Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 11.0s

Alternatives: 14

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.3276139956030086e+225
  2. n = -1.0574993779863043e+56

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, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. div-sub N/A

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

   14. metadata-eval N/A

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

   15. pow-sub N/A

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

   16. lift-pow.f64 N/A

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

   17. metadata-eval N/A

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

   18. pow-flip N/A

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

   19. pow1/2 N/A

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

   20. lift-sqrt.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 99.7

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. sqrt-prod N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lower-*.f64 99.5

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

   14. lift-sqrt.f64 N/A

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

   15. pow1/2 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. pow1/2 N/A

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

   20. lift-sqrt.f64 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. times-frac N/A

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

   5. *-commutative N/A

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

   6. associate-*r/ N/A

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

   7. lower-/.f64 N/A

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

10. Applied rewrites 99.7%

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

Alternative 2: 42.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ t_1 := \mathsf{fma}\left(k, n, n \cdot k\right)\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{t\_1}{k \cdot k}}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(2 \cdot \frac{n}{k}\right) \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{t\_1}}{\left|k\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0
         (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0))))
        (t_1 (fma k n (* n k))))
   (if (<= t_0 0.0)
     (sqrt (/ t_1 (* k k)))
     (if (<= t_0 2e+154)
       (sqrt (* (* 2.0 (/ n k)) (PI)))
       (/ (sqrt t_1) (fabs k))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   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 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.2%

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

   11. Applied rewrites 10.2%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 2.00000000000000007e154

   1. Initial program 98.9%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 94.8

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

   5. Applied rewrites 94.8%

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

   7. Applied rewrites 95.2%

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

   9. Applied rewrites 95.2%

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

   if 2.00000000000000007e154 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.8%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 3.5

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

   5. Applied rewrites 3.5%

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

   7. Applied rewrites 3.5%

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

   9. Applied rewrites 3.5%

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

   11. Applied rewrites 8.2%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(k, n, n \cdot k\right)}}{\color{blue}{\left|k\right|}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(k, n, n \cdot k\right)}{k \cdot k}}\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(2 \cdot \frac{n}{k}\right) \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(k, n, n \cdot k\right)}}{\left|k\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<=
      (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))
      0.0)
   (sqrt 0.0)
   (/ (sqrt (* (* n 2.0) (PI))) (sqrt k))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 0.0

   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

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   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 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 48.7

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

   5. Applied rewrites 48.7%

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

   7. Applied rewrites 65.5%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(2 \cdot \frac{n}{k}\right) \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(k, n, n \cdot k\right)}}{\left|k\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<=
      (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))
      2e+154)
   (sqrt (* (* 2.0 (/ n k)) (PI)))
   (/ (sqrt (fma k n (* n k))) (fabs k))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 2.00000000000000007e154

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 57.0

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

   5. Applied rewrites 57.0%

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

   7. Applied rewrites 57.3%

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

   9. Applied rewrites 57.3%

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

   if 2.00000000000000007e154 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.8%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 3.5

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

   5. Applied rewrites 3.5%

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

   7. Applied rewrites 3.5%

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

   9. Applied rewrites 3.5%

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

   11. Applied rewrites 8.2%

       \[\leadsto \frac{\sqrt{\mathsf{fma}\left(k, n, n \cdot k\right)}}{\color{blue}{\left|k\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 2 \cdot 10^{+154}:\\ \;\;\;\;\sqrt{\left(2 \cdot \frac{n}{k}\right) \cdot \mathsf{PI}\left(\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(k, n, n \cdot k\right)}}{\left|k\right|}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.5% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval 99.5

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-pow.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. lift-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. div-sub N/A

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

   14. metadata-eval N/A

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

   15. pow-sub N/A

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

   16. lift-pow.f64 N/A

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

   17. metadata-eval N/A

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

   18. pow-flip N/A

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

   19. pow1/2 N/A

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

   20. lift-sqrt.f64 N/A

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

6. Applied rewrites 99.7%

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

   1. lift-*.f64 N/A

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

   2. *-rgt-identity 99.7

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

   3. lift-sqrt.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. sqrt-prod N/A

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

   11. lift-sqrt.f64 N/A

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

   12. lift-sqrt.f64 N/A

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

   13. lower-*.f64 99.5

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

   14. lift-sqrt.f64 N/A

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

   15. pow1/2 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. pow1/2 N/A

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

   20. lift-sqrt.f64 99.5

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

8. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. sqrt-unprod N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lower-sqrt.f64 99.7

