Average Error: 0.5 → 0.4
Time: 8.9s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}} \cdot \frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}} \cdot \frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\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
 (*
  (/ 1.0 (pow (* (* 2.0 PI) n) (/ k 4.0)))
  (/ (/ (sqrt (* (* 2.0 PI) n)) (pow (* (* 2.0 PI) n) (/ k 4.0))) (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 (1.0 / pow(((2.0 * ((double) M_PI)) * n), (k / 4.0))) * ((sqrt((2.0 * ((double) M_PI)) * n) / pow(((2.0 * ((double) M_PI)) * n), (k / 4.0))) / 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_7650.5

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

  5. Applied pow-sub_binary64_8360.4

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

  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_7600.4

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

  9. Applied sqrt-prod_binary64_7760.4

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

  10. Applied sqr-pow_binary64_7320.5

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

  11. Applied *-un-lft-identity_binary64_7600.5

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

  12. Applied times-frac_binary64_7660.4

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

  13. Applied times-frac_binary64_7660.4

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

  14. Simplified0.4

    \[\leadsto \color{blue}{\frac{1}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}}} \cdot \frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{\frac{k}{2}}{2}\right)}}}{\sqrt{k}}\]

  15. Simplified0.4

    \[\leadsto \frac{1}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}} \cdot \color{blue}{\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}}}{\sqrt{k}}}\]

  16. Final simplification0.4

    \[\leadsto \frac{1}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}} \cdot \frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{4}\right)}}}{\sqrt{k}}\]

Reproduce

herbie shell --seed 2020295 
(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))))