Migdal et al, Equation (51)

Percentage Accurate: 99.4% → 99.4%

Time: 12.4s

Alternatives: 12

Speedup: 0.9×

• Could not converge on a ground truth

  1. k = 1.6885034394725313e+25
  2. n = -2.8119324055132327e+281

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, 0.6× speedup

Math

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

FPCore

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

Derivation

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. 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. inv-pow N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval 99.1

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

4. Applied rewrites 99.1%

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

6. Applied rewrites 99.3%

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

Alternative 2: 42.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)}\\ \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(n, k, k \cdot n\right)}{k \cdot k}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n}\\ \end{array} \end{array} \]

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)
     (sqrt (/ (fma n k (* k n)) (* k k)))
     (if (<= t_0 5e+153)
       (sqrt (* (/ (* n (PI)) k) 2.0))
       (* (sqrt (/ 2.0 k)) (sqrt n))))))

Derivation

1. Split input into 3 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.2%

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

   11. Applied rewrites 14.0%

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

   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)))) < 5.00000000000000018e153

   1. Initial program 98.6%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 94.3

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

   5. Applied rewrites 94.3%

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

   7. Applied rewrites 94.6%

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

   if 5.00000000000000018e153 < (*.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 98.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 3.3

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

   5. Applied rewrites 3.3%

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

   7. Applied rewrites 3.3%

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

   9. Applied rewrites 3.3%

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

   11. Applied rewrites 7.9%

       \[\leadsto \sqrt{\frac{2}{k}} \cdot \color{blue}{\sqrt{n}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 0:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(n, k, k \cdot n\right)}{k \cdot k}}\\ \mathbf{elif}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.1% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 3.2

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

   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%

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

   11. Applied rewrites 14.0%

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

   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 98.7%

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

   3. Taylor expanded in k around 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 50.5

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 65.6%

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

4. Final simplification 52.9%

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

Alternative 4: 53.2% accurate, 0.5× speedup

Math

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

FPCore

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 100.0%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 3.2

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

   5. Applied rewrites 3.2%

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

   7. Applied rewrites 3.2%

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

   9. Applied rewrites 3.2%

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

   11. Applied rewrites 14.0%

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

   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 98.7%

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

   3. Taylor expanded in k around 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 50.5

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

   5. Applied rewrites 50.5%

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

   7. Applied rewrites 50.6%

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

   9. Applied rewrites 65.6%

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

4. Final simplification 52.9%

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

Alternative 5: 40.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<=
      (* (pow (sqrt k) -1.0) (pow (* (* 2.0 (PI)) n) (/ (- 1.0 k) 2.0)))
      5e+153)
   (sqrt (* (/ (* n (PI)) k) 2.0))
   (* (sqrt (/ 2.0 k)) (sqrt n))))

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 k)) (pow.f64 (*.f64 (*.f64 #s(literal 2 binary64) (PI.f64)) n) (/.f64 (-.f64 #s(literal 1 binary64) k) #s(literal 2 binary64)))) < 5.00000000000000018e153

   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 59.1

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

   5. Applied rewrites 59.1%

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

   7. Applied rewrites 59.3%

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

   if 5.00000000000000018e153 < (*.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 98.8%

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

   3. Taylor expanded in k around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-PI.f64 3.3

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

   5. Applied rewrites 3.3%

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

   7. Applied rewrites 3.3%

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

   9. Applied rewrites 3.3%

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

   11. Applied rewrites 7.9%

       \[\leadsto \sqrt{\frac{2}{k}} \cdot \color{blue}{\sqrt{n}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{\left(\sqrt{k}\right)}^{-1} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\sqrt{\frac{n \cdot \mathsf{PI}\left(\right)}{k} \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{2}{k}} \cdot \sqrt{n}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 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 99.0%

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

3. Taylor expanded in k around inf

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

5. Applied rewrites 99.1%

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

6. Final simplification 99.1%

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

Alternative 7: 99.4% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

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. 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. inv-pow N/A

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

   3. lift-sqrt.f64 N/A

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

   4. pow1/2 N/A

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

   5. pow-pow N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval 99.1

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

4. Applied rewrites 99.1%

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

6. Applied rewrites 99.3%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. metadata-eval N/A

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

   4. sqrt-pow2 N/A

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

   5. lift-sqrt.f64 N/A

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

   6. inv-pow N/A

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

   7. lift-/.f64 N/A

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

   8. lift-sqrt.f64 N/A

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

   9. pow1/2 N/A

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

   10. lift-pow.f64 N/A

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

   11. pow-sub N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. *-commutative N/A

