Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 9.0s

Alternatives: 12

Speedup: 1.1×

• Could not converge on a ground truth

  1. k = 2.1493687872377688e+127
  2. n = -8.880360063529309e+37

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

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

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

   3. associate-*r* N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. exp-to-pow N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.7%

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

8. Final simplification 99.7%

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

Alternative 2: 85.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}}\\ \mathbf{if}\;t\_0 \leq 4 \cdot 10^{-100}:\\ \;\;\;\;\left(\sqrt{\frac{k}{\frac{k}{\mathsf{PI}\left(\right)} \cdot k}} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;t\_0 \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(0 \cdot n\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (pow (* (* (PI) 2.0) n) (/ (- 1.0 k) 2.0)) (/ 1.0 (sqrt k)))))
   (if (<= t_0 4e-100)
     (* (* (sqrt (/ k (* (/ k (PI)) k))) (sqrt n)) (sqrt 2.0))
     (if (<= t_0 1e+274)
       (/ (sqrt (* (* (PI) n) 2.0)) (sqrt k))
       (/ (pow (* 0.0 n) (fma k -0.5 0.5)) (sqrt k))))))

Derivation

1. Split input into 3 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 4.0000000000000001e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 7.1

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 7.1%

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

   9. Applied rewrites 48.9%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{\frac{\mathsf{fma}\left(0, \frac{k}{\mathsf{PI}\left(\right)}, k\right)}{k \cdot \frac{k}{\mathsf{PI}\left(\right)}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.9%

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

   if 4.0000000000000001e-100 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 9.99999999999999921e273

   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 78.0

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

   5. Applied rewrites 78.0%

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

   7. Applied rewrites 97.6%

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

   if 9.99999999999999921e273 < (*.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. 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. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 100.0

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

   4. Applied rewrites 100.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \leq 4 \cdot 10^{-100}:\\ \;\;\;\;\left(\sqrt{\frac{k}{\frac{k}{\mathsf{PI}\left(\right)} \cdot k}} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{elif}\;{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \leq 10^{+274}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(0 \cdot n\right)}^{\left(\mathsf{fma}\left(k, -0.5, 0.5\right)\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \leq 4 \cdot 10^{-100}:\\ \;\;\;\;\left(\sqrt{\frac{k}{\frac{k}{\mathsf{PI}\left(\right)} \cdot k}} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<=
      (* (pow (* (* (PI) 2.0) n) (/ (- 1.0 k) 2.0)) (/ 1.0 (sqrt k)))
      4e-100)
   (* (* (sqrt (/ k (* (/ k (PI)) k))) (sqrt n)) (sqrt 2.0))
   (/ (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)))) < 4.0000000000000001e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 7.1

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 7.1%

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

   9. Applied rewrites 48.9%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{\frac{\mathsf{fma}\left(0, \frac{k}{\mathsf{PI}\left(\right)}, k\right)}{k \cdot \frac{k}{\mathsf{PI}\left(\right)}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.9%

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

   if 4.0000000000000001e-100 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 67.5%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \leq 4 \cdot 10^{-100}:\\ \;\;\;\;\left(\sqrt{\frac{k}{\frac{k}{\mathsf{PI}\left(\right)} \cdot k}} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \leq 4 \cdot 10^{-100}:\\ \;\;\;\;\left(\sqrt{\frac{k}{k \cdot k} \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<=
      (* (pow (* (* (PI) 2.0) n) (/ (- 1.0 k) 2.0)) (/ 1.0 (sqrt k)))
      4e-100)
   (* (* (sqrt (* (/ k (* k k)) (PI))) (sqrt n)) (sqrt 2.0))
   (/ (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)))) < 4.0000000000000001e-100

   1. Initial program 99.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 7.1

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

   5. Applied rewrites 7.1%

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

   7. Applied rewrites 7.1%

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

   9. Applied rewrites 48.9%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{\frac{\mathsf{fma}\left(0, \frac{k}{\mathsf{PI}\left(\right)}, k\right)}{k \cdot \frac{k}{\mathsf{PI}\left(\right)}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 48.9%

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

   if 4.0000000000000001e-100 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 54.0

