Average Error: 0.5 → 0.4
Time: 12.2s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{\sqrt{2 \cdot \left(n \cdot \pi\right)}}{\sqrt{k}}}{{\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{\sqrt{2 \cdot \left(n \cdot \pi\right)}}{\sqrt{k}}}{{\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{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 (* 2.0 (* n PI))) (sqrt k)) (pow (sqrt (* n (* 2.0 PI))) 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(2.0 * (n * ((double) M_PI))) / sqrt(k)) / pow(sqrt(n * (2.0 * ((double) M_PI))), 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

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

  4. Simplified0.5

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

  6. Applied pow-sub_binary64_8360.4

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

  7. Applied associate-*r/_binary64_7020.4

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

  8. Simplified0.4

    \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{k}} \cdot \sqrt{\left(n \cdot \pi\right) \cdot 2}}}{{\left(\sqrt{n \cdot \left(2 \cdot \pi\right)}\right)}^{k}}\]
  9. Using strategy rm

  10. Applied sqrt-div_binary64_7770.5

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

  11. Applied associate-*l/_binary64_7030.4

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

  12. Simplified0.4

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

  13. Final simplification0.4

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

Reproduce

herbie shell --seed 2021032 
(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))))