Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 7.4s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 4.344514377267187e+144
  2. n = -3.290307298511045e+220

Specification

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0))))

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0))))

Alternative 1: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n}}{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (sqrt (* (* 2.0 (PI)) n)) (* (pow (* (* 2.0 n) (PI)) (* 0.5 k)) (sqrt k))))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. div-sub N/A

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

   9. metadata-eval N/A

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

   10. pow-sub N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}}{\sqrt{k} \cdot {\left({\left(n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-pow N/A

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

   4. *-commutative N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

   \[\leadsto \frac{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n}}{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}} \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\\ \frac{\sqrt{t\_0}}{\sqrt{{t\_0}^{k} \cdot k}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 n) (PI)))) (/ (sqrt t_0) (sqrt (* (pow t_0 k) k)))))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. associate-/r/ N/A

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

   5. metadata-eval N/A

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

   6. pow-unpow N/A

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

   7. remove-double-neg N/A

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

   8. neg-sub0 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\frac{1}{2}}\right)}^{\color{blue}{\left(0 - \left(\mathsf{neg}\left(\left(1 - k\right)\right)\right)\right)}} \]

   9. pow-sub N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\frac{1}{2}}\right)}^{0}}{{\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\frac{1}{2}}\right)}^{\left(\mathsf{neg}\left(\left(1 - k\right)\right)\right)}}} \]

   10. metadata-eval N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. unpow1/2 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 N/A

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

   21. neg-sub0 N/A

       \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\color{blue}{\left(0 - \left(1 - k\right)\right)}}} \]

   22. lift--.f64 N/A

       \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\left(0 - \color{blue}{\left(1 - k\right)}\right)}} \]

   23. sub-neg N/A

       \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(k\right)\right)\right)}\right)}} \]

4. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\left(k - 1\right)}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. frac-times N/A

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

   5. metadata-eval N/A

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

   6. lift-pow.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

      \[\leadsto \frac{1}{\sqrt{k} \cdot {\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\color{blue}{\left(k + \left(\mathsf{neg}\left(1\right)\right)\right)}}} \]

   9. metadata-eval N/A

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

   10. unpow-prod-up N/A

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

   11. lift-sqrt.f64 N/A

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

   12. sqrt-pow2 N/A

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

   13. sqrt-pow1 N/A

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

   14. lift-pow.f64 N/A

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

   15. unpow1/2 N/A

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

   16. lift-pow.f64 N/A

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

   17. inv-pow N/A

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}}{\sqrt{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}} \]

7. Final simplification 99.6%

   \[\leadsto \frac{\sqrt{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}}{\sqrt{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k} \cdot k}} \]
8. Add Preprocessing

Alternative 3: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{1}{k}} \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (* (sqrt (/ 1.0 k)) (pow (* (* 2.0 n) (PI)) (fma -0.5 k 0.5))))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot e^{\frac{1}{2} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(1 - k\right)\right)}} \]
4. Step-by-step derivation

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

   \[\leadsto \sqrt{\frac{1}{k}} \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)} \]
7. Add Preprocessing

Alternative 4: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{\mathsf{PI}\left(\right) \cdot n}}{\sqrt{0.5 \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.06e-20)
   (/ (sqrt (* (PI) n)) (sqrt (* 0.5 k)))
   (sqrt (/ (pow (* (* 2.0 n) (PI)) (- 1.0 k)) k))))

Derivation

1. Split input into 2 regimes

2. if k < 1.06e-20

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in k around 0

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

      6. *-commutative N/A

         \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

      7. lower-*.f64 N/A

         \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

      8. lower-PI.f64 71.7

         \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot n}{k}} \]

   5. Applied rewrites 71.7%

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}}} \]
   6. Step-by-step derivation

   7. Applied rewrites 71.8%

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}}} \]
   8. Step-by-step derivation

   9. Applied rewrites 72.3%

      \[\leadsto \sqrt{\frac{n}{k} \cdot 2} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \]
   10. Step-by-step derivation

   11. Applied rewrites 99.5%

       \[\leadsto \frac{\sqrt{\mathsf{PI}\left(\right) \cdot n}}{\color{blue}{\sqrt{0.5 \cdot k}}} \]

   if 1.06e-20 < k

   1. Initial program 99.7%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. associate-/r/ N/A

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

      5. metadata-eval N/A

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

      6. pow-unpow N/A

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

      7. remove-double-neg N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\frac{1}{2}}\right)}^{\color{blue}{\left(0 - \left(\mathsf{neg}\left(\left(1 - k\right)\right)\right)\right)}} \]

      9. pow-sub N/A

         \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{{\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\frac{1}{2}}\right)}^{0}}{{\left({\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\frac{1}{2}}\right)}^{\left(\mathsf{neg}\left(\left(1 - k\right)\right)\right)}}} \]

      10. metadata-eval N/A

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

      11. lower-/.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 N/A

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

      21. neg-sub0 N/A

          \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\color{blue}{\left(0 - \left(1 - k\right)\right)}}} \]

      22. lift--.f64 N/A

          \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\left(0 - \color{blue}{\left(1 - k\right)}\right)}} \]

      23. sub-neg N/A

          \[\leadsto \frac{1}{\sqrt{k}} \cdot \frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\left(0 - \color{blue}{\left(1 + \left(\mathsf{neg}\left(k\right)\right)\right)}\right)}} \]

   4. Applied rewrites 99.7%

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{1}{{\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\left(k - 1\right)}}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. frac-times N/A

