Average Error: 0.5 → 0.5
Time: 10.8s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot \frac{\frac{\sqrt{n}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
\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{\pi}\right) \cdot \frac{\frac{\sqrt{n}}{{\left(n \cdot \left(2 \cdot \pi\right)\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{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) (sqrt PI))
  (/ (/ (sqrt n) (pow (* n (* 2.0 PI)) (/ k 2.0))) (sqrt 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) * sqrt((double) M_PI)) * ((sqrt(n) / pow((n * (2.0 * ((double) M_PI))), (k / 2.0))) / sqrt(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. Using strategy rm

  4. Applied div-sub_binary64_7560.5

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

  5. Applied pow-sub_binary64_8240.4

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

  6. Simplified0.4

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

  8. Applied *-un-lft-identity_binary64_7510.4

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

  9. Applied sqrt-prod_binary64_7660.4

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

  10. Applied *-un-lft-identity_binary64_7510.4

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

  11. Applied sqrt-prod_binary64_7660.6

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

  12. Applied times-frac_binary64_7570.6

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

  13. Applied times-frac_binary64_7570.6

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

  14. Simplified0.6

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

  16. Applied sqrt-prod_binary64_7660.5

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

  17. Final simplification0.5

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

Reproduce

herbie shell --seed 2020288 
(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))))