Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.5%

Time: 10.0s

Alternatives: 10

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 4.9445636975602833e+278
  2. n = -5.893479535827748e+237

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: 99.5% 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

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

FPCore

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

Derivation

1. Initial program 99.3%

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

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-PI.f64 N/A

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

   6. add-sqr-sqrt N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lift-PI.f64 N/A

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

   12. lower-sqrt.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-sqrt.f64 99.3

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

4. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. *-commutative N/A

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 98.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{0}^{\left(\mathsf{fma}\left(k, -0.25, 0.5\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0
         (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_0 0.0)
     0.0
     (if (<= t_0 5e+298)
       (/ (sqrt (* (* (PI) 2.0) n)) (sqrt k))
       (pow 0.0 (fma k -0.25 0.5))))))

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%

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0} \]

   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)))) < 5.0000000000000003e298

   1. Initial program 98.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 94.6%

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

   if 5.0000000000000003e298 < (*.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 inf

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

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

   5. Applied rewrites 100.0%

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

   7. Applied rewrites 100.0%

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{{0}^{\left(\mathsf{fma}\left(k, -0.25, 0.5\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+298}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;{0}^{\left(\mathsf{fma}\left(k, -0.25, 0.5\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.8% accurate, 0.5× speedup

Math

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

FPCore

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

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0} \]

   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 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 68.7%

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

4. Final simplification 76.0%

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

Alternative 4: 73.8% accurate, 0.5× speedup

Math

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

FPCore

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

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0} \]

   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 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 51.8%

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

   9. Applied rewrites 68.2%

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

4. Final simplification 75.7%

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

Alternative 5: 62.4% accurate, 0.5× speedup

Math

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

FPCore

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

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0} \]

   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 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 52.1%

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

4. Final simplification 63.3%

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

Alternative 6: 62.5% accurate, 0.5× speedup

Math

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

FPCore

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

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0} \]

   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 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 52.1%

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

4. Final simplification 63.3%

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

Alternative 7: 62.4% accurate, 0.5× speedup

Math

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

FPCore

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

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

   9. Applied rewrites 100.0%

      \[\leadsto \color{blue}{0} \]

   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 99.1%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   7. Applied rewrites 52.1%

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

   9. Applied rewrites 51.6%

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

4. Final simplification 63.0%

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

Alternative 8: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in k around inf

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

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 9: 99.5% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in k around inf

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

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.4%

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

Alternative 10: 26.6% accurate, 152.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (k n) :precision binary64 0.0)

C

double code(double k, double n) {
	return 0.0;
}

Fortran

real(8) function code(k, n)
    real(8), intent (in) :: k
    real(8), intent (in) :: n
    code = 0.0d0
end function

Java

public static double code(double k, double n) {
	return 0.0;
}

Python

def code(k, n):
	return 0.0

Julia

function code(k, n)
	return 0.0
end

MATLAB

function tmp = code(k, n)
	tmp = 0.0;
end

Wolfram

code[k_, n_] := 0.0

Derivation

1. Initial program 99.3%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 40.6

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

5. Applied rewrites 40.6%

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

7. Applied rewrites 40.6%

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

9. Applied rewrites 25.3%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024340 
(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))))