normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 8.2s

Alternatives: 5

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

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

FPCore

(FPCore (u1 u2)
 :precision binary64
 (fma (sqrt (* (log u1) -0.05555555555555555)) (cos (* u2 (* (PI) 2.0))) 0.5))

Derivation

1. Initial program 99.3%

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

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

   4. lower-*.f64 N/A

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

   5. lower-log.f64 98.9

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 98.9

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

4. Applied rewrites 98.9%

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

6. Applied rewrites 99.7%

   \[\leadsto \color{blue}{\sqrt{\left(0.027777777777777776 \cdot \left(\log u1 \cdot -2\right)\right) \cdot {\cos \left(u2 \cdot \left(\mathsf{PI}\left(\right) \cdot 2\right)\right)}^{2}}} + 0.5 \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Final simplification 99.7%

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

Alternative 2: 98.8% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

   \[\left(\frac{1}{6} \cdot {\left(-2 \cdot \log u1\right)}^{0.5}\right) \cdot \cos \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) + 0.5 \]
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 99.4

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-PI.f64 98.9

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

8. Applied rewrites 98.9%

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

9. Final simplification 98.9%

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

Alternative 3: 98.8% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

5. Taylor expanded in u2 around 0

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

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-PI.f64 N/A

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

   11. lower-PI.f64 98.8

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 4: 98.8% accurate, 2.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 99.3%

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

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lower-fma.f64 N/A

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

4. Applied rewrites 99.3%

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

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

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

   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. lower-*.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-PI.f64 N/A

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

   10. lower-PI.f64 98.8

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

7. Applied rewrites 98.8%

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

8. Final simplification 98.8%

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

Alternative 5: 98.5% 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 99.3%

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

3. Taylor expanded in u2 around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-log.f64 0.0

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

5. Applied rewrites 0.0%

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

7. Applied rewrites 98.4%

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

9. Applied rewrites 98.7%

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

10. Final simplification 98.7%

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

Reproduce

herbie shell --seed 2024298 
(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))