Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.6%

Time: 10.2s

Alternatives: 7

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.2597827617707477e+282
  2. n = -4.259473379265416e+172

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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\\ \frac{\sqrt{t\_0}}{{t\_0}^{\left(0.5 \cdot k\right)} \cdot \sqrt{k}} \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (PI)) n)))
   (/ (sqrt t_0) (* (pow t_0 (* 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 99.8%

   \[\leadsto \color{blue}{\frac{\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}}{\sqrt{k} \cdot {\left({\left(n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}^{k}\right)}^{0.5}}} \]
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. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

7. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (sqrt (* (* 2.0 (PI)) n)) (- 1.0 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-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. rem-exp-log N/A

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

   3. lift-/.f64 N/A

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

   4. frac-2neg N/A

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

   5. distribute-frac-neg N/A

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

   6. pow-neg N/A

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

   7. lower-/.f64 N/A

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

   8. rem-exp-log N/A

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

   9. distribute-neg-frac2 N/A

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

   10. lift-/.f64 N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 N/A

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

   18. lift-/.f64 N/A

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

   19. clear-num N/A

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

   20. associate-/r/ N/A

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

   21. metadata-eval N/A

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

4. Applied rewrites 99.4%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{1}{{\left(n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lower-/.f64 99.4

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

6. Applied rewrites 99.5%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (/ (pow (* (* 2.0 (PI)) n) (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 99.8%

   \[\leadsto \color{blue}{\frac{\sqrt{n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)}}{\sqrt{k} \cdot {\left({\left(n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}^{k}\right)}^{0.5}}} \]
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. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. associate-/r* N/A

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

8. Applied rewrites 99.5%

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

Alternative 4: 50.0% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 (/ (sqrt (* (* 2.0 (PI)) n)) (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. 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 32.5

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

5. Applied rewrites 32.5%

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

7. Applied rewrites 44.9%

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

Alternative 5: 50.0% 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 32.5

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

5. Applied rewrites 32.5%

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

7. Applied rewrites 44.9%

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

8. Final simplification 44.9%

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

Alternative 6: 38.6% accurate, 4.8× speedup

Math

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

FPCore

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

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 32.5

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

5. Applied rewrites 32.5%

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

7. Applied rewrites 32.5%

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

9. Applied rewrites 32.5%

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

Alternative 7: 38.6% accurate, 4.8× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 (sqrt (* (/ (PI) k) (* 2.0 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 32.5

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

5. Applied rewrites 32.5%

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

7. Applied rewrites 32.5%

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

9. Applied rewrites 32.5%

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

10. Final simplification 32.5%

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

Reproduce

herbie shell --seed 2024256 
(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))))