Average Error: 0.5 → 0.4
Time: 10.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\frac{\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot \sqrt{n}}{{\left(2 \cdot \pi\right)}^{\left(\frac{k}{2}\right)} \cdot {n}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\frac{\left(\sqrt{2} \cdot \sqrt{\pi}\right) \cdot \sqrt{n}}{{\left(2 \cdot \pi\right)}^{\left(\frac{k}{2}\right)} \cdot {n}^{\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 (* 2.0 PI) (/ k 2.0)) (pow n (/ 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((2.0 * ((double) M_PI)), (k / 2.0)) * pow(n, (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 unpow-prod-down_binary64_8390.6

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

  6. Applied div-sub_binary64_7650.6

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

  7. Applied pow-sub_binary64_8360.5

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

  8. Applied div-sub_binary64_7650.5

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

  9. Applied pow-sub_binary64_8360.5

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

  10. Applied frac-times_binary64_7700.5

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

  11. Simplified0.5

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

  13. Applied sqrt-prod_binary64_7760.4

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

  14. Final simplification0.4

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

Reproduce

herbie shell --seed 2020346 
(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))))