Average Error: 0.4 → 0.3
Time: 6.8s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\begin{array}{l} t_0 := 2 \cdot \left(n \cdot \pi\right)\\ \sqrt{{t_0}^{\left(k \cdot -0.5\right)}} \cdot \left(\frac{\sqrt{t_0}}{\sqrt{k}} \cdot {t_0}^{\left(k \cdot -0.25\right)}\right) \end{array} \]
\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 := 2 \cdot \left(n \cdot \pi\right)\\
\sqrt{{t_0}^{\left(k \cdot -0.5\right)}} \cdot \left(\frac{\sqrt{t_0}}{\sqrt{k}} \cdot {t_0}^{\left(k \cdot -0.25\right)}\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 (* 2.0 (* n PI))))
   (*
    (sqrt (pow t_0 (* k -0.5)))
    (* (/ (sqrt t_0) (sqrt k)) (pow t_0 (* k -0.25))))))
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 * (n * ((double) M_PI));
	return sqrt(pow(t_0, (k * -0.5))) * ((sqrt(t_0) / sqrt(k)) * pow(t_0, (k * -0.25)));
}

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 0.4

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

  2. Simplified0.3

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

  3. Applied *-un-lft-identity_binary640.3

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

  4. Applied sqrt-prod_binary640.3

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

  5. Applied fma-udef_binary640.3

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

  6. Applied unpow-prod-up_binary640.3

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

  7. Applied times-frac_binary640.3

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

  8. Simplified0.3

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

  9. Simplified0.3

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

  10. Applied add-sqr-sqrt_binary640.3

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

  11. Applied associate-*l*_binary640.3

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

  12. Simplified0.3

    \[\leadsto \sqrt{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(k \cdot -0.5\right)}} \cdot \color{blue}{\left(\frac{\sqrt{2 \cdot \left(n \cdot \pi\right)}}{\sqrt{k}} \cdot {\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(-0.25 \cdot k\right)}\right)} \]

  13. Final simplification0.3

    \[\leadsto \sqrt{{\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(k \cdot -0.5\right)}} \cdot \left(\frac{\sqrt{2 \cdot \left(n \cdot \pi\right)}}{\sqrt{k}} \cdot {\left(2 \cdot \left(n \cdot \pi\right)\right)}^{\left(k \cdot -0.25\right)}\right) \]

Reproduce

herbie shell --seed 2022067 
(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))))