normal distribution

Percentage Accurate: 99.4% → 99.7%

Time: 6.0s

Alternatives: 4

Speedup: 1.5×

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: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (* (sqrt (* -0.05555555555555555 (log u1))) (cos (* u2 (* (PI) 2.0))))
  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-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. sqr-abs-rev N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left(\left|{\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{4}}}\right|\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. fabs-sqr N/A

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

   9. pow2 N/A

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

   10. pow-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\color{blue}{\frac{1}{4}}}{2}\right)}\right)}^{\left(2 \cdot 2\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{\left(2 \cdot 2\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   14. metadata-eval N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{\left(2 \cdot 2\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   20. metadata-eval 98.5

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{0.125}\right)}^{\color{blue}{4}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

4. Applied rewrites 98.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(\log u1 \cdot -2\right)}^{0.125}\right)}^{4}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\sqrt{-0.05555555555555555 \cdot \log u1} \cdot \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right) + 0.5} \]
7. Add Preprocessing

Alternative 2: 99.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right) \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* -0.05555555555555555 (log u1))) (cos (* u2 (* (PI) 2.0))) 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-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. sqr-abs-rev N/A

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

   4. pow2 N/A

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

   5. metadata-eval N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\left(\left|{\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{4}}}\right|\right)}^{2}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. fabs-sqr N/A

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

   9. pow2 N/A

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

   10. pow-pow N/A

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

   11. lower-pow.f64 N/A

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

   12. metadata-eval N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\left(\frac{\color{blue}{\frac{1}{4}}}{2}\right)}\right)}^{\left(2 \cdot 2\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   13. metadata-eval N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(-2 \cdot \log u1\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{\left(2 \cdot 2\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   14. metadata-eval N/A

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

   15. lower-pow.f64 N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

   18. lower-*.f64 N/A

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

   19. metadata-eval N/A

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\color{blue}{\frac{1}{8}}}\right)}^{\left(2 \cdot 2\right)}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   20. metadata-eval 98.5

       \[\leadsto \left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{0.125}\right)}^{\color{blue}{4}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]

4. Applied rewrites 98.5%

   \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(\log u1 \cdot -2\right)}^{0.125}\right)}^{4}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
5. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2}} \]

   2. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} + \frac{1}{2} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}\right)} \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   4. lift-/.f64 N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   5. metadata-eval N/A

      \[\leadsto \left(\color{blue}{\frac{1}{6}} \cdot {\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   6. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}^{4}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   7. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{6} \cdot {\color{blue}{\left({\left(\log u1 \cdot -2\right)}^{\frac{1}{8}}\right)}}^{4}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   8. pow-pow N/A

      \[\leadsto \left(\frac{1}{6} \cdot \color{blue}{{\left(\log u1 \cdot -2\right)}^{\left(\frac{1}{8} \cdot 4\right)}}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + \frac{1}{2} \]

   9. metadata-eval N/A

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

   10. pow1/2 N/A

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

   11. lift-sqrt.f64 N/A

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

   12. *-commutative N/A

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

   13. lift-*.f64 N/A

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

   14. lower-fma.f64 99.4

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

6. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{-0.05555555555555555 \cdot \log u1}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right)} \]
7. Add Preprocessing

Alternative 3: 98.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log \left({u1}^{-0.05555555555555555}\right)} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (sqrt (log (pow u1 -0.05555555555555555))) 0.5))

C

double code(double u1, double u2) {
	return sqrt(log(pow(u1, -0.05555555555555555))) + 0.5;
}

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(u1, u2)
use fmin_fmax_functions
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt(log((u1 ** (-0.05555555555555555d0)))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt(Math.log(Math.pow(u1, -0.05555555555555555))) + 0.5;
}

Python

def code(u1, u2):
	return math.sqrt(math.log(math.pow(u1, -0.05555555555555555))) + 0.5

Julia

function code(u1, u2)
	return Float64(sqrt(log((u1 ^ -0.05555555555555555))) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt(log((u1 ^ -0.05555555555555555))) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[Log[N[Power[u1, -0.05555555555555555], $MachinePrecision]], $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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\sqrt{{\left(\log u1 \cdot -2\right)}^{1} \cdot 0.027777777777777776}} + 0.5 \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{{\left(\log u1 \cdot -2\right)}^{1} \cdot \frac{1}{36}}} + \frac{1}{2} \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{\color{blue}{{\left(\log u1 \cdot -2\right)}^{1}} \cdot \frac{1}{36}} + \frac{1}{2} \]

