Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 11.8s

Alternatives: 16

Speedup: 0.9×

• Could not converge on a ground truth

  1. k = 4.799715519731572e+259
  2. n = -2.0146239605162058e+52

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

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

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (PI) (* n 2.0))))
   (/ (sqrt t_0) (* (pow t_0 (/ k 2.0)) (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. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*l* N/A

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

   4. count-2-rev N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f64 99.5

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

4. Applied rewrites 99.5%

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 58.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{n}{k} \cdot 2\\ t_1 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ t_2 := n \cdot \mathsf{PI}\left(\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;{\left(t\_0 \cdot t\_0\right)}^{0.25}\\ \mathbf{elif}\;t\_1 \leq 10^{+273}:\\ \;\;\;\;\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{t\_2}} \cdot \sqrt{\sqrt{k \cdot \left(t\_2 \cdot k\right)}}\right) \cdot \sqrt{2}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (/ n k) 2.0))
        (t_1
         (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0))))
        (t_2 (* n (PI))))
   (if (<= t_1 0.0)
     (pow (* t_0 t_0) 0.25)
     (if (<= t_1 1e+273)
       (/ (sqrt (* (PI) (* n 2.0))) (sqrt k))
       (/
        (* (* (sqrt (sqrt t_2)) (sqrt (sqrt (* k (* t_2 k))))) (sqrt 2.0))
        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%

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

   9. Applied rewrites 3.2%

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

   11. Applied rewrites 20.3%

       \[\leadsto {\left(\left(\frac{n}{k} \cdot 2\right) \cdot \left(\frac{n}{k} \cdot 2\right)\right)}^{\color{blue}{0.25}} \]

   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)))) < 9.99999999999999945e272

   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 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 95.9%

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

   if 9.99999999999999945e272 < (*.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 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

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \mathsf{fma}\left(\log \left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left({k}^{3} \cdot \mathsf{PI}\left(\right)\right) \cdot n}, -0.5, \sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot k}\right)}{k}} \]

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 2.2%

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

   10. Applied rewrites 25.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;{\left(\left(\frac{n}{k} \cdot 2\right) \cdot \left(\frac{n}{k} \cdot 2\right)\right)}^{0.25}\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 10^{+273}:\\ \;\;\;\;\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{n \cdot \mathsf{PI}\left(\right)}} \cdot \sqrt{\sqrt{k \cdot \left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot k\right)}}\right) \cdot \sqrt{2}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ t_1 := n \cdot \mathsf{PI}\left(\right)\\ t_2 := t\_1 \cdot k\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(k \cdot n, \mathsf{PI}\left(\right), t\_2\right)}{k \cdot k}}\\ \mathbf{elif}\;t\_0 \leq 10^{+273}:\\ \;\;\;\;\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{t\_1}} \cdot \sqrt{\sqrt{k \cdot t\_2}}\right) \cdot \sqrt{2}}{k}\\ \end{array} \end{array} \]

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 (* n (PI)))
        (t_2 (* t_1 k)))
   (if (<= t_0 0.0)
     (sqrt (/ (fma (* k n) (PI) t_2) (* k k)))
     (if (<= t_0 1e+273)
       (/ (sqrt (* (PI) (* n 2.0))) (sqrt k))
       (/ (* (* (sqrt (sqrt t_1)) (sqrt (sqrt (* k t_2)))) (sqrt 2.0)) 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%

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

   9. Applied rewrites 10.0%

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

   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)))) < 9.99999999999999945e272

   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 74.3

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

   5. Applied rewrites 74.3%

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

   7. Applied rewrites 95.9%

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

   if 9.99999999999999945e272 < (*.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 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

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]

   5. Applied rewrites 63.6%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \mathsf{fma}\left(\log \left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left({k}^{3} \cdot \mathsf{PI}\left(\right)\right) \cdot n}, -0.5, \sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot k}\right)}{k}} \]

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 2.2%

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

   10. Applied rewrites 25.4%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(k \cdot n, \mathsf{PI}\left(\right), \left(n \cdot \mathsf{PI}\left(\right)\right) \cdot k\right)}{k \cdot k}}\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 10^{+273}:\\ \;\;\;\;\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{n \cdot \mathsf{PI}\left(\right)}} \cdot \sqrt{\sqrt{k \cdot \left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot k\right)}}\right) \cdot \sqrt{2}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.9% accurate, 0.5× speedup

