Average Error: 0.5 → 0.4
Time: 8.0s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{\sqrt{k}}}{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{\sqrt{k}}}{{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\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 PI) n)) (sqrt k)) (pow (sqrt (* (* 2.0 PI) n)) 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 * ((double) M_PI)) * n) / sqrt(k)) / pow(sqrt((2.0 * ((double) M_PI)) * n), 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. Simplified0.4

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

  4. Applied clear-num_binary64_7590.5

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

  5. Simplified0.5

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

  7. Applied pow-sub_binary64_8360.4

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

  8. Applied associate-/r/_binary64_7060.4

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

  9. Applied associate-/r*_binary64_7040.4

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

  10. Simplified0.4

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

  11. Final simplification0.4

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

Reproduce

herbie shell --seed 2021009 
(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))))