Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 10.8s

Alternatives: 11

Speedup: 0.6×

• could not determine a ground truth

  1. k = 1.9188910670559148e+171
  2. n = -9.889343581980647e-29

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: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

   1. lift-*.f64 N/A

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

   2. *-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 99.4

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.7%

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

7. Final simplification 99.7%

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

Alternative 2: 97.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0}^{\left(-0.125 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0
         (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_0 0.0)
     0.0
     (if (<= t_0 5e+273)
       (* (sqrt (* 2.0 n)) (sqrt (/ (PI) k)))
       (/ (pow 0.0 (* -0.125 k)) (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)))) < 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

   4. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}}{\sqrt{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(0 \cdot n\right)}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      3. mul0-lft N/A

         \[\leadsto \frac{{\color{blue}{0}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      4. pow-base-0 100.0

         \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 4.99999999999999961e273

   1. Initial program 99.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 69.6

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

   5. Applied rewrites 69.6%

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

   7. Applied rewrites 69.7%

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

   9. Applied rewrites 97.1%

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

   if 4.99999999999999961e273 < (*.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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]

   5. Taylor expanded in k around inf

      \[\leadsto \frac{{\left(0 \cdot n\right)}^{\color{blue}{\left(\frac{-1}{8} \cdot k\right)}}}{\sqrt{k}} \]
   6. Step-by-step derivation

      1. lower-*.f64 100.0

         \[\leadsto \frac{{\left(0 \cdot n\right)}^{\color{blue}{\left(-0.125 \cdot k\right)}}}{\sqrt{k}} \]

   7. Applied rewrites 100.0%

      \[\leadsto \frac{{\left(0 \cdot n\right)}^{\color{blue}{\left(-0.125 \cdot k\right)}}}{\sqrt{k}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+273}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{0}^{\left(-0.125 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot n\\ t_1 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{t\_0}} \cdot \sqrt{\sqrt{k \cdot \left(t\_0 \cdot k\right)}}\right) \cdot \sqrt{2}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (PI) n))
        (t_1
         (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_1 0.0)
     0.0
     (if (<= t_1 4e+272)
       (* (sqrt (* 2.0 n)) (sqrt (/ (PI) k)))
       (/
        (* (* (sqrt (sqrt t_0)) (sqrt (sqrt (* k (* t_0 k))))) (sqrt 2.0))
        k)))))

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}}{\sqrt{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(0 \cdot n\right)}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      3. mul0-lft N/A

         \[\leadsto \frac{{\color{blue}{0}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      4. pow-base-0 100.0

         \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 4.0000000000000003e272

   1. Initial program 99.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 97.0%

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

   if 4.0000000000000003e272 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64))))

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 3.7%

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

   10. Applied rewrites 31.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 4 \cdot 10^{+272}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{\sqrt{\mathsf{PI}\left(\right) \cdot n}} \cdot \sqrt{\sqrt{k \cdot \left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot k\right)}}\right) \cdot \sqrt{2}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\mathsf{PI}\left(\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot k\right) \cdot 2}}{k}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0
         (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))))
   (if (<= t_0 0.0)
     0.0
     (if (<= t_0 4e+136)
       (sqrt (* (* 2.0 n) (/ (PI) k)))
       (/ (sqrt (* (* (* (PI) n) k) 2.0)) k)))))

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}}{\sqrt{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(0 \cdot n\right)}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      3. mul0-lft N/A

         \[\leadsto \frac{{\color{blue}{0}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      4. pow-base-0 100.0

         \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

   if 0.0 < (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 4.00000000000000023e136

   1. Initial program 98.9%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 95.8

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

   5. Applied rewrites 95.8%

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

   7. Applied rewrites 96.0%

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

   9. Applied rewrites 96.1%

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

   if 4.00000000000000023e136 < (*.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 99.6%

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

   3. Taylor expanded in k around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in k around 0

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

   8. Applied rewrites 41.4%

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

   10. Applied rewrites 41.5%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;0\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 4 \cdot 10^{+136}:\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot \frac{\mathsf{PI}\left(\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{PI}\left(\right) \cdot n\right) \cdot k\right) \cdot 2}}{k}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.8% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}}{\sqrt{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(0 \cdot n\right)}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      3. mul0-lft N/A

         \[\leadsto \frac{{\color{blue}{0}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      4. pow-base-0 100.0

