Average Error: 0.5 → 0.5
Time: 10.7s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[{\left(\sqrt{2 \cdot \left(n \cdot \pi\right)}\right)}^{\left(1 - k\right)} \cdot {k}^{-0.5}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\sqrt{2 \cdot \left(n \cdot \pi\right)}\right)}^{\left(1 - k\right)} \cdot {k}^{-0.5}
(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 (sqrt (* 2.0 (* n PI))) (- 1.0 k)) (pow k -0.5)))
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(sqrt(2.0 * (n * ((double) M_PI))), (1.0 - k)) * pow(k, -0.5);
}

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. Taylor expanded around 0 3.4

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

  4. Simplified0.5

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

  6. Applied inv-pow_binary64_8450.5

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

  7. Applied sqrt-pow1_binary64_7780.5

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

  8. Simplified0.5

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

  9. Final simplification0.5

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

Reproduce

herbie shell --seed 2021046 
(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))))