Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 5.6s

Alternatives: 7

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.9806138088036667e+290
  2. n = -2.8655401845566854e+188

Specification

Math

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

FPCore

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 99.2

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

5. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. lift-PI.f64 N/A

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

   16. lift-sqrt.f64 99.2

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

7. Applied rewrites 99.2%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.2

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 99.2

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

9. Applied rewrites 99.2%

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

Alternative 2: 98.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot n\\ \mathbf{if}\;k \leq 1.02:\\ \;\;\;\;\frac{1}{\sqrt{k}} \cdot \left(\sqrt{t\_0} \cdot \sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t\_0 \cdot 2\right)}^{\left(-0.5 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (PI) n)))
   (if (<= k 1.02)
     (* (/ 1.0 (sqrt k)) (* (sqrt t_0) (sqrt 2.0)))
     (/ (pow (* t_0 2.0) (* -0.5 k)) (sqrt k)))))

Derivation

1. Split input into 2 regimes

2. if k < 1.02

   1. Initial program 98.3%

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

   3. Taylor expanded in k around 0

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

      1. sqrt-unprod N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-PI.f64 95.9

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

   5. Applied rewrites 95.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-PI.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. sqrt-unprod N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sqrt.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift-PI.f64 N/A

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

      15. lower-sqrt.f64 96.5

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

   7. Applied rewrites 96.5%

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

   if 1.02 < 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 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 100.0

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

   5. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-PI.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-PI.f64 N/A

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

      16. lift-sqrt.f64 100.0

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

   7. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-lft-identity 100.0

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

      3. lift-*.f64 N/A

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

      4. lift-PI.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 100.0

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

   9. Applied rewrites 100.0%

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

   10. Taylor expanded in k around inf

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

       1. lower-*.f64 98.5

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

   12. Applied rewrites 98.5%

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

Alternative 3: 48.8% accurate, 2.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 47.8

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

5. Applied rewrites 47.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. sqrt-unprod N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-sqrt.f64 48.1

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

7. Applied rewrites 48.1%

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

Alternative 4: 48.7% accurate, 2.8× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (* (/ 1.0 (sqrt k)) (* (sqrt (+ (PI) (PI))) (sqrt n))))

Derivation

1. Initial program 99.2%

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

3. Taylor expanded in k around 0

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

   1. sqrt-unprod N/A

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

   2. *-commutative N/A

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

   3. *-commutative N/A

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

   4. associate-*l* N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 47.8

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

5. Applied rewrites 47.8%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. sqrt-prod N/A

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

   4. lower-*.f64 N/A

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

   5. lower-sqrt.f64 N/A

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

   6. lower-sqrt.f64 48.0

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

7. Applied rewrites 48.0%

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

   1. lift-PI.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. count-2-rev N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]

   5. lower-+.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]

   6. lift-PI.f64 N/A

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]

   7. lift-PI.f64 48.0

      \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]

9. Applied rewrites 48.0%

   \[\leadsto \frac{1}{\sqrt{k}} \cdot \left(\sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \]
10. Add Preprocessing

Alternative 5: 48.9% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-pow.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. lift--.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. associate-*l/ N/A

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

   11. lower-/.f64 N/A

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

4. Applied rewrites 99.2%

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

5. Taylor expanded in k around 0

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

   1. *-lft-identity N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-pow N/A

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

   6. lift-*.f64 N/A

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

   7. lift-PI.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. sqrt-unprod N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

7. Applied rewrites 47.9%

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

8. Final simplification 47.9%

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

Alternative 6: 38.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 38.4

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

5. Applied rewrites 38.4%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-/.f64 38.4

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

7. Applied rewrites 38.4%

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

Alternative 7: 38.0% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.2%

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

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

   2. lower-sqrt.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lift-PI.f64 38.4

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

5. Applied rewrites 38.4%

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

   1. lift-/.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lift-PI.f64 38.4

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

7. Applied rewrites 38.4%

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

Reproduce

herbie shell --seed 2025056 
(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))))