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 99.7

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 99.7

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

10. Applied rewrites 99.7%

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

Alternative 6: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

   1. sinh-+-cosh-rev N/A

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

   2. +-commutative N/A

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

   3. distribute-rgt-out-- N/A

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

   4. fp-cancel-sub-sign N/A

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-in N/A

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

   7. distribute-rgt-out-- N/A

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

   8. fp-cancel-sub-sign N/A

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

5. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.5

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

7. Applied rewrites 99.5%

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

Alternative 7: 52.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{+163}:\\ \;\;\;\;\frac{\sqrt{\left(n \cdot 2\right) \cdot \mathsf{PI}\left(\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(k, n, n \cdot k\right)}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.4e+163)
   (/ (sqrt (* (* n 2.0) (PI))) (sqrt k))
   (sqrt (/ (fma k n (* n k)) (* k k)))))

Derivation

1. Split input into 2 regimes

2. if k < 3.4000000000000001e163

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 64.6%

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

   if 3.4000000000000001e163 < 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

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 2.7%

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

   9. Applied rewrites 2.7%

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

   11. Applied rewrites 10.2%

       \[\leadsto \sqrt{\frac{\mathsf{fma}\left(k, n, n \cdot k\right)}{k \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{\mathsf{PI}\left(\right) \cdot n} \cdot \sqrt{\frac{2}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(k, n, n \cdot k\right)}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.4e+163)
   (* (sqrt (* (PI) n)) (sqrt (/ 2.0 k)))
   (sqrt (/ (fma k n (* n k)) (* k k)))))

Derivation

1. Split input into 2 regimes

2. if k < 3.4000000000000001e163

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 48.3%

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

   9. Applied rewrites 64.6%

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

   if 3.4000000000000001e163 < 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

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 2.7%

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

   9. Applied rewrites 2.7%

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

   11. Applied rewrites 10.2%

       \[\leadsto \sqrt{\frac{\mathsf{fma}\left(k, n, n \cdot k\right)}{k \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 3.4 \cdot 10^{+163}:\\ \;\;\;\;\sqrt{n} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(k, n, n \cdot k\right)}{k \cdot k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 3.4e+163)
   (* (sqrt n) (sqrt (* (/ (PI) k) 2.0)))
   (sqrt (/ (fma k n (* n k)) (* k k)))))

Derivation

1. Split input into 2 regimes

2. if k < 3.4000000000000001e163

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 48.1

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

   5. Applied rewrites 48.1%

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

   7. Applied rewrites 48.3%

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

   9. Applied rewrites 64.6%

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

   if 3.4000000000000001e163 < 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

      \[\leadsto \color{blue}{\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k}} \cdot \sqrt{2}} \]
   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 2.7

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

   5. Applied rewrites 2.7%

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

   7. Applied rewrites 2.7%

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

   9. Applied rewrites 2.7%

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

   11. Applied rewrites 10.2%

       \[\leadsto \sqrt{\frac{\mathsf{fma}\left(k, n, n \cdot k\right)}{k \cdot k}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 37.7% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.0

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

5. Applied rewrites 37.0%

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

7. Applied rewrites 37.1%

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

9. Applied rewrites 37.1%

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

Alternative 11: 37.6% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.0

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

5. Applied rewrites 37.0%

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

7. Applied rewrites 37.1%

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

9. Applied rewrites 37.1%

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

11. Applied rewrites 37.1%

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

Alternative 12: 9.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{n \cdot \frac{2}{k}} \end{array} \]

FPCore

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

C

double code(double k, double n) {
	return sqrt((n * (2.0 / k)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(k, n)
use fmin_fmax_functions
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = sqrt((n * (2.0d0 / k)))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt((n * (2.0 / k)));
}

Python

def code(k, n):
	return math.sqrt((n * (2.0 / k)))

Julia

function code(k, n)
	return sqrt(Float64(n * Float64(2.0 / k)))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt((n * (2.0 / k)));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(n * N[(2.0 / k), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.0

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

5. Applied rewrites 37.0%

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

7. Applied rewrites 37.1%

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

9. Applied rewrites 37.1%

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

11. Applied rewrites 9.1%

    \[\leadsto \sqrt{n \cdot \frac{2}{k}} \]
12. Add Preprocessing

Alternative 13: 4.8% accurate, 7.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.0

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

5. Applied rewrites 37.0%

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

7. Applied rewrites 37.1%

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

9. Applied rewrites 37.1%

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

11. Applied rewrites 4.6%

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

Alternative 14: 4.8% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \end{array} \]

FPCore

(FPCore (k n) :precision binary64 (sqrt (+ (PI) (PI))))

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 37.0

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

5. Applied rewrites 37.0%

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

7. Applied rewrites 37.1%

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

9. Applied rewrites 37.1%

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

11. Applied rewrites 4.6%

    \[\leadsto \sqrt{\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024353 
(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))))