   3. unpow1 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot \frac{1}{36}} + \frac{1}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot \frac{1}{36}} + \frac{1}{2} \]

   5. associate-*l* N/A

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot \frac{1}{36}\right)}} + \frac{1}{2} \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\log u1 \cdot \color{blue}{\frac{-1}{18}}} + \frac{1}{2} \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\frac{-1}{18} \cdot \log u1}} + \frac{1}{2} \]

   8. lift-log.f64 N/A

      \[\leadsto \sqrt{\frac{-1}{18} \cdot \color{blue}{\log u1}} + \frac{1}{2} \]

   9. log-pow-rev N/A

      \[\leadsto \sqrt{\color{blue}{\log \left({u1}^{\frac{-1}{18}}\right)}} + \frac{1}{2} \]

   10. lower-log.f64 N/A

       \[\leadsto \sqrt{\color{blue}{\log \left({u1}^{\frac{-1}{18}}\right)}} + \frac{1}{2} \]

   11. lower-pow.f64 98.5

       \[\leadsto \sqrt{\log \color{blue}{\left({u1}^{-0.05555555555555555}\right)}} + 0.5 \]

6. Applied rewrites 98.5%

   \[\leadsto \sqrt{\color{blue}{\log \left({u1}^{-0.05555555555555555}\right)}} + 0.5 \]
7. Add Preprocessing

Alternative 4: 98.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{\log u1 \cdot -0.05555555555555555} + 0.5 \end{array} \]

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+ (sqrt (* (log u1) -0.05555555555555555)) 0.5))

C

double code(double u1, double u2) {
	return sqrt((log(u1) * -0.05555555555555555)) + 0.5;
}

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(u1, u2)
use fmin_fmax_functions
    real(8), intent (in) :: u1
    real(8), intent (in) :: u2
    code = sqrt((log(u1) * (-0.05555555555555555d0))) + 0.5d0
end function

Java

public static double code(double u1, double u2) {
	return Math.sqrt((Math.log(u1) * -0.05555555555555555)) + 0.5;
}

Python

def code(u1, u2):
	return math.sqrt((math.log(u1) * -0.05555555555555555)) + 0.5

Julia

function code(u1, u2)
	return Float64(sqrt(Float64(log(u1) * -0.05555555555555555)) + 0.5)
end

MATLAB

function tmp = code(u1, u2)
	tmp = sqrt((log(u1) * -0.05555555555555555)) + 0.5;
end

Wolfram

code[u1_, u2_] := N[(N[Sqrt[N[(N[Log[u1], $MachinePrecision] * -0.05555555555555555), $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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\sqrt{{\left(\log u1 \cdot -2\right)}^{1} \cdot 0.027777777777777776}} + 0.5 \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{{\left(\log u1 \cdot -2\right)}^{1} \cdot \frac{1}{36}}} + \frac{1}{2} \]

   2. lift-pow.f64 N/A

      \[\leadsto \sqrt{\color{blue}{{\left(\log u1 \cdot -2\right)}^{1}} \cdot \frac{1}{36}} + \frac{1}{2} \]

   3. unpow1 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot \frac{1}{36}} + \frac{1}{2} \]

   4. lift-*.f64 N/A

      \[\leadsto \sqrt{\color{blue}{\left(\log u1 \cdot -2\right)} \cdot \frac{1}{36}} + \frac{1}{2} \]

   5. associate-*l* N/A

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot \left(-2 \cdot \frac{1}{36}\right)}} + \frac{1}{2} \]

   6. metadata-eval N/A

      \[\leadsto \sqrt{\log u1 \cdot \color{blue}{\frac{-1}{18}}} + \frac{1}{2} \]

   7. lift-*.f64 98.5

      \[\leadsto \sqrt{\color{blue}{\log u1 \cdot -0.05555555555555555}} + 0.5 \]

6. Applied rewrites 98.5%

   \[\leadsto \sqrt{\color{blue}{\log u1 \cdot -0.05555555555555555}} + 0.5 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(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))