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

      5. metadata-eval N/A

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

      6. lift-pow.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

         \[\leadsto \frac{1}{\sqrt{k} \cdot {\left(\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}\right)}^{\color{blue}{\left(k + \left(\mathsf{neg}\left(1\right)\right)\right)}}} \]

      9. metadata-eval N/A

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

      10. unpow-prod-up N/A

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

      11. lift-sqrt.f64 N/A

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

      12. sqrt-pow2 N/A

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

      13. sqrt-pow1 N/A

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

      14. lift-pow.f64 N/A

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

      15. unpow1/2 N/A

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

      16. lift-pow.f64 N/A

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

      17. inv-pow N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}}{\sqrt{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}} \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\sqrt{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}}{\sqrt{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}} \]

      2. lift-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}}}{\sqrt{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}} \]

      3. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}}{\color{blue}{\sqrt{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}} \]

      4. sqrt-undiv N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}} \]

      6. lift-*.f64 N/A

         \[\leadsto \sqrt{\frac{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{k \cdot {\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}} \]

      7. *-commutative N/A

         \[\leadsto \sqrt{\frac{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}{\color{blue}{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k} \cdot k}}} \]

      8. associate-/r* N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}{k}}} \]

      9. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{\frac{\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)}{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{k}}}{k}}} \]

   8. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\sqrt{\frac{{\left(\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\right)}^{\left(1 - k\right)}}{k}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 1.06 \cdot 10^{-20}:\\ \;\;\;\;\frac{\sqrt{\mathsf{PI}\left(\right) \cdot n}}{\sqrt{0.5 \cdot k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\mathsf{fma}\left(-0.5, k, 0.5\right)\right)}}{\sqrt{k}} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* (* 2.0 n) (PI)) (fma -0.5 k 0.5)) (sqrt k)))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift--.f64 N/A

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

   8. div-sub N/A

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

   9. metadata-eval N/A

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

   10. pow-sub N/A

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

   11. associate-/l/ N/A

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

   12. lower-/.f64 N/A

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

4. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\frac{\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}}{\sqrt{k} \cdot {\left({\left(n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}^{k}\right)}^{0.5}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-pow N/A

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

   4. *-commutative N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-*.f64 99.7

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

6. Applied rewrites 99.7%

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

8. Applied rewrites 99.5%

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

Alternative 6: 48.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{\sqrt{\mathsf{PI}\left(\right) \cdot n}}{\sqrt{0.5 \cdot k}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (/ (sqrt (* (PI) n)) (sqrt (* 0.5 k))))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   8. lower-PI.f64 37.8

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot n}{k}} \]

5. Applied rewrites 37.8%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}}} \]
6. Step-by-step derivation

7. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}}} \]
8. Step-by-step derivation

9. Applied rewrites 38.1%

   \[\leadsto \sqrt{\frac{n}{k} \cdot 2} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \]
10. Step-by-step derivation

11. Applied rewrites 51.4%

    \[\leadsto \frac{\sqrt{\mathsf{PI}\left(\right) \cdot n}}{\color{blue}{\sqrt{0.5 \cdot k}}} \]
12. Add Preprocessing

Alternative 7: 48.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{n} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (* (sqrt (/ (* 2.0 (PI)) k)) (sqrt n)))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   8. lower-PI.f64 37.8

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot n}{k}} \]

5. Applied rewrites 37.8%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}}} \]
6. Step-by-step derivation

7. Applied rewrites 51.4%

   \[\leadsto \sqrt{n} \cdot \color{blue}{\sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{k}}} \]

8. Final simplification 51.4%

   \[\leadsto \sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{n} \]
9. Add Preprocessing

Alternative 8: 37.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{k}{n}}} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (/ (* 2.0 (PI)) (/ k n))))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   8. lower-PI.f64 37.8

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot n}{k}} \]

5. Applied rewrites 37.8%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}}} \]
6. Step-by-step derivation

7. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}}} \]
8. Step-by-step derivation

9. Applied rewrites 37.9%

   \[\leadsto \sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{\frac{k}{n}}} \]
10. Add Preprocessing

Alternative 9: 37.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{k} \cdot n} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* (/ (* 2.0 (PI)) k) n)))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   8. lower-PI.f64 37.8

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot n}{k}} \]

5. Applied rewrites 37.8%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}}} \]
6. Step-by-step derivation

7. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}}} \]
8. Step-by-step derivation

9. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\sqrt{\frac{2 \cdot \mathsf{PI}\left(\right)}{k} \cdot n}} \]
10. Add Preprocessing

Alternative 10: 37.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (* (/ (* (PI) n) k) 2.0)))

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \color{blue}{\sqrt{2}} \cdot \sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{2} \cdot \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   5. lower-/.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\color{blue}{\frac{n \cdot \mathsf{PI}\left(\right)}{k}}} \]

   6. *-commutative N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   7. lower-*.f64 N/A

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right) \cdot n}}{k}} \]

   8. lower-PI.f64 37.8

      \[\leadsto \sqrt{2} \cdot \sqrt{\frac{\color{blue}{\mathsf{PI}\left(\right)} \cdot n}{k}} \]

5. Applied rewrites 37.8%

   \[\leadsto \color{blue}{\sqrt{2} \cdot \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k}}} \]
6. Step-by-step derivation

7. Applied rewrites 37.9%

   \[\leadsto \color{blue}{\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{n}{k}}} \]

8. Taylor expanded in k around 0

   \[\leadsto \sqrt{2 \cdot \frac{n \cdot \mathsf{PI}\left(\right)}{k}} \]
9. Step-by-step derivation

10. Applied rewrites 37.9%

    \[\leadsto \sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024308 
(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))))