Average Error: 0.5 → 0.4
Time: 7.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot {\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{\left(-k\right)}\right) \cdot \sqrt{\frac{1}{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\left(\sqrt{n \cdot \left(2 \cdot \pi\right)} \cdot {\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{\left(-k\right)}\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
 (*
  (* (sqrt (* n (* 2.0 PI))) (pow (sqrt (* n (* 2.0 PI))) (- 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 (sqrt(n * (2.0 * ((double) M_PI))) * pow(sqrt(n * (2.0 * ((double) M_PI))), -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 3.5

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

  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. Using strategy rm

  6. Applied *-un-lft-identity_binary640.5

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

  7. Applied cancel-sign-sub-inv_binary640.5

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

  8. Applied unpow-prod-up_binary640.4

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

  9. Final simplification0.4

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

Reproduce

herbie shell --seed 2021176 
(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))))