normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 10.0s

Alternatives: 7

Speedup: 1.4×

Specification

\[\left(0 \leq u1 \land u1 \leq 1\right) \land \left(0 \leq u2 \land u2 \leq 1\right)\]

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 (PI)) u2)))
  0.5))

Initial Program: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 (PI)) u2)))
  0.5))

Alternative 1: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \left(0.16666666666666666 \cdot \sqrt{2}\right), \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* (* (PI) 2.0) u2)) (* 0.16666666666666666 (sqrt 2.0)))
  (sqrt (- (log u1)))
  0.5))

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)}^{3} + {\frac{1}{2}}^{3}}{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \frac{1}{2}\right)}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \frac{1}{2}\right)}{{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)}^{3} + {\frac{1}{2}}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \frac{1}{2}\right)}{{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)}^{3} + {\frac{1}{2}}^{3}}}} \]

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}, 0.5\right)}}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt{2} \cdot 0.16666666666666666\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \sqrt{-\log u1}, 0.5\right)} \]

7. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(\cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right) \cdot \left(0.16666666666666666 \cdot \sqrt{2}\right), \sqrt{-\log u1}, 0.5\right) \]
8. Add Preprocessing

Alternative 2: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right), 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (sqrt (* -2.0 (log u1))) (cos (* (* (PI) 2.0) u2)))
  0.16666666666666666
  0.5))

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} + \frac{1}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left({\left(-2 \cdot \log u1\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)} + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left({\left(-2 \cdot \log u1\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \frac{1}{6}} + \frac{1}{2} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(-2 \cdot \log u1\right)}^{\frac{1}{2}} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right), \frac{1}{6}, \frac{1}{2}\right)} \]

4. Applied rewrites 99.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) \cdot \sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]

5. Final simplification 99.4%

   \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1} \cdot \cos \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right), 0.16666666666666666, 0.5\right) \]
6. Add Preprocessing

Alternative 3: 99.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -0.3333333333333333 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 0.16666666666666666\right) \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (*
   (fma (* u2 u2) (* -0.3333333333333333 (* (PI) (PI))) 0.16666666666666666)
   (sqrt 2.0))
  (sqrt (- (log u1)))
  0.5))

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. flip3-+ N/A

      \[\leadsto \color{blue}{\frac{{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)}^{3} + {\frac{1}{2}}^{3}}{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \frac{1}{2}\right)}} \]

   3. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \frac{1}{2}\right)}{{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)}^{3} + {\frac{1}{2}}^{3}}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) + \left(\frac{1}{2} \cdot \frac{1}{2} - \left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right) \cdot \frac{1}{2}\right)}{{\left(\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{\frac{1}{2}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\right)}^{3} + {\frac{1}{2}}^{3}}}} \]

4. Applied rewrites 99.3%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left(\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.16666666666666666 \cdot \sqrt{\log u1 \cdot -2}, 0.5\right)}}} \]

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\sqrt{2} \cdot 0.16666666666666666\right) \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \sqrt{-\log u1}, 0.5\right)} \]

7. Taylor expanded in u2 around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{3} \cdot \left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sqrt{2}\right)\right) + \frac{1}{6} \cdot \sqrt{2}}, \sqrt{-\log u1}, \frac{1}{2}\right) \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left({u2}^{2} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sqrt{2}\right)\right) \cdot \frac{-1}{3}} + \frac{1}{6} \cdot \sqrt{2}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   2. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot \left(\left({\mathsf{PI}\left(\right)}^{2} \cdot \sqrt{2}\right) \cdot \frac{-1}{3}\right)} + \frac{1}{6} \cdot \sqrt{2}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{fma}\left({u2}^{2} \cdot \color{blue}{\left(\frac{-1}{3} \cdot \left({\mathsf{PI}\left(\right)}^{2} \cdot \sqrt{2}\right)\right)} + \frac{1}{6} \cdot \sqrt{2}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left({u2}^{2} \cdot \color{blue}{\left(\left(\frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) \cdot \sqrt{2}\right)} + \frac{1}{6} \cdot \sqrt{2}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   5. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left({u2}^{2} \cdot \left(\frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right) \cdot \sqrt{2}} + \frac{1}{6} \cdot \sqrt{2}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   6. distribute-rgt-out N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot \left({u2}^{2} \cdot \left(\frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{6}\right)}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot \left({u2}^{2} \cdot \left(\frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{6}\right)}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   8. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{2}} \cdot \left({u2}^{2} \cdot \left(\frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \color{blue}{\mathsf{fma}\left({u2}^{2}, \frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right)}, \sqrt{-\log u1}, \frac{1}{2}\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(\color{blue}{u2 \cdot u2}, \frac{-1}{3} \cdot {\mathsf{PI}\left(\right)}^{2}, \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   12. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{3}}, \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{{\mathsf{PI}\left(\right)}^{2} \cdot \frac{-1}{3}}, \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{-1}{3}, \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(u2 \cdot u2, \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{-1}{3}, \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   16. lower-PI.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{-1}{3}, \frac{1}{6}\right), \sqrt{-\log u1}, \frac{1}{2}\right) \]

