Average Error: 0.5 → 0.6
Time: 6.7s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\]
↓
\[\frac{\sqrt{{\left(\sqrt{n} \cdot \left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)\right)}^{\left(\frac{1 - k}{2}\right)}}}{\frac{\sqrt{k}}{\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}}}\]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\sqrt{{\left(\sqrt{n} \cdot \left(\left(2 \cdot \pi\right) \cdot \sqrt{n}\right)\right)}^{\left(\frac{1 - k}{2}\right)}}}{\frac{\sqrt{k}}{\sqrt{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - 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
 (/
  (sqrt (pow (* (sqrt n) (* (* 2.0 PI) (sqrt n))) (/ (- 1.0 k) 2.0)))
  (/ (sqrt k) (sqrt (pow (* (* 2.0 PI) n) (/ (- 1.0 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 sqrt(pow((sqrt(n) * ((2.0 * ((double) M_PI)) * sqrt(n))), ((1.0 - k) / 2.0))) / (sqrt(k) / sqrt(pow(((2.0 * ((double) M_PI)) * n), ((1.0 - 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

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. Simplified0.4

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

  4. Applied add-sqr-sqrt_binary640.5

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

  5. Applied associate-*r*_binary640.5

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

  7. Applied add-sqr-sqrt_binary640.6

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

  8. Applied associate-/l*_binary640.6

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

  9. Simplified0.6

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

  10. Final simplification0.6

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

Reproduce

herbie shell --seed 2020253 
(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))))