Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 10.4s

Alternatives: 11

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.0319252806277913e+160
  2. n = -7.807021105051012e+300

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: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)\\ \sqrt{\frac{1}{k}} \cdot \left(\sqrt{t\_0} \cdot {t\_0}^{\left(-0.5 \cdot k\right)}\right) \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (PI) (* n 2.0))))
   (* (sqrt (/ 1.0 k)) (* (sqrt t_0) (pow t_0 (* -0.5 k))))))

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 99.7%

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (PI) (* n 2.0))))
   (/ (* (pow t_0 (* (- 0.5) k)) (sqrt t_0)) (sqrt k))))

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 99.6

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

6. Applied rewrites 99.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. flip-+ N/A

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

   5. *-commutative N/A

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

   6. *-commutative N/A

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

   7. swap-sqr N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{{\left(\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(\frac{\frac{1}{2} \cdot \frac{1}{2} - \left(k \cdot k\right) \cdot \color{blue}{\frac{1}{4}}}{\frac{1}{2} - \frac{-1}{2} \cdot k}\right)}}{\sqrt{k}} \]

   9. metadata-eval N/A

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

   10. swap-sqr N/A

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

   11. metadata-eval N/A

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

   12. div-inv N/A

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

   13. metadata-eval N/A

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

   14. div-inv N/A

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

   15. *-commutative N/A

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

   16. cancel-sign-sub-inv N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. div-inv N/A

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

   20. frac-2neg N/A

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

   21. flip-- N/A

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

   22. sub-neg N/A

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

8. Applied rewrites 99.8%

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

9. Final simplification 99.8%

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

Alternative 3: 98.2% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 74.0

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

   5. Applied rewrites 74.0%

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

   7. Applied rewrites 74.2%

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

   9. Applied rewrites 97.5%

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

   if 1 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 100.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. lower-*.f64 100.0

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

   7. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.9%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 99.6

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 5: 49.6% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 34.8%

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

9. Applied rewrites 45.3%

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

10. Final simplification 45.3%

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

Alternative 6: 49.5% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 34.8%

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

9. Applied rewrites 45.3%

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

10. Final simplification 45.3%

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

Alternative 7: 38.3% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 34.8%

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

9. Applied rewrites 34.8%

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

Alternative 8: 38.3% 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 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 34.8%

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

Alternative 9: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 34.8%

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

9. Applied rewrites 34.8%

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

Alternative 10: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 34.8%

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

9. Applied rewrites 34.8%

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

10. Final simplification 34.8%

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

Alternative 11: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.6%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 34.7

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

5. Applied rewrites 34.7%

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

7. Applied rewrites 34.8%

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

9. Applied rewrites 34.8%

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

10. Final simplification 34.8%

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

Reproduce

herbie shell --seed 2024248 
(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))))