Math

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

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 (/ (fma (* k n) (PI) (* (* n (PI)) k)) (* k k)))
   (/ (sqrt (* (PI) (* n 2.0))) (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%

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

   9. Applied rewrites 10.0%

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

   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))))

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 46.8

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

   5. Applied rewrites 46.8%

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

   7. Applied rewrites 60.2%

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

4. Final simplification 47.1%

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

Alternative 5: 49.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot k\right) \cdot 2}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<=
      (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))
      5e+142)
   (sqrt (* (/ (* (PI) n) k) 2.0))
   (/ (sqrt (* (* (* n (PI)) k) 2.0)) 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)))) < 5.0000000000000001e142

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 54.9

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

   5. Applied rewrites 54.9%

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

   7. Applied rewrites 55.1%

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

   if 5.0000000000000001e142 < (*.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

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot \left(\sqrt{{k}^{3} \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)\right) + \sqrt{k \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{2}}{k}} \]

   5. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\frac{\sqrt{2} \cdot \mathsf{fma}\left(\log \left(\left(2 \cdot n\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\left({k}^{3} \cdot \mathsf{PI}\left(\right)\right) \cdot n}, -0.5, \sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot k}\right)}{k}} \]

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 30.8%

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

   10. Applied rewrites 30.9%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+142}:\\ \;\;\;\;\sqrt{\frac{\mathsf{PI}\left(\right) \cdot n}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot k\right) \cdot 2}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

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

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 7: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

5. Applied rewrites 99.6%

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

6. Final simplification 99.6%

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

Alternative 8: 95.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 2.3 \cdot 10^{+15}:\\ \;\;\;\;\frac{\sqrt{\mathsf{PI}\left(\right) \cdot \left(n \cdot 2\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 2.3e+15)
   (/ (sqrt (* (PI) (* n 2.0))) (sqrt k))
   (/ (pow (* -2.0 n) (/ (- 1.0 k) 2.0)) (sqrt k))))

Derivation

1. Split input into 2 regimes

2. if k < 2.3e15

   1. Initial program 99.1%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 71.3

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

   5. Applied rewrites 71.3%

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

   7. Applied rewrites 92.1%

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

   if 2.3e15 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*l* N/A

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

      4. count-2-rev N/A

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

      5. lift-PI.f64 N/A

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

      6. add-sqr-sqrt N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-PI.f64 N/A

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

      12. lower-sqrt.f64 N/A

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

      13. lift-PI.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. *-rgt-identity 100.0

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

      3. lift-sqrt.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*r* N/A

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

      7. lift-*.f64 N/A

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

      8. sqrt-prod N/A

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

      9. lift-sqrt.f64 N/A

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

      10. lift-sqrt.f64 N/A

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

      11. lower-*.f64 100.0

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

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

Alternative 9: 49.4% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 (/ (sqrt (* (PI) (* n 2.0))) (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 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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 45.3%

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

Alternative 10: 49.4% accurate, 3.6× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 (* (sqrt (* n (PI))) (sqrt (/ 2.0 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 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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.5%

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

9. Applied rewrites 45.3%

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

Alternative 11: 49.4% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.5%

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

9. Applied rewrites 45.3%

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

Alternative 12: 38.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.5%

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

Alternative 13: 38.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.5%

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

9. Applied rewrites 35.5%

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

Alternative 14: 38.1% accurate, 5.1× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 (sqrt (* (+ (PI) (PI)) (/ n 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 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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.5%

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

9. Applied rewrites 35.5%

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

11. Applied rewrites 35.5%

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

Alternative 15: 9.2% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

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 / k) * 2.0d0))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(n / k), $MachinePrecision] * 2.0), $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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.5%

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

9. Applied rewrites 35.5%

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

11. Applied rewrites 8.7%

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

Alternative 16: 5.1% accurate, 5.6× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 (sqrt (* (/ 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 35.4

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

5. Applied rewrites 35.4%

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

7. Applied rewrites 35.5%

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

9. Applied rewrites 35.5%

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

11. Applied rewrites 4.9%

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

Reproduce

herbie shell --seed 2024364 
(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))))