Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.0%

Time: 9.8s

Alternatives: 9

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 4.326533336808854e+306
  2. n = -5.652156362596694e+150

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: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.1 \cdot 10^{-61}:\\ \;\;\;\;\sqrt{\frac{\mathsf{PI}\left(\right)}{k}} \cdot \sqrt{n \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.1e-61)
   (* (sqrt (/ (PI) k)) (sqrt (* n 2.0)))
   (sqrt (/ (pow (* (* n (PI)) 2.0) (- 1.0 k)) k))))

Derivation

1. Split input into 2 regimes

2. if k < 1.10000000000000004e-61

   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 76.8

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

   5. Applied rewrites 76.8%

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 77.1%

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

   11. Applied rewrites 99.5%

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

   if 1.10000000000000004e-61 < k

   1. Initial program 99.4%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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. frac-2neg N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-unpow N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-pow.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 N/A

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in k around inf

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

      1. *-commutative N/A

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

   7. Applied rewrites 99.4%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

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. 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. frac-2neg N/A

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

   4. distribute-frac-neg N/A

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

   5. pow-neg N/A

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

   6. rem-exp-log N/A

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

   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

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

4. Applied rewrites 99.3%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \color{blue}{\frac{1}{{\left(n \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}^{\left(-0.5 \cdot \left(1 - k\right)\right)}}} \]
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 99.4%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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 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 99.3%

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

7. Applied rewrites 99.4%

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

Alternative 4: 48.5% 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 99.3%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 38.5%

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

9. Applied rewrites 38.5%

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

11. Applied rewrites 46.5%

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

Alternative 5: 48.5% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

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 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 38.5%

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

9. Applied rewrites 46.5%

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

Alternative 6: 48.5% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

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 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 38.5%

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

9. Applied rewrites 46.5%

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

Alternative 7: 37.1% 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 99.3%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 38.5%

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

Alternative 8: 37.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

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 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 38.5%

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

9. Applied rewrites 38.5%

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

Alternative 9: 37.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

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 38.4

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

5. Applied rewrites 38.4%

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

7. Applied rewrites 38.5%

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

9. Applied rewrites 38.5%

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

11. Applied rewrites 38.5%

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

Reproduce

herbie shell --seed 2024309 
(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))))