         \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

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

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 47.9%

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

   9. Applied rewrites 66.4%

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

4. Final simplification 74.8%

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

Alternative 6: 73.8% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}}{\sqrt{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(0 \cdot n\right)}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      3. mul0-lft N/A

         \[\leadsto \frac{{\color{blue}{0}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      4. pow-base-0 100.0

         \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

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

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 47.9%

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

   9. Applied rewrites 66.4%

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

4. Final simplification 74.8%

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

Alternative 7: 62.8% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   4. Applied rewrites 1.6%

      \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\color{blue}{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}}{\sqrt{k}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{{\color{blue}{\left(0 \cdot n\right)}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      3. mul0-lft N/A

         \[\leadsto \frac{{\color{blue}{0}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

      4. pow-base-0 100.0

         \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   6. Applied rewrites 100.0%

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

   7. Taylor expanded in k around 0

      \[\leadsto \color{blue}{0} \]
   8. Step-by-step derivation

   9. Applied rewrites 100.0%

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

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

   1. Initial program 99.3%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 47.8

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

   5. Applied rewrites 47.8%

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

   7. Applied rewrites 47.9%

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

   9. Applied rewrites 47.9%

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

4. Final simplification 60.9%

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

Alternative 8: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

5. Applied rewrites 99.5%

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

6. Taylor expanded in k around inf

   \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot e^{\frac{1}{2} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(1 - k\right)\right)}} \]
7. 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. *-commutative N/A

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

   4. exp-prod N/A

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

   5. unpow1/2 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. exp-to-pow N/A

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

   8. lower-pow.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower-PI.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower-sqrt.f64 N/A

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

   15. lower-/.f64 99.5

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

8. Applied rewrites 99.5%

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

9. Final simplification 99.5%

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

Alternative 9: 99.4% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

3. Taylor expanded in k around inf

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

5. Applied rewrites 99.5%

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

6. Final simplification 99.5%

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

Alternative 10: 99.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 1.25 \cdot 10^{-21}:\\ \;\;\;\;\sqrt{2 \cdot n} \cdot \sqrt{\frac{\mathsf{PI}\left(\right)}{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{\left(\left(n \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 1.25e-21)
   (* (sqrt (* 2.0 n)) (sqrt (/ (PI) k)))
   (sqrt (/ (pow (* (* n (PI)) 2.0) (- 1.0 k)) k))))

Derivation

1. Split input into 2 regimes

2. if k < 1.24999999999999993e-21

   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 70.0

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

   5. Applied rewrites 70.0%

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

   7. Applied rewrites 70.2%

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

   9. Applied rewrites 99.5%

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

   if 1.24999999999999993e-21 < k

   1. Initial program 99.7%

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

   3. Taylor expanded in k around inf

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in k around inf

      \[\leadsto \color{blue}{\sqrt{\frac{1}{k}} \cdot e^{\frac{1}{2} \cdot \left(\log \left(2 \cdot \left(n \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \left(1 - k\right)\right)}} \]
   7. 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. *-commutative N/A

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

      4. exp-prod N/A

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

      5. unpow1/2 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. exp-to-pow N/A

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

      8. lower-pow.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-PI.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-sqrt.f64 N/A

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

      15. lower-/.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in k around inf

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

   11. Applied rewrites 99.7%

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

Alternative 11: 26.6% accurate, 152.0× speedup

Math

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

FPCore

(FPCore (k n) :precision binary64 0.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(k, n):
	return 0.0

Julia

function code(k, n)
	return 0.0
end

MATLAB

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

Wolfram

code[k_, n_] := 0.0

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 26.0%

   \[\leadsto \color{blue}{\frac{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}}} \]
5. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \frac{\color{blue}{{\left(0 \cdot n\right)}^{\left(\frac{1 - k}{8}\right)}}}{\sqrt{k}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{{\color{blue}{\left(0 \cdot n\right)}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

   3. mul0-lft N/A

      \[\leadsto \frac{{\color{blue}{0}}^{\left(\frac{1 - k}{8}\right)}}{\sqrt{k}} \]

   4. pow-base-0 26.7

      \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

6. Applied rewrites 26.7%

   \[\leadsto \frac{\color{blue}{0}}{\sqrt{k}} \]

7. Taylor expanded in k around 0

   \[\leadsto \color{blue}{0} \]
8. Step-by-step derivation

9. Applied rewrites 26.7%

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

Reproduce

herbie shell --seed 2024363 
(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))))