Average Error: 0.5 → 0.4
Time: 8.9s
Precision: binary64
\[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
↓
\[{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \left(2 \cdot \pi\right)\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}} \]
\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \pi\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}
↓
{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \left(2 \cdot \pi\right)\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}}

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 PI) n) (/ (- 1.0 k) 2.0))))
↓
(FPCore (k n)
 :precision binary64
 (*
  (pow (* (* (cbrt n) (cbrt n)) (* (cbrt n) (* 2.0 PI))) (* k -0.5))
  (/ (sqrt (* n (* 2.0 PI))) (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) {
	return pow(((cbrt(n) * cbrt(n)) * (cbrt(n) * (2.0 * ((double) M_PI)))), (k * -0.5)) * (sqrt(n * (2.0 * ((double) M_PI))) / 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. Simplified0.5

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

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

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

  4. Applied sqrt-prod_binary640.5

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

  5. Applied fma-udef_binary640.5

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

  6. Applied unpow-prod-up_binary640.4

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

  7. Applied times-frac_binary640.4

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

  8. Simplified0.4

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

  9. Simplified0.4

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

  10. Applied add-cube-cbrt_binary640.4

    \[\leadsto {\left(\color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \sqrt[3]{n}\right)} \cdot \left(2 \cdot \pi\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}} \]

  11. Applied associate-*l*_binary640.4

    \[\leadsto {\color{blue}{\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \left(2 \cdot \pi\right)\right)\right)}}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}} \]

  12. Simplified0.4

    \[\leadsto {\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \color{blue}{\left(\left(2 \cdot \pi\right) \cdot \sqrt[3]{n}\right)}\right)}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}} \]

  13. Final simplification0.4

    \[\leadsto {\left(\left(\sqrt[3]{n} \cdot \sqrt[3]{n}\right) \cdot \left(\sqrt[3]{n} \cdot \left(2 \cdot \pi\right)\right)\right)}^{\left(k \cdot -0.5\right)} \cdot \frac{\sqrt{n \cdot \left(2 \cdot \pi\right)}}{\sqrt{k}} \]

Reproduce

herbie shell --seed 2021307 
(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))))