normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 9.1s

Alternatives: 3

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: 99.6% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* -0.05555555555555555 (log u1))) (cos (* (* 2.0 (PI)) u2)) 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. lower-fma.f64 99.4

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

   7. lift-pow.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 99.4

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 99.4

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

   13. lift-/.f64 N/A

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

   14. metadata-eval 99.4

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f64 99.4

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

   18. lift-*.f64 N/A

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

   19. *-commutative N/A

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

   20. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

6. Applied rewrites 99.3%

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

   1. unpow1 N/A

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

   2. metadata-eval N/A

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

   3. pow-prod-up N/A

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

   4. pow-prod-down N/A

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

   5. unpow2 N/A

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

   6. lift-pow.f64 N/A

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

   7. lower-pow.f64 99.3

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

   8. lift-pow.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. unpow-prod-down N/A

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

   12. lower-*.f64 N/A

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

   13. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(\color{blue}{\frac{1}{36}} \cdot {\left(\sqrt{\log u1 \cdot -2}\right)}^{2}\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}} \]

   14. lift-sqrt.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(\frac{1}{36} \cdot {\color{blue}{\left(\sqrt{\log u1 \cdot -2}\right)}}^{2}\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}} \]

   15. sqrt-pow2 N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(\frac{1}{36} \cdot \color{blue}{{\left(\log u1 \cdot -2\right)}^{\left(\frac{2}{2}\right)}}\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}} \]

   16. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(\frac{1}{36} \cdot {\left(\log u1 \cdot -2\right)}^{\color{blue}{1}}\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}} \]

   17. unpow1 99.5

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(0.027777777777777776 \cdot \color{blue}{\left(\log u1 \cdot -2\right)}\right)}^{0.5}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right)}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(\frac{1}{36} \cdot \color{blue}{\left(\log u1 \cdot -2\right)}\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}} \]

   19. *-commutative N/A

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(\frac{1}{36} \cdot \color{blue}{\left(-2 \cdot \log u1\right)}\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}} \]

   20. lower-*.f64 99.5

       \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left({\left(0.027777777777777776 \cdot \color{blue}{\left(-2 \cdot \log u1\right)}\right)}^{0.5}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right)}} \]

8. Applied rewrites 99.5%

   \[\leadsto \frac{1}{\frac{1}{\mathsf{fma}\left(\color{blue}{{\left(0.027777777777777776 \cdot \left(-2 \cdot \log u1\right)\right)}^{0.5}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right)}} \]
9. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{\mathsf{fma}\left({\left(\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}}} \]

   2. lift-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\mathsf{fma}\left({\left(\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)\right)}^{\frac{1}{2}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right)}}} \]

   3. remove-double-div 99.6

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(0.027777777777777776 \cdot \left(-2 \cdot \log u1\right)\right)}^{0.5}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), 0.5\right)} \]

   4. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{\left(\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)\right)}^{\frac{1}{2}}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   5. unpow1/2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{1}{36} \cdot \left(-2 \cdot \log u1\right)}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{36} \cdot \color{blue}{\left(-2 \cdot \log u1\right)}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   8. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(\frac{1}{36} \cdot -2\right) \cdot \log u1}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\frac{-1}{18}} \cdot \log u1}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\sqrt{\frac{-1}{18} \cdot \log u1}}, \cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right), \frac{1}{2}\right) \]

   11. lower-*.f64 99.6

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

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{-1}{18} \cdot \log u1}, \cos \color{blue}{\left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}, \frac{1}{2}\right) \]

   13. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{-1}{18} \cdot \log u1}, \cos \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot u2\right)}, \frac{1}{2}\right) \]

   14. lower-*.f64 99.6

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

   15. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{-1}{18} \cdot \log u1}, \cos \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot u2\right), \frac{1}{2}\right) \]

   16. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{-1}{18} \cdot \log u1}, \cos \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right), \frac{1}{2}\right) \]

   17. lower-*.f64 99.6

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

10. Applied rewrites 99.6%

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

Alternative 2: 98.9% accurate, 1.3× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (+
  (*
   (* (pow 6.0 -1.0) (* (sqrt 2.0) (sqrt (- (log u1)))))
   (fma (* (* (PI) (PI)) -2.0) (* u2 u2) 1.0))
  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. 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. rem-square-sqrt N/A

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

   3. unpow2 N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. rem-square-sqrt N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. lower-PI.f64 N/A

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

   16. lower-PI.f64 N/A

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

   17. unpow2 N/A

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

   18. rem-square-sqrt N/A

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

   19. unpow2 N/A

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

   20. lower-*.f64 98.4

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

8. Applied rewrites 98.4%

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

9. Final simplification 98.4%

   \[\leadsto \left({6}^{-1} \cdot \left(\sqrt{2} \cdot \sqrt{-\log u1}\right)\right) \cdot \mathsf{fma}\left(\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -2, u2 \cdot u2, 1\right) + 0.5 \]
10. Add Preprocessing

Alternative 3: 98.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-log.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 97.2%

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

9. Applied rewrites 97.2%

   \[\leadsto \frac{1}{\color{blue}{{\left(\mathsf{fma}\left(\sqrt{-2 \cdot \log u1}, 0.16666666666666666, 0.5\right)\right)}^{-1}}} \]
10. Step-by-step derivation

11. Applied rewrites 97.4%

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

Reproduce

herbie shell --seed 2024319 
(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))