Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 9.5s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.3749323331035776e+176
  2. n = -3.979433204618148e+63

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, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow-prod-up N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. /-rgt-identity 99.3

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.3

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.3

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

8. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. sqrt-prod N/A

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

   9. lift-sqrt.f64 N/A

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

   10. div-inv N/A

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

   11. times-frac N/A

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

10. Applied rewrites 99.3%

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

Alternative 2: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow-prod-up N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. /-rgt-identity 99.3

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.3

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.3

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

8. Applied rewrites 99.3%

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

10. Applied rewrites 99.3%

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

11. Final simplification 99.3%

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

Alternative 3: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow-prod-up N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lift-*.f64 N/A

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

   6. *-rgt-identity N/A

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

   7. times-frac N/A

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

   8. lower-*.f64 N/A

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

6. Applied rewrites 99.3%

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

   1. lift-/.f64 N/A

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

   2. /-rgt-identity 99.3

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 99.3

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.3

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

8. Applied rewrites 99.3%

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

Alternative 4: 98.3% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

2. if k < 1

   1. Initial program 98.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 76.9

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

   5. Applied rewrites 76.9%

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

   7. Applied rewrites 77.1%

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

   9. Applied rewrites 96.8%

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

   if 1 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around inf

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

      1. lower-*.f64 99.2

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

   5. Applied rewrites 99.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 99.2

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

      6. lift-*.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*l* N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 99.2

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

   7. Applied rewrites 99.2%

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

Alternative 5: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
2. Add Preprocessing
3. 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. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. unpow-prod-up N/A

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

   9. lower-*.f64 N/A

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 99.3

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

6. Applied rewrites 99.1%

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

Alternative 6: 49.6% 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 99.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
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 42.4

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

5. Applied rewrites 42.4%

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

7. Applied rewrites 42.5%

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

9. Applied rewrites 53.1%

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

Alternative 7: 49.6% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
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 42.4

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

5. Applied rewrites 42.4%

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

7. Applied rewrites 42.5%

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

9. Applied rewrites 42.5%

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

11. Applied rewrites 53.0%

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

Alternative 8: 38.0% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
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 42.4

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

5. Applied rewrites 42.4%

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

7. Applied rewrites 42.5%

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

9. Applied rewrites 42.5%

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

11. Applied rewrites 42.5%

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

Alternative 9: 38.0% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
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 42.4

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

5. Applied rewrites 42.4%

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

7. Applied rewrites 42.5%

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

9. Applied rewrites 42.5%

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

Alternative 10: 37.9% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

   \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
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 42.4

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

5. Applied rewrites 42.4%

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

7. Applied rewrites 42.5%

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

9. Applied rewrites 42.5%

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

Reproduce

herbie shell --seed 2024318 
(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))))