Average Error: 0.5 → 0.5
Time: 15.3s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \left(2 \cdot \pi\right) \cdot n\\ \frac{\frac{{t_0}^{\left(k \cdot -0.5\right)}}{\sqrt{k}}}{{t_0}^{-0.5}} \end{array} \]
\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 := \left(2 \cdot \pi\right) \cdot n\\
\frac{\frac{{t_0}^{\left(k \cdot -0.5\right)}}{\sqrt{k}}}{{t_0}^{-0.5}}
\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 (* (* 2.0 PI) n)))
   (/ (/ (pow t_0 (* k -0.5)) (sqrt k)) (pow t_0 -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) {
	double t_0 = (2.0 * ((double) M_PI)) * n;
	return (pow(t_0, (k * -0.5)) / sqrt(k)) / pow(t_0, -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

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

  3. Applied fma-udef_binary640.5

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

  4. Applied unpow-prod-up_binary640.4

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

  5. Applied associate-/l*_binary640.5

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

  6. Simplified0.5

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

  7. Applied div-inv_binary640.5

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

  8. Applied associate-/r*_binary640.5

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

  9. Applied pow1_binary640.5

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

  10. Applied pow1_binary640.5

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

  11. Applied pow-prod-down_binary640.5

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

  12. Applied pow1_binary640.5

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

  13. Applied pow-prod-down_binary640.5

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

  14. Applied sqrt-pow1_binary640.5

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

  15. Applied pow-flip_binary640.5

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

  16. Final simplification0.5

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

Reproduce

herbie shell --seed 2022097 
(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))))