   17. lower-PI.f64 99.4

       \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right) \cdot -0.3333333333333333, 0.16666666666666666\right), \sqrt{-\log u1}, 0.5\right) \]

9. Applied rewrites 99.4%

   \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{2} \cdot \mathsf{fma}\left(u2 \cdot u2, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.3333333333333333, 0.16666666666666666\right)}, \sqrt{-\log u1}, 0.5\right) \]

10. Final simplification 99.4%

    \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(u2 \cdot u2, -0.3333333333333333 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right), 0.16666666666666666\right) \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \]
11. Add Preprocessing

Alternative 4: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \left(\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (*
   (fma (* (* u2 u2) -2.0) (* (PI) (PI)) 1.0)
   (* (sqrt (* -2.0 (log u1))) 0.16666666666666666))
  0.5))

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u1 around inf

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{\log \left(\frac{1}{u1}\right)} \cdot \sqrt{2}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\color{blue}{\sqrt{2}} \cdot \sqrt{\log \left(\frac{1}{u1}\right)}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lower-sqrt.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \color{blue}{\sqrt{\log \left(\frac{1}{u1}\right)}}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. log-rec N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{\mathsf{neg}\left(\log u1\right)}}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lower-neg.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{\color{blue}{-\log u1}}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. lower-log.f64 99.5

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\color{blue}{\log u1}}\right)\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

5. Applied rewrites 99.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{\left(\sqrt{2} \cdot \sqrt{-\log u1}\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

6. Taylor expanded in u2 around 0

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \color{blue}{\left(1 + -2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right)\right)} + \frac{1}{2} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \color{blue}{\left(-2 \cdot \left({u2}^{2} \cdot {\mathsf{PI}\left(\right)}^{2}\right) + 1\right)} + \frac{1}{2} \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \left(\color{blue}{\left(-2 \cdot {u2}^{2}\right) \cdot {\mathsf{PI}\left(\right)}^{2}} + 1\right) + \frac{1}{2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(-2 \cdot {u2}^{2}, {\mathsf{PI}\left(\right)}^{2}, 1\right)} + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot -2}, {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   5. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{{u2}^{2} \cdot -2}, {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   7. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\left(u2 \cdot u2\right)} \cdot -2, {\mathsf{PI}\left(\right)}^{2}, 1\right) + \frac{1}{2} \]

   8. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) + \frac{1}{2} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)}, 1\right) + \frac{1}{2} \]

   10. lower-PI.f64 N/A

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right), 1\right) + \frac{1}{2} \]

   11. lower-PI.f64 99.3

       \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}, 1\right) + 0.5 \]

8. Applied rewrites 99.3%

   \[\leadsto \left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right)} + 0.5 \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) + \frac{1}{2} \]

   2. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) + \frac{1}{2} \]

   3. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) + \frac{1}{2} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-\log u1}\right) \cdot \frac{1}{6}\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) + \frac{1}{2} \]

   5. lower-*.f64 99.3

      \[\leadsto \color{blue}{\left(\left(\sqrt{2} \cdot \sqrt{-\log u1}\right) \cdot 0.16666666666666666\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) + 0.5 \]

10. Applied rewrites 99.2%

    \[\leadsto \color{blue}{\left(\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666\right)} \cdot \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) + 0.5 \]

11. Final simplification 99.2%

    \[\leadsto \mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -2, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), 1\right) \cdot \left(\sqrt{-2 \cdot \log u1} \cdot 0.16666666666666666\right) + 0.5 \]
12. Add Preprocessing

Alternative 5: 98.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (* 0.16666666666666666 (sqrt 2.0)) (sqrt (- (log u1))) 0.5))

C

double code(double u1, double u2) {
	return fma((0.16666666666666666 * sqrt(2.0)), sqrt(-log(u1)), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(0.16666666666666666 * sqrt(2.0)), sqrt(Float64(-log(u1))), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(0.16666666666666666 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{\log u1}\right)} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right) \cdot \sqrt{\log u1}} + \frac{1}{2} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{-2}, \sqrt{\log u1}, \frac{1}{2}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot \sqrt{-2}}, \sqrt{\log u1}, \frac{1}{2}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}, \sqrt{\log u1}, \frac{1}{2}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{-2}, \color{blue}{\sqrt{\log u1}}, \frac{1}{2}\right) \]

