Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 8.9s

Alternatives: 8

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 3.701759686057461e+294
  2. n = -1.6171628511370469e+121

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

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

FPCore

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

Derivation

1. Initial program 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.0

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in k around inf

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

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

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

   2. +-commutative N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. cancel-sign-sub N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. +-commutative N/A

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

9. Applied rewrites 99.0%

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

11. Applied rewrites 99.3%

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

Alternative 2: 74.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\frac{{4}^{\left(\frac{1 - k}{4}\right)}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{n} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k}}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= (* (/ 1.0 (sqrt k)) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0))) 0.0)
   (/ (pow 4.0 (/ (- 1.0 k) 4.0)) (sqrt k))
   (* (* (sqrt n) (sqrt (/ (PI) k))) (sqrt 2.0))))

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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lower-/.f64 100.0

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 100.0

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. count-2-rev N/A

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

      5. distribute-rgt-in N/A

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

      6. rem-square-sqrt N/A

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

      7. lift-sqrt.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f64 N/A

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

      11. *-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 100.0

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

      14. lift-pow.f64 N/A

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

      15. sqr-pow N/A

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

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{{4}^{\left(\frac{1 - k}{4}\right)}}}{\sqrt{k}} \]

   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 98.7%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 48.9

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

   5. Applied rewrites 48.9%

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

   7. Applied rewrites 48.9%

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

   9. Applied rewrites 62.1%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 38.9

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

5. Applied rewrites 38.9%

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

7. Applied rewrites 4.9%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{k}}} \]

8. Taylor expanded in k around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

10. Applied rewrites 99.0%

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

Alternative 4: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.0

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.0

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.0

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

4. Applied rewrites 99.0%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-PI.f64 N/A

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

   5. rem-square-sqrt N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-sqrt.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 98.9

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

6. Applied rewrites 98.9%

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

7. Taylor expanded in k around inf

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

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

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

   2. +-commutative N/A

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

   3. *-lft-identity N/A

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

   4. metadata-eval N/A

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

   5. cancel-sign-sub N/A

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

   6. *-lft-identity N/A

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

   7. metadata-eval N/A

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

   8. cancel-sign-sub N/A

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

   9. +-commutative N/A

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

9. Applied rewrites 99.0%

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

Alternative 5: 49.1% accurate, 2.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 38.9

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

5. Applied rewrites 38.9%

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

7. Applied rewrites 38.9%

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

9. Applied rewrites 49.2%

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

Alternative 6: 49.2% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. 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 38.9

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

5. Applied rewrites 38.9%

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

7. Applied rewrites 48.8%

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

9. Applied rewrites 48.9%

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

Alternative 7: 37.7% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 38.9

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

5. Applied rewrites 38.9%

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

7. Applied rewrites 48.8%

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

9. Applied rewrites 39.0%

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

11. Applied rewrites 39.1%

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

Alternative 8: 5.1% accurate, 6.9× speedup

Math

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

FPCore

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

C

double code(double k, double n) {
	return sqrt((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((2.0d0 / k))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.0%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 38.9

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

5. Applied rewrites 38.9%

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

7. Applied rewrites 4.9%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{k}}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(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))))