Average Error: 0.5 → 0.5
Time: 16.2s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{1}{\frac{\sqrt{k}}{{\left(\sqrt{\sqrt{n}} \cdot \sqrt{\sqrt{n} \cdot \left(\pi \cdot 2\right)}\right)}^{\left(1 - k\right)}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{1}{\frac{\sqrt{k}}{{\left(\sqrt{\sqrt{n}} \cdot \sqrt{\sqrt{n} \cdot \left(\pi \cdot 2\right)}\right)}^{\left(1 - k\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
 (/
  1.0
  (/
   (sqrt k)
   (pow (* (sqrt (sqrt n)) (sqrt (* (sqrt n) (* PI 2.0)))) (- 1.0 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 / (sqrt(k) / pow((sqrt(sqrt(n)) * sqrt(sqrt(n) * (((double) M_PI) * 2.0))), (1.0 - 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 clear-num_binary640.5

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

  5. Simplified0.5

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

  7. Applied add-sqr-sqrt_binary640.6

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

  8. Applied associate-*l*_binary640.6

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

  9. Simplified0.6

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

  11. Applied sqrt-prod_binary640.5

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

  12. Simplified0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2021174 
(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))))