Migdal et al, Equation (51)

Average Error: 0.5 → 0.4

Time: 5.2s

Precision: binary64

Math

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

↓

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

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
 (let* ((t_0 (* n (* PI 2.0))))
   (* (/ (sqrt t_0) (sqrt k)) (pow t_0 (* k -0.5)))))

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

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

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

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(n * Float64(pi * 2.0))
	return Float64(Float64(sqrt(t_0) / sqrt(k)) * (t_0 ^ Float64(k * -0.5)))
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 = n * (pi * 2.0);
	tmp = (sqrt(t_0) / sqrt(k)) * (t_0 ^ (k * -0.5));
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[(n * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[k], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, N[(k * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

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 0.5

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

3. Applied egg-rr 0.4

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

4. Taylor expanded in k around inf 0.5

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

5. Simplified 0.5

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

6. Applied egg-rr 0.4

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

7. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022192 
(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))))