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

   5. Applied rewrites 54.0%

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

   7. Applied rewrites 67.5%

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

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \cdot \frac{1}{\sqrt{k}} \leq 4 \cdot 10^{-100}:\\ \;\;\;\;\left(\sqrt{\frac{k}{k \cdot k} \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.8% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (PI) n)))
   (if (<= (* (pow (* (* (PI) 2.0) n) (/ (- 1.0 k) 2.0)) (/ 1.0 (sqrt k))) 0.0)
     (sqrt (/ (fma t_0 k (* t_0 k)) (* k k)))
     (/ (sqrt (* t_0 2.0)) (sqrt k)))))

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.2%

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

   11. Applied rewrites 10.2%

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

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

   1. Initial program 99.4%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 54.4

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

   5. Applied rewrites 54.4%

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

   7. Applied rewrites 67.7%

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

4. Final simplification 54.9%

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

Alternative 6: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

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

   3. associate-*r* N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. exp-to-pow N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

   13. lower-PI.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. lower-/.f64 99.6

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

5. Applied rewrites 99.6%

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

7. Applied rewrites 99.7%

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

9. Applied rewrites 99.6%

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

10. Final simplification 99.6%

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

Alternative 7: 66.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{\frac{k \cdot k}{\left(\frac{k}{\mathsf{PI}\left(\right)} \cdot k\right) \cdot k}} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{k}{k \cdot k} \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 8.5e+104)
   (/ (sqrt (* (* (PI) n) 2.0)) (sqrt k))
   (if (<= k 1.32e+154)
     (* (* (sqrt (/ (* k k) (* (* (/ k (PI)) k) k))) (sqrt n)) (sqrt 2.0))
     (* (* (sqrt (* (/ k (* k k)) (PI))) (sqrt n)) (sqrt 2.0)))))

Derivation

1. Split input into 3 regimes

2. if k < 8.4999999999999999e104

   1. Initial program 99.3%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 57.9

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

   5. Applied rewrites 57.9%

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

   7. Applied rewrites 72.0%

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

   if 8.4999999999999999e104 < k < 1.31999999999999998e154

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 2.9

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

   5. Applied rewrites 2.9%

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

   7. Applied rewrites 2.9%

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

   9. Applied rewrites 2.9%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{\frac{\mathsf{fma}\left(0, \frac{k}{\mathsf{PI}\left(\right)}, k\right)}{k \cdot \frac{k}{\mathsf{PI}\left(\right)}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 64.2%

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

   if 1.31999999999999998e154 < k

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 2.6

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

   5. Applied rewrites 2.6%

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

   7. Applied rewrites 2.7%

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

   9. Applied rewrites 45.7%

      \[\leadsto \sqrt{2} \cdot \left(\sqrt{n} \cdot \sqrt{\frac{\mathsf{fma}\left(0, \frac{k}{\mathsf{PI}\left(\right)}, k\right)}{k \cdot \frac{k}{\mathsf{PI}\left(\right)}}}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 45.7%

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

4. Final simplification 65.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 8.5 \cdot 10^{+104}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2}}{\sqrt{k}}\\ \mathbf{elif}\;k \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\left(\sqrt{\frac{k \cdot k}{\left(\frac{k}{\mathsf{PI}\left(\right)} \cdot k\right) \cdot k}} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\frac{k}{k \cdot k} \cdot \mathsf{PI}\left(\right)} \cdot \sqrt{n}\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.9% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 43.0

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

5. Applied rewrites 43.0%

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

7. Applied rewrites 53.3%

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

8. Final simplification 53.3%

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

Alternative 9: 49.9% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 43.0

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

5. Applied rewrites 43.0%

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

7. Applied rewrites 43.0%

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

9. Applied rewrites 53.3%

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

10. Final simplification 53.3%

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

Alternative 10: 49.9% accurate, 3.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 43.0

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

5. Applied rewrites 43.0%

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

7. Applied rewrites 43.0%

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

9. Applied rewrites 53.2%

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

10. Final simplification 53.2%

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

Alternative 11: 38.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 43.0

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

5. Applied rewrites 43.0%

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

7. Applied rewrites 43.0%

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

9. Applied rewrites 43.1%

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

10. Final simplification 43.1%

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

Alternative 12: 38.1% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-PI.f64 43.0

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

5. Applied rewrites 43.0%

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

7. Applied rewrites 43.0%

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

9. Applied rewrites 43.1%

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

11. Applied rewrites 43.1%

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

12. Final simplification 43.1%

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

Reproduce

herbie shell --seed 2024312 
(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))))