normal distribution

Percentage Accurate: 99.4% → 99.6%

Time: 4.2s

Alternatives: 9

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, 1.3× speedup

Math

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

FPCore

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

4. Applied rewrites 99.7%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sin-+PI/2-rev N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-PI.f64 99.7

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

6. Applied rewrites 99.7%

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

7. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-PI.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lift-PI.f64 99.7

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

9. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left(\left(0.16666666666666666 \cdot \sqrt{2}\right) \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\mathsf{PI}\left(\right) \cdot u2, -2, \mathsf{PI}\left(\right) \cdot 0.5\right)\right)}, \sqrt{-\log u1}, 0.5\right) \]
10. 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 99.4%

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

4. Applied rewrites 99.7%

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

Alternative 3: 99.5% accurate, 1.4× speedup

Math

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

FPCore

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

4. Applied rewrites 99.5%

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

Alternative 4: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

4. Applied rewrites 99.4%

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

   1. lift-cos.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. sin-+PI/2-rev N/A

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

   6. lower-sin.f64 N/A

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

   7. *-commutative N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. lift-PI.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lift-PI.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift-fma.f64 N/A

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

   2. *-commutative N/A

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

   3. lift-PI.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lift-PI.f64 99.4

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

8. Applied rewrites 99.4%

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

Alternative 5: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

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 99.4%

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

Alternative 6: 99.4% accurate, 1.4× speedup

Math

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

FPCore

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

4. Applied rewrites 99.4%

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

Alternative 7: 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 99.4%

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

4. Applied rewrites 99.7%

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

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 99.1

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

7. Applied rewrites 99.1%

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

Alternative 8: 98.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.4%

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

3. Taylor expanded in u2 around 0

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

   1. 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 98.9

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

5. Applied rewrites 98.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-log.f64 N/A

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

   4. sqrt-prod N/A

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

   5. metadata-eval N/A

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

   6. sqrt-unprod N/A

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

   7. associate-*r* N/A

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

   8. sqrt-unprod N/A

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

   9. *-commutative N/A

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

   10. mul-1-neg N/A

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

   11. lift-log.f64 N/A

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

   12. lift-neg.f64 N/A

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

   13. lift-sqrt.f64 N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-*.f64 99.0

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

7. Applied rewrites 99.0%

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

Alternative 9: 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 99.4%

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

3. Taylor expanded in u2 around 0

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

   1. 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 98.9

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

5. Applied rewrites 98.9%

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

Reproduce

herbie shell --seed 2025073 
(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))