\right)} \cdot \frac{\sqrt{\pi \cdot \color{blue}{\left(2 \cdot n\right)}}}{\sqrt{k}} \]

7. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	26176

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

Alternative 2
Accuracy	99.3%
Cost	20036

\[\begin{array}{l} \mathbf{if}\;k \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sqrt{\frac{k}{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}}}\\ \end{array} \]

Alternative 3
Accuracy	99.1%
Cost	19908

\[\begin{array}{l} \mathbf{if}\;k \leq 5.4 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(n \cdot \left(\pi \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]

Alternative 4
Accuracy	99.3%
Cost	19904

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

Alternative 5
Accuracy	67.1%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;k \leq 4 \cdot 10^{+101}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;{\left({\left(\frac{n}{k} \cdot \left(\pi \cdot 2\right)\right)}^{1.5}\right)}^{0.3333333333333333}\\ \end{array} \]

Alternative 6
Accuracy	67.1%
Cost	19780

\[\begin{array}{l} \mathbf{if}\;k \leq 4.6 \cdot 10^{+225}:\\ \;\;\;\;\frac{\sqrt{2 \cdot n}}{\sqrt{\frac{k}{\pi}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(n \cdot \frac{2}{\frac{k}{\pi}}\right)}^{1.5}}\\ \end{array} \]

Alternative 7
Accuracy	65.7%
Cost	19584

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

Alternative 8
Accuracy	65.7%
Cost	19584

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

Alternative 9
Accuracy	49.6%
Cost	13184

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

Alternative 10
Accuracy	49.6%
Cost	13184

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

Reproduce

herbie shell --seed 2023129 
(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))))