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

   16. associate-*l* N/A

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

   17. lift-PI.f64 N/A

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

   18. metadata-eval N/A

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

   19. lift-/.f64 N/A

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

   20. div-sub N/A

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

8. Applied rewrites 99.3%

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

Alternative 8: 95.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;k \leq 7.5 \cdot 10^{+15}:\\ \;\;\;\;\sqrt{\frac{\mathsf{PI}\left(\right)}{k} \cdot 2} \cdot \sqrt{n}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(-2 \cdot n\right)}^{\left(0.5 - 0.25 \cdot k\right)}}{\sqrt{k}}\\ \end{array} \end{array} \]

FPCore

(FPCore (k n)
 :precision binary64
 (if (<= k 7.5e+15)
   (* (sqrt (* (/ (PI) k) 2.0)) (sqrt n))
   (/ (pow (* -2.0 n) (- 0.5 (* 0.25 k))) (sqrt k))))

Derivation

1. Split input into 2 regimes

2. if k < 7.5e15

   1. Initial program 98.2%

      \[\frac{1}{\sqrt{k}} \cdot {\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot n\right)}^{\left(\frac{1 - k}{2}\right)} \]
   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 70.8

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

   5. Applied rewrites 70.8%

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

   7. Applied rewrites 71.0%

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

   9. Applied rewrites 92.2%

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

   if 7.5e15 < 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. 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. inv-pow N/A

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

      3. lift-sqrt.f64 N/A

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

      4. pow1/2 N/A

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

      5. pow-pow N/A

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

      6. lower-pow.f64 N/A

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

      7. metadata-eval 100.0

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

   4. Applied rewrites 100.0%

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

   6. Applied rewrites 100.0%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. metadata-eval N/A

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

      4. sqrt-pow2 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. inv-pow N/A

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

      7. lift-/.f64 N/A

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

      8. lift-sqrt.f64 N/A

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

      9. pow1/2 N/A

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

      10. lift-pow.f64 N/A

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

      11. pow-sub N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. associate-*l* N/A

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

      17. lift-PI.f64 N/A

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

      18. metadata-eval N/A

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

      19. lift-/.f64 N/A

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

      20. div-sub N/A

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

   8. Applied rewrites 100.0%

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

   10. Applied rewrites 100.0%

       \[\leadsto \color{blue}{\frac{{\left(-2 \cdot n\right)}^{\left(0.5 - 0.25 \cdot k\right)}}{\sqrt{k}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

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 38.8

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

5. Applied rewrites 38.8%

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

7. Applied rewrites 39.0%

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

9. Applied rewrites 39.0%

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

Alternative 10: 38.3% accurate, 4.8× speedup

Math

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

FPCore

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

Derivation

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 38.8

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

5. Applied rewrites 38.8%

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

7. Applied rewrites 39.0%

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

9. Applied rewrites 38.9%

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

Alternative 11: 9.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{n}{k} \cdot 2} \end{array} \]

FPCore

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

C

double code(double k, double n) {
	return sqrt(((n / k) * 2.0));
}

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 = sqrt(((n / k) * 2.0d0))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt(((n / k) * 2.0));
}

Python

def code(k, n):
	return math.sqrt(((n / k) * 2.0))

Julia

function code(k, n)
	return sqrt(Float64(Float64(n / k) * 2.0))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(((n / k) * 2.0));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(n / k), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

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 38.8

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

5. Applied rewrites 38.8%

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

7. Applied rewrites 39.0%

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

9. Applied rewrites 38.9%

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

11. Applied rewrites 9.5%

    \[\leadsto \color{blue}{\sqrt{\frac{n}{k} \cdot 2}} \]
12. Add Preprocessing

Alternative 12: 9.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\frac{2}{k} \cdot n} \end{array} \]

FPCore

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

C

double code(double k, double n) {
	return sqrt(((2.0 / k) * n));
}

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 = sqrt(((2.0d0 / k) * n))
end function

Java

public static double code(double k, double n) {
	return Math.sqrt(((2.0 / k) * n));
}

Python

def code(k, n):
	return math.sqrt(((2.0 / k) * n))

Julia

function code(k, n)
	return sqrt(Float64(Float64(2.0 / k) * n))
end

MATLAB

function tmp = code(k, n)
	tmp = sqrt(((2.0 / k) * n));
end

Wolfram

code[k_, n_] := N[Sqrt[N[(N[(2.0 / k), $MachinePrecision] * n), $MachinePrecision]], $MachinePrecision]

Derivation

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 38.8

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

5. Applied rewrites 38.8%

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

7. Applied rewrites 39.0%

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

9. Applied rewrites 38.9%

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

11. Applied rewrites 9.5%

    \[\leadsto \sqrt{\frac{2}{k} \cdot n} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(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))))