Average Error: 0.5 → 0.4
Time: 12.4s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := n \cdot \left(\pi \cdot 2\right)\\ \frac{{t_0}^{\left(k \cdot -0.5\right)} \cdot \sqrt{t_0}}{\sqrt{k}} \end{array} \]
\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 := n \cdot \left(\pi \cdot 2\right)\\
\frac{{t_0}^{\left(k \cdot -0.5\right)} \cdot \sqrt{t_0}}{\sqrt{k}}
\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 (* n (* PI 2.0))))
   (/ (* (pow t_0 (* k -0.5)) (sqrt t_0)) (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 = n * (((double) M_PI) * 2.0);
	return (pow(t_0, (k * -0.5)) * sqrt(t_0)) / 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. Simplified0.4

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

  3. Applied add-sqr-sqrt_binary640.6

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

  4. Applied fma-udef_binary640.6

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

  5. Applied unpow-prod-up_binary640.5

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

  6. Applied times-frac_binary640.5

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

  7. Simplified0.5

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

  8. Simplified0.5

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

  9. Applied *-un-lft-identity_binary640.5

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

  10. Applied sqrt-prod_binary640.5

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

  11. Applied sqrt-prod_binary640.5

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

  12. Applied *-un-lft-identity_binary640.5

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

  13. Applied times-frac_binary640.5

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

  14. Applied associate-*l*_binary640.5

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

  15. Simplified0.4

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

  16. Final simplification0.4

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

Reproduce

herbie shell --seed 2021280 
(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))))