Average Error: 0.5 → 0.6
Time: 12.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[{\left(\sqrt{2} \cdot \sqrt{n \cdot \pi}\right)}^{\left(1 - k\right)} \cdot \sqrt{\frac{1}{k}} \]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\sqrt{2} \cdot \sqrt{n \cdot \pi}\right)}^{\left(1 - k\right)} \cdot \sqrt{\frac{1}{k}}

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (* (pow (* (sqrt 2.0) (sqrt (* n PI))) (- 1.0 k)) (sqrt (/ 1.0 k))))
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 pow((sqrt(2.0) * sqrt(n * ((double) M_PI))), (1.0 - k)) * sqrt(1.0 / k);
}

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Simplified0.5

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

  3. Taylor expanded around 0 3.4

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

  4. Simplified0.5

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

  5. Taylor expanded around inf 0.6

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

  6. Simplified0.6

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

  7. Final simplification0.6

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

Reproduce

herbie shell --seed 2021196 
(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))))