Average Error: 0.5 → 0.5
Time: 16.5s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[\frac{\sqrt{\frac{1}{k}} \cdot e^{-0.5 \cdot \left(k \cdot \left(\log \left(2 \cdot \pi\right) - \log \left(\frac{1}{n}\right)\right)\right)}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{-0.5}} \]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
\frac{\sqrt{\frac{1}{k}} \cdot e^{-0.5 \cdot \left(k \cdot \left(\log \left(2 \cdot \pi\right) - \log \left(\frac{1}{n}\right)\right)\right)}}{{\left(\left(2 \cdot \pi\right) \cdot n\right)}^{-0.5}}

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (/
  (*
   (sqrt (/ 1.0 k))
   (exp (* -0.5 (* k (- (log (* 2.0 PI)) (log (/ 1.0 n)))))))
  (pow (* (* 2.0 PI) n) -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) {
	return (sqrt(1.0 / k) * exp(-0.5 * (k * (log(2.0 * ((double) M_PI)) - log(1.0 / n))))) / pow(((2.0 * ((double) M_PI)) * n), -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. Simplified0.4

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

  3. Applied fma-udef_binary640.4

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

  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 pow1/2_binary640.5

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

  10. Applied pow-flip_binary640.5

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

  11. Simplified0.5

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

  12. Taylor expanded in n around inf 0.5

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

  13. Final simplification0.5

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

Reproduce

herbie shell --seed 2022020 
(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))))