Migdal et al, Equation (51)

Percentage Accurate: 99.5% → 99.6%

Time: 10.4s

Alternatives: 9

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 1.5756395813402617e+265
  2. n = -5.857935109445434e+140

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: 99.6% accurate, 0.9× speedup

Math

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

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 99.4%

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

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

6. Applied rewrites 99.6%

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

7. Final simplification 99.6%

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

Alternative 2: 60.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\sqrt{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{k \cdot k}} \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<=
      (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))
      2e-62)
   (sqrt (* (sqrt (/ (* (PI) (PI)) (* k k))) (* 2.0 n)))
   (/ (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)))) < 2.0000000000000001e-62

   1. Initial program 99.6%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 17.7

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

   5. Applied rewrites 17.7%

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

   7. Applied rewrites 17.6%

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

   9. Applied rewrites 17.7%

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

   11. Applied rewrites 59.1%

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

   if 2.0000000000000001e-62 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 69.5%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 2 \cdot 10^{-62}:\\ \;\;\;\;\sqrt{\sqrt{\frac{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}{k \cdot k}} \cdot \left(2 \cdot n\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.8% accurate, 0.5× speedup

Math

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

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 (* (/ (* n (+ k k)) (* k k)) (PI)))
   (/ (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 3.1

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 3.1%

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

   9. Applied rewrites 13.8%

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

   11. Applied rewrites 13.8%

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

   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 99.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 70.8%

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

4. Final simplification 56.6%

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

Alternative 4: 51.8% accurate, 0.5× speedup

Math

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

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 (* (/ (* n (+ k k)) (* k k)) (PI)))
   (* (sqrt (/ (PI) k)) (sqrt (* 2.0 n)))))

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 3.1

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 3.1%

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

   9. Applied rewrites 13.8%

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

   11. Applied rewrites 13.8%

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

   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 99.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 58.2%

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

   9. Applied rewrites 70.8%

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

4. Final simplification 56.5%

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

Alternative 5: 51.8% accurate, 0.5× speedup

Math

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

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 (* (/ (* n (+ k k)) (* k k)) (PI)))
   (* (sqrt (* 2.0 (/ (PI) k))) (sqrt n))))

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 3.1

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 3.1%

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

   9. Applied rewrites 13.8%

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

   11. Applied rewrites 13.8%

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

   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 99.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 58.2%

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

   9. Applied rewrites 70.8%

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

4. Final simplification 56.5%

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

Alternative 6: 40.5% accurate, 0.5× speedup

Math

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

FPCore

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

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 3.1

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

   5. Applied rewrites 3.1%

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

   7. Applied rewrites 3.1%

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

   9. Applied rewrites 13.8%

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

   11. Applied rewrites 13.8%

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

   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 99.2%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 58.1

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

   5. Applied rewrites 58.1%

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

   7. Applied rewrites 58.2%

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

4. Final simplification 47.1%

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

Alternative 7: 99.5% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

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 inf

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

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

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

   2. +-commutative N/A

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

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

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

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

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-in N/A

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

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

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

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

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.6%

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

8. Final simplification 99.6%

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

Alternative 8: 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 99.4%

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

3. Taylor expanded in k around inf

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

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

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

   2. +-commutative N/A

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

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

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

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

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

   5. mul-1-neg N/A

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

   6. distribute-rgt-in N/A

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

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

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

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

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

5. Applied rewrites 99.4%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.4

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

7. Applied rewrites 99.4%

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

Alternative 9: 37.5% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

   \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
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 44.4

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

5. Applied rewrites 44.4%

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

7. Applied rewrites 44.5%

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

Reproduce

herbie shell --seed 2024347 
(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))))