Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.6%

Time: 9.7s

Alternatives: 11

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.5087617310695174e+85
  2. n = -9.147949230723108e+58

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: 99.5% 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, 1.0× speedup

Math

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

FPCore

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

Derivation

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. div-inv N/A

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

   4. metadata-eval N/A

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

   5. pow-unpow N/A

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

   6. unpow1/2 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-pow.f64 99.4

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.4

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lift-sqrt.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. sqrt-pow1 N/A

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

   8. lift--.f64 N/A

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

   9. div-sub N/A

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

   10. metadata-eval N/A

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

   11. pow-sub N/A

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

   12. pow1/2 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. associate-/r* N/A

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

   15. associate-/l/ N/A

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

6. Applied rewrites 99.7%

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

Alternative 2: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{k \cdot {\left(\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(k - 1\right)}}\right)}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.2e-30)
   (* (sqrt (* (* 2.0 (PI)) n)) (pow k -0.5))
   (pow (sqrt (* k (pow (* (* n 2.0) (PI)) (- k 1.0)))) -1.0)))

Derivation

1. Split input into 2 regimes

2. if k < 3.2e-30

   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. 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 72.3

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

   5. Applied rewrites 72.3%

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

   7. Applied rewrites 99.5%

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

   if 3.2e-30 < 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. div-inv N/A

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

      4. metadata-eval N/A

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

      5. pow-unpow N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-pow.f64 99.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 99.4

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

   6. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-sqrt.f64 N/A

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

      3. sqrt-pow2 N/A

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

      4. div-inv N/A

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

      5. metadata-eval N/A

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

      6. pow-unpow N/A

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

      7. unpow1/2 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-pow.f64 99.4

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 99.4

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

   8. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lower-sqrt.f64 N/A

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

      8. div-inv N/A

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

      9. lift-pow.f64 N/A

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

      10. lift--.f64 N/A

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

      11. pow-sub N/A

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

      12. unpow1 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-PI.f64 N/A

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

   10. Applied rewrites 99.3%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 3.2 \cdot 10^{-30}:\\ \;\;\;\;\sqrt{\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{k \cdot {\left(\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)}^{\left(k - 1\right)}}\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

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. 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 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\\ \mathbf{if}\;k \leq 2.2 \cdot 10^{-31}:\\ \;\;\;\;\sqrt{t\_0} \cdot {k}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* 2.0 (PI)) n)))
   (if (<= k 2.2e-31)
     (* (sqrt t_0) (pow k -0.5))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

Derivation

1. Split input into 2 regimes

2. if k < 2.2000000000000001e-31

   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 71.8

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

   5. Applied rewrites 71.8%

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

   7. Applied rewrites 99.5%

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

   if 2.2000000000000001e-31 < 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. div-inv N/A

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

      4. metadata-eval N/A

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

      5. pow-unpow N/A

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

      6. unpow1/2 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-pow.f64 99.4

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 99.4

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-sqrt.f64 N/A

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

      7. sqrt-undiv N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-/.f64 98.8

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 98.8

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

      13. lift-*.f64 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f64 98.8

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

   6. Applied rewrites 98.8%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

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. div-inv N/A

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

   4. metadata-eval N/A

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

   5. pow-unpow N/A

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

   6. unpow1/2 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-pow.f64 99.4

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.4

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift--.f64 N/A

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

   3. pow-sub N/A

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

   4. unpow1 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. pow1/2 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. pow1/2 N/A

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

   9. pow-pow N/A

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

   10. pow-div N/A

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

   11. lower-pow.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower-*.f64 99.4

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

8. Applied rewrites 99.4%

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

Alternative 6: 49.3% accurate, 1.2× speedup

Math

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

FPCore

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

Derivation

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. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 48.8%

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

Alternative 7: 49.2% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

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. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 48.8%

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

Alternative 8: 37.7% accurate, 4.0× speedup

Math

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

FPCore

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

Derivation

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. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.3%

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

9. Applied rewrites 37.3%

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

Alternative 9: 37.7% 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 99.4%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.3%

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

Alternative 10: 37.7% 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 99.4%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.3%

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

9. Applied rewrites 37.3%

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

Alternative 11: 37.7% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

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. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.2

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

5. Applied rewrites 37.2%

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

7. Applied rewrites 37.3%

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

9. Applied rewrites 37.2%

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

Reproduce

herbie shell --seed 2024296 
(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))))