Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.5%

Time: 9.3s

Alternatives: 8

Speedup: 1.0×

• Could not converge on a ground truth

  1. k = 6.553465669540798e+287
  2. n = -8.141007027565467e+145

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, 0.7× speedup

Math

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

FPCore

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

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

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

   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. lower-*.f64 99.5

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.5

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.5

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

   10. lift-/.f64 N/A

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

   11. inv-pow N/A

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

   12. lift-sqrt.f64 N/A

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

   13. pow1/2 N/A

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

   14. pow-pow N/A

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

   15. lower-pow.f64 N/A

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

   16. metadata-eval 99.6

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

4. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. *-lft-identity N/A

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

   5. lower-/.f64 99.6

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 99.6

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 99.6

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

4. Applied rewrites 99.6%

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

Alternative 3: 99.4% accurate, 1.1× speedup

Math

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

FPCore

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

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-pow.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. lift-*.f64 N/A

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

   3. unpow-prod-down N/A

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

   4. lower-*.f64 N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-pow.f64 76.7

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

4. Applied rewrites 76.7%

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

5. Taylor expanded in k around inf

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

7. Applied rewrites 99.5%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot n\right) \cdot 2\\ \mathbf{if}\;k \leq 1.35 \cdot 10^{-29}:\\ \;\;\;\;\frac{\sqrt{t\_0}}{\sqrt{k}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{t\_0}^{\left(1 - k\right)}}{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (let* ((t_0 (* (* (PI) n) 2.0)))
   (if (<= k 1.35e-29)
     (/ (sqrt t_0) (sqrt k))
     (sqrt (/ (pow t_0 (- 1.0 k)) k)))))

Derivation

1. Split input into 2 regimes

2. if k < 1.35000000000000011e-29

   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

   5. Applied rewrites 70.7%

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

   7. Applied rewrites 99.5%

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

   if 1.35000000000000011e-29 < 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

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in k around inf

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

   8. Applied rewrites 98.9%

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

Alternative 5: 49.6% 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

5. Applied rewrites 39.5%

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

7. Applied rewrites 53.5%

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

Alternative 6: 49.6% accurate, 3.6× speedup

Math

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

FPCore

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

5. Applied rewrites 39.5%

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

7. Applied rewrites 39.6%

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

9. Applied rewrites 39.7%

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

11. Applied rewrites 53.2%

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

Alternative 7: 36.9% accurate, 4.8× speedup

Math

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

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

5. Applied rewrites 39.5%

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

7. Applied rewrites 39.6%

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

9. Applied rewrites 39.7%

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

Alternative 8: 36.9% accurate, 4.8× speedup

Math

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

FPCore

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

5. Applied rewrites 39.5%

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

7. Applied rewrites 39.6%

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

9. Applied rewrites 39.7%

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

11. Applied rewrites 39.6%

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

Reproduce

herbie shell --seed 2025019 
(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))))