   8. lower-log.f64 0.0

      \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-2}, \sqrt{\color{blue}{\log u1}}, 0.5\right) \]

5. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-2}, \sqrt{\log u1}, 0.5\right)} \]

7. Applied rewrites 99.1%

   \[\leadsto \mathsf{fma}\left(\sqrt{2} \cdot 0.16666666666666666, \color{blue}{\sqrt{-\log u1}}, 0.5\right) \]

8. Final simplification 99.1%

   \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{2}, \sqrt{-\log u1}, 0.5\right) \]
9. Add Preprocessing

Alternative 6: 98.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-\log u1} \cdot 0.16666666666666666, \sqrt{2}, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (* (sqrt (- (log u1))) 0.16666666666666666) (sqrt 2.0) 0.5))

C

double code(double u1, double u2) {
	return fma((sqrt(-log(u1)) * 0.16666666666666666), sqrt(2.0), 0.5);
}

Julia

function code(u1, u2)
	return fma(Float64(sqrt(Float64(-log(u1))) * 0.16666666666666666), sqrt(2.0), 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[(N[Sqrt[(-N[Log[u1], $MachinePrecision])], $MachinePrecision] * 0.16666666666666666), $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision] + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{\log u1}\right)} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right) \cdot \sqrt{\log u1}} + \frac{1}{2} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{-2}, \sqrt{\log u1}, \frac{1}{2}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot \sqrt{-2}}, \sqrt{\log u1}, \frac{1}{2}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}, \sqrt{\log u1}, \frac{1}{2}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{-2}, \color{blue}{\sqrt{\log u1}}, \frac{1}{2}\right) \]

   8. lower-log.f64 0.0

      \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-2}, \sqrt{\color{blue}{\log u1}}, 0.5\right) \]

5. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-2}, \sqrt{\log u1}, 0.5\right)} \]
6. Step-by-step derivation

7. Applied rewrites 99.0%

   \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-\log u1}, \color{blue}{\sqrt{2}}, 0.5\right) \]

8. Final simplification 99.0%

   \[\leadsto \mathsf{fma}\left(\sqrt{-\log u1} \cdot 0.16666666666666666, \sqrt{2}, 0.5\right) \]
9. Add Preprocessing

Alternative 7: 98.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666, 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* -2.0 (log u1))) 0.16666666666666666 0.5))

C

double code(double u1, double u2) {
	return fma(sqrt((-2.0 * log(u1))), 0.16666666666666666, 0.5);
}

Julia

function code(u1, u2)
	return fma(sqrt(Float64(-2.0 * log(u1))), 0.16666666666666666, 0.5)
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(-2.0 * N[Log[u1], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.16666666666666666 + 0.5), $MachinePrecision]

Derivation

1. Initial program 99.4%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
2. Add Preprocessing

3. Taylor expanded in u2 around 0

   \[\leadsto \color{blue}{\frac{1}{2} + \frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{6} \cdot \left(\sqrt{\log u1} \cdot \sqrt{-2}\right) + \frac{1}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{6} \cdot \color{blue}{\left(\sqrt{-2} \cdot \sqrt{\log u1}\right)} + \frac{1}{2} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot \sqrt{-2}\right) \cdot \sqrt{\log u1}} + \frac{1}{2} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{-2}, \sqrt{\log u1}, \frac{1}{2}\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{6} \cdot \sqrt{-2}}, \sqrt{\log u1}, \frac{1}{2}\right) \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot \color{blue}{\sqrt{-2}}, \sqrt{\log u1}, \frac{1}{2}\right) \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{6} \cdot \sqrt{-2}, \color{blue}{\sqrt{\log u1}}, \frac{1}{2}\right) \]

   8. lower-log.f64 0.0

      \[\leadsto \mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-2}, \sqrt{\color{blue}{\log u1}}, 0.5\right) \]

5. Applied rewrites 0.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.16666666666666666 \cdot \sqrt{-2}, \sqrt{\log u1}, 0.5\right)} \]
6. Step-by-step derivation

7. Applied rewrites 98.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{\log u1 \cdot -2}, 0.16666666666666666, 0.5\right)} \]

8. Final simplification 98.9%

   \[\leadsto \mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666, 0.5\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024248 
(FPCore (u1 u2)
  :name "normal distribution"
  :precision binary64
  :pre (and (and (<= 0.0 u1) (<= u1 1.0)) (and (<= 0.0 u2) (<= u2 1.0)))
  (+ (* (* (/ 1.0 6.0) (pow (* -2.0 (log u1)) 0.5)) (cos (* (* 2.0 (PI)) u2))) 0.5))