Average Error: 0.5 → 0.4
Time: 13.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{n \cdot \left(2 \cdot \pi\right)}\\ \frac{-t_0}{{t_0}^{k} \cdot \left(-\sqrt{k}\right)} \end{array} \]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\begin{array}{l}
t_0 := \sqrt{n \cdot \left(2 \cdot \pi\right)}\\
\frac{-t_0}{{t_0}^{k} \cdot \left(-\sqrt{k}\right)}
\end{array}

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (let* ((t_0 (sqrt (* n (* 2.0 PI)))))
   (/ (- t_0) (* (pow t_0 k) (- (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) {
	double t_0 = sqrt(n * (2.0 * ((double) M_PI)));
	return -t_0 / (pow(t_0, k) * -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 div-sub_binary640.5

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

  5. Applied pow-sub_binary640.4

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

  6. Applied associate-/l/_binary640.4

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

  7. Simplified0.4

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

  9. Applied frac-2neg_binary640.4

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

  10. Simplified0.4

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

  11. Simplified0.4

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

  12. Final simplification0.4

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

Reproduce

herbie shell --seed 2021196 
(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))))