normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 4.5s

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

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

FPCore

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-sqrt.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-PI.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*l* N/A

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

   9. *-commutative N/A

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

   10. *-commutative N/A

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

   11. *-commutative N/A

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

   12. *-commutative N/A

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

   13. *-commutative N/A

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

   14. *-commutative N/A

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

   15. *-commutative N/A

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

6. Applied rewrites 99.5%

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

Alternative 2: 99.6% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

Alternative 3: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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. lift-/.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

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

Alternative 4: 98.6% accurate, 1.5× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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. lift-/.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lift-log.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. associate-*r* N/A

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

   14. lower-fma.f64 N/A

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. *-commutative N/A

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

   7. lift-*.f64 N/A

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

   8. lift-PI.f64 98.8

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

7. Applied rewrites 98.8%

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

Alternative 5: 98.7% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. remove-double-neg N/A

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

   5. log-rec N/A

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

   6. mul-1-neg N/A

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

   7. associate-*l* N/A

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

   8. metadata-eval N/A

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

   9. pow1/2 N/A

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

   10. *-commutative N/A

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

   11. sqrt-unprod N/A

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

   12. metadata-eval N/A

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

   13. sqrt-unprod N/A

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

   14. lower-*.f64 N/A

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

   15. lower-sqrt.f64 N/A

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

   16. log-rec N/A

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

   17. lower-neg.f64 N/A

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

   18. lift-log.f64 N/A

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

   19. sqrt-unprod N/A

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

   20. metadata-eval N/A

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

   21. lower-sqrt.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. *-commutative N/A

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

   7. cos-2 N/A

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

   8. lower--.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-cos.f64 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f64 N/A

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

   17. lift-PI.f64 N/A

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

   18. lower-*.f64 N/A

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower--.f64 N/A

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

   5. mul-1-neg N/A

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

   6. lower-neg.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 N/A

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

   10. lift-PI.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 98.8

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

9. Applied rewrites 98.8%

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

10. Final simplification 98.8%

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

Alternative 6: 98.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

4. Applied rewrites 99.5%

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

5. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lift-sqrt.f64 98.2

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

7. Applied rewrites 98.2%

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

Alternative 7: 98.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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. metadata-eval N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. sqrt-unprod N/A

      \[\leadsto \sqrt{\log u1 \cdot -2} \cdot \frac{1}{6} + \frac{1}{2} \]

   5. *-commutative N/A

      \[\leadsto \sqrt{-2 \cdot \log u1} \cdot \frac{1}{6} + \frac{1}{2} \]

   6. unpow1/2 N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow1/2 N/A

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

   9. lower-sqrt.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. lift-log.f64 N/A

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

   13. metadata-eval 98.1

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

5. Applied rewrites 98.1%

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

Reproduce

herbie shell --seed 2025052 
(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))