Average Error: 0.5 → 0.4
Time: 5.0min
Precision: binary64
Cost: 33152
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{k}^{-0.5} \cdot \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left({\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{2}\right)}^{\left(\frac{k}{2}\right)}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{k}^{-0.5} \cdot \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left({\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{2}\right)}^{\left(\frac{k}{2}\right)}}
(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (*
  (pow k -0.5)
  (/
   (sqrt (* (* 2.0 PI) n))
   (pow (pow (sqrt (* (* 2.0 PI) n)) 2.0) (/ k 2.0)))))
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 pow(k, -0.5) * (sqrt((2.0 * ((double) M_PI)) * n) / pow(pow(sqrt((2.0 * ((double) M_PI)) * n), 2.0), (k / 2.0)));
}

Error

Bits error versus k

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error0.4
Cost20288
\[{k}^{-0.5} \cdot \frac{\sqrt{2 \cdot \left(\pi \cdot n\right)}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
Alternative 2
Error0.4
Cost20224
\[\frac{\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}}{\sqrt{k}}\]
Alternative 3
Error0.5
Cost19968
\[{k}^{-0.5} \cdot {\left({\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}\right)}^{0.5}\]
Alternative 4
Error0.5
Cost13632
\[{k}^{-0.5} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
Alternative 5
Error0.5
Cost13568
\[\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\]
Alternative 6
Error1.1
Cost13889
\[\begin{array}{l} \mathbf{if}\;k \leq 2.7008463207183678 \cdot 10^{-74}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\frac{\pi \cdot -2}{\frac{-1}{n}}\right)}^{\left(1 - k\right)}}{k}}\\ \end{array}\]
Alternative 7
Error1.1
Cost13761
\[\begin{array}{l} \mathbf{if}\;k \leq 6.003961771686958 \cdot 10^{-75}:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(1 - k\right)}}{k}}\\ \end{array}\]
Alternative 8
Error1.7
Cost13633
\[\begin{array}{l} \mathbf{if}\;k \leq 3.042568944369596:\\ \;\;\;\;{k}^{-0.5} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot n}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 9
Error1.7
Cost13569
\[\begin{array}{l} \mathbf{if}\;k \leq 3.157038681566503:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 10
Error11.9
Cost7169
\[\begin{array}{l} \mathbf{if}\;k \leq 3.157038681566503:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(\pi \cdot n\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 11
Error11.9
Cost7169
\[\begin{array}{l} \mathbf{if}\;k \leq 2.9280992071726892:\\ \;\;\;\;\sqrt{\frac{2 \cdot \pi}{\frac{k}{n}}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 12
Error40.2
Cost385
\[\begin{array}{l} \mathbf{if}\;k \leq 3.157038681566503:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]
Alternative 13
Error60.5
Cost64
\[1\]

Error

Time

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. Using strategy rm

  3. Applied pow1/2_binary64_11810.5

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

  4. Applied pow-flip_binary64_11750.5

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

  5. Simplified0.5

    \[\leadsto {k}^{\color{blue}{-0.5}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
  6. Using strategy rm

  7. Applied div-sub_binary64_11060.5

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

  8. Applied pow-sub_binary64_11770.4

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

  9. Simplified0.4

    \[\leadsto {k}^{-0.5} \cdot \frac{\color{blue}{\sqrt{\left(2 \cdot \pi\right) \cdot n}}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{k}{2}\right)}}\]
  10. Using strategy rm

  11. Applied add-sqr-sqrt_binary64_11230.4

    \[\leadsto {k}^{-0.5} \cdot \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\color{blue}{\left(\sqrt{\left(2 \cdot \pi\right) \cdot n} \cdot \sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}}^{\left(\frac{k}{2}\right)}}\]
  12. Using strategy rm

  13. Applied pow2_binary64_11820.4

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

  14. Simplified0.4

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

  15. Final simplification0.4

    \[\leadsto {k}^{-0.5} \cdot \frac{\sqrt{\left(2 \cdot \pi\right) \cdot n}}{{\left({\left(\sqrt{\left(2 \cdot \pi\right) \cdot n}\right)}^{2}\right)}^{\left(\frac{k}{2}\right)}}\]

Reproduce

herbie shell --seed 2021065 
(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))))