Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 14.0s

Alternatives: 4

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.204963938664249e+287
  2. n = -1.7902171144842386e+168

Specification

Math

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

FPCore

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

Initial Program: 99.4% accurate, 1.0× speedup

Math

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

FPCore

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

Alternative 1: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0

   \[\leadsto \frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\color{blue}{\left(\frac{1}{2} + \frac{-1}{2} \cdot k\right)}} \]
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 99.1

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

5. Applied rewrites 99.1%

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

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

7. Applied rewrites 99.1%

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

   1. lift-*.f64 N/A

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

   2. *-lft-identity 99.1

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

   3. lift-*.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 99.1

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

9. Applied rewrites 99.1%

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

    1. lift-*.f64 N/A

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

    2. count-2-rev N/A

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

    3. lower-+.f64 99.1

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

11. Applied rewrites 99.1%

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

Alternative 2: 49.1% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. unpow-prod-down N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-PI.f64 N/A

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

   13. lift-/.f64 N/A

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

   14. lift--.f64 N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-/.f64 N/A

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

   17. lift--.f64 79.8

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

4. Applied rewrites 79.8%

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

5. Taylor expanded in k around 0

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

   1. lower-sqrt.f64 78.8

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

7. Applied rewrites 78.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. metadata-eval N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-div N/A

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

   6. lift-*.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. lift-PI.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift--.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*r* N/A

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

   13. lower-*.f64 N/A

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

9. Applied rewrites 78.8%

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

10. Taylor expanded in k around 0

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

    1. sqrt-unprod N/A

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

    2. lower-sqrt.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-/.f64 N/A

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

    5. lift-PI.f64 55.0

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

12. Applied rewrites 55.0%

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

Alternative 3: 38.2% 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 99.1%

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

3. Taylor expanded in k around 0

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

   1. 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 41.4

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

5. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2}} \]
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 41.4

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

7. Applied rewrites 41.4%

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

Alternative 4: 38.2% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.1%

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

3. Taylor expanded in k around 0

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

   1. 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 41.4

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

5. Applied rewrites 41.4%

   \[\leadsto \color{blue}{\sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2}} \]
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 41.4

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

7. Applied rewrites 41.4%

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

   1. lift-*.f64 N/A

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

   2. lift-PI.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lower-*.f64 41.4

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

9. Applied rewrites 41.4%

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

Reproduce

herbie shell --seed 2025043 
(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))))