Average Error: 0.5 → 0.5
Time: 8.0s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := \sqrt{n \cdot \pi}\\ \left({\left(2 \cdot \left(t_0 \cdot t_0\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \sqrt{2 \cdot \left(n \cdot \pi\right)}\right) \cdot {k}^{-0.5} \end{array} \]
\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 := \sqrt{n \cdot \pi}\\
\left({\left(2 \cdot \left(t_0 \cdot t_0\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \sqrt{2 \cdot \left(n \cdot \pi\right)}\right) \cdot {k}^{-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 (sqrt (* n PI))))
   (*
    (* (pow (* 2.0 (* t_0 t_0)) (* k -0.5)) (sqrt (* 2.0 (* n PI))))
    (pow k -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 = sqrt(n * ((double) M_PI));
	return (pow((2.0 * (t_0 * t_0)), (k * -0.5)) * sqrt(2.0 * (n * ((double) M_PI)))) * pow(k, -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. Simplified0.4

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

  6. Simplified0.4

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

  7. Applied div-inv_binary640.5

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

  8. Applied pow1/2_binary640.5

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

  9. Applied pow-flip_binary640.5

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

  10. Simplified0.5

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

  11. Applied add-sqr-sqrt_binary640.5

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

  12. Final simplification0.5

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

Reproduce

herbie shell --seed 2022117 
(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))))