normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 7.9s

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

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-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. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-log.f64 98.9

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

Alternative 2: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-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. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-log.f64 98.9

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

9. Applied rewrites 99.4%

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

Alternative 3: 99.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (cos (* u2 (* (PI) 2.0))) 0.16666666666666666)
  (sqrt (* (log u1) -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. unpow1/2 N/A

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

   3. lower-sqrt.f64 99.4

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

   14. lift-/.f64 N/A

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

   15. metadata-eval 99.4

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

6. Applied rewrites 99.4%

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

Alternative 4: 99.0% accurate, 2.2× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.4%

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

   1. lift-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. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lower-exp.f64 N/A

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

   7. lower-log.f64 98.9

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 98.9

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

4. Applied rewrites 98.9%

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

5. Taylor expanded in u1 around inf

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

7. Applied rewrites 99.6%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 98.3%

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

Alternative 5: 98.8% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (* (fma (* (* -2.0 u2) u2) (* (PI) (PI)) 1.0) 0.16666666666666666)
  (sqrt (* (log u1) -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. unpow1/2 N/A

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

   3. lower-sqrt.f64 99.4

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

   14. lift-/.f64 N/A

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

   15. metadata-eval 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. associate-*r* N/A

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

   6. lower-*.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-PI.f64 98.1

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

9. Applied rewrites 98.1%

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

Alternative 6: 98.8% accurate, 2.4× speedup

Math

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma
  (fma (* (* u2 u2) -0.3333333333333333) (* (PI) (PI)) 0.16666666666666666)
  (sqrt (* (log u1) -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. unpow1/2 N/A

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

   3. lower-sqrt.f64 99.4

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 99.4

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

4. Applied rewrites 99.4%

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

   1. lift-+.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-*.f64 99.4

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 99.4

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

   14. lift-/.f64 N/A

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

   15. metadata-eval 99.4

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

6. Applied rewrites 99.4%

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

7. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-PI.f64 98.1

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

9. Applied rewrites 98.1%

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

Alternative 7: 98.4% 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

real(8) function code(u1, u2)
    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 98.1%

   \[\leadsto \color{blue}{\sqrt{{\left(\log u1 \cdot -2\right)}^{1} \cdot 0.027777777777777776}} + 0.5 \]
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. metadata-eval N/A

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

   8. lower-*.f64 N/A

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

   9. metadata-eval 98.1

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

6. Applied rewrites 98.1%

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

Reproduce

herbie shell --seed 2024337 
(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))