Migdal et al, Equation (51)

Average Error: 0.5 → 0.4

Time: 7.1s

Precision: binary64

Math

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

↓

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

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

Error

Bits error versus k

Bits error versus n

Derivation

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 0.4

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

3. Applied egg-rr 0.4

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

4. Taylor expanded in k around inf 0.5

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

5. Simplified 0.5

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

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022134 
(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))))