Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.7% → 98.3%

Time: 11.0s

Alternatives: 8

Speedup: 1.8×

Specification

\[\left(\left(cosTheta\_i > 0.9999 \land cosTheta\_i \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq u1 \land u1 \leq 1\right)\right) \land \left(2.328306437 \cdot 10^{-10} \leq u2 \land u2 \leq 1\right)\]

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))

Initial Program: 57.7% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))

Alternative 1: 98.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({t\_0}^{2}, t\_0 \cdot u2, u2 \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))))
   (*
    (sqrt (+ (- (log1p (* (- u1) u1))) (log1p u1)))
    (sin (fma (pow t_0 2.0) (* t_0 u2) (* u2 (PI)))))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lift--.f32 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. fp-cancel-sub-sign-inv N/A

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

   13. lower-log1p.f32 N/A

       \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)}\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   15. lower-neg.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   16. lower-log1p.f32 98.4

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \color{blue}{\mathsf{log1p}\left(u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.4%

   \[\leadsto \sqrt{\color{blue}{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

   5. lift-PI.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right) \]

   6. add-cube-cbrt N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right) \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right)} + \mathsf{PI}\left(\right) \cdot u2\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   9. pow2 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   10. lower-pow.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   11. lift-PI.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   12. lower-cbrt.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2}, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   14. lift-PI.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   17. lower-*.f32 98.5

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Add Preprocessing

Alternative 2: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{\mathsf{PI}\left(\right)}\\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({t\_0}^{2}, t\_0 \cdot u2, u2 \cdot \mathsf{PI}\left(\right)\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (cbrt (PI))))
   (*
    (sqrt (- (log1p (- u1))))
    (sin (fma (pow t_0 2.0) (* t_0 u2) (* u2 (PI)))))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. *-lft-identity N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. distribute-lft-neg-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-neg.f32 98.4

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

   5. lift-PI.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right) \]

   6. add-cube-cbrt N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right) \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right)} + \mathsf{PI}\left(\right) \cdot u2\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   9. pow2 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   10. lower-pow.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   11. lift-PI.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   12. lower-cbrt.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2}, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   14. lift-PI.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   17. lower-*.f32 98.5

       \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]
7. Add Preprocessing

Alternative 3: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (- (log1p (- u1)))) (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f32 N/A

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

   2. lift--.f32 N/A

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

   3. *-lft-identity N/A

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

   4. fp-cancel-sub-sign-inv N/A

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

   5. distribute-lft-neg-in N/A

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

   6. *-lft-identity N/A

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

   7. lower-log1p.f32 N/A

      \[\leadsto \sqrt{-\color{blue}{\mathsf{log1p}\left(\mathsf{neg}\left(u1\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-neg.f32 98.4

      \[\leadsto \sqrt{-\mathsf{log1p}\left(\color{blue}{-u1}\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.4%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Add Preprocessing

Alternative 4: 93.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
  (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

   \[\leadsto \sqrt{\color{blue}{u1 \cdot \left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) + 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right)\right) \cdot u1} + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right), u1, 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   6. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{u1 \cdot \left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) + \frac{1}{2}}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. *-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{3} + \frac{1}{4} \cdot u1\right) \cdot u1} + \frac{1}{2}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{3} + \frac{1}{4} \cdot u1, u1, \frac{1}{2}\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   9. +-commutative N/A

      \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{4} \cdot u1 + \frac{1}{3}}, u1, \frac{1}{2}\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. lower-fma.f32 93.8

       \[\leadsto \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right)}, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Applied rewrites 93.8%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Add Preprocessing

Alternative 5: 91.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \sqrt{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (*
  (sqrt (* (fma (fma 0.3333333333333333 u1 0.5) u1 1.0) u1))
  (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   2. lower-*.f32 N/A

      \[\leadsto \sqrt{\color{blue}{\left(1 + u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right)\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   3. +-commutative N/A

      \[\leadsto \sqrt{\color{blue}{\left(u1 \cdot \left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) + 1\right)} \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   4. *-commutative N/A

      \[\leadsto \sqrt{\left(\color{blue}{\left(\frac{1}{2} + \frac{1}{3} \cdot u1\right) \cdot u1} + 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   5. lower-fma.f32 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f32 91.7

      \[\leadsto \sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right)}, u1, 1\right) \cdot u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Applied rewrites 91.7%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, u1, 0.5\right), u1, 1\right) \cdot u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
6. Add Preprocessing

Alternative 6: 87.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (fma (* 0.5 u1) u1 u1)) (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 87.5

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

5. Applied rewrites 87.5%

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

7. Applied rewrites 87.6%

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

Alternative 7: 87.8% accurate, 1.7× speedup

Math

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

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt (* (fma 0.5 u1 1.0) u1)) (sin (* (PI) (+ u2 u2)))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lift--.f32 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. fp-cancel-sub-sign-inv N/A

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

   13. lower-log1p.f32 N/A

       \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)}\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   15. lower-neg.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   16. lower-log1p.f32 98.4

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \color{blue}{\mathsf{log1p}\left(u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.4%

   \[\leadsto \sqrt{\color{blue}{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
5. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]

   2. lift-*.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)} \cdot u2\right) \]

   3. associate-*l* N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(2 \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   4. count-2-rev N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right)} \]

   5. lift-PI.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right) \]

   6. add-cube-cbrt N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\left(\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right)} \cdot u2 + \mathsf{PI}\left(\right) \cdot u2\right) \]

   7. associate-*l* N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}\right) \cdot \left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2\right)} + \mathsf{PI}\left(\right) \cdot u2\right) \]

   8. lower-fma.f32 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left(\sqrt[3]{\mathsf{PI}\left(\right)} \cdot \sqrt[3]{\mathsf{PI}\left(\right)}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right)} \]

   9. pow2 N/A

      \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   10. lower-pow.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   11. lift-PI.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   12. lower-cbrt.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   13. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2}, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   14. lift-PI.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\color{blue}{\mathsf{PI}\left(\right)}} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   15. lower-cbrt.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \color{blue}{\sqrt[3]{\mathsf{PI}\left(\right)}} \cdot u2, \mathsf{PI}\left(\right) \cdot u2\right)\right) \]

   16. *-commutative N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

   17. lower-*.f32 98.5

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, \color{blue}{u2 \cdot \mathsf{PI}\left(\right)}\right)\right) \]

6. Applied rewrites 98.5%

   \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)} \cdot \sin \color{blue}{\left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, u2 \cdot \mathsf{PI}\left(\right)\right)\right)} \]

7. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f32 87.5

      \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right)} \cdot u1} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]

9. Applied rewrites 87.5%

   \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1}} \cdot \sin \left(\mathsf{fma}\left({\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{2}, \sqrt[3]{\mathsf{PI}\left(\right)} \cdot u2, u2 \cdot \mathsf{PI}\left(\right)\right)\right) \]
10. Step-by-step derivation

    1. lift-fma.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. associate-*r* N/A

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

    4. *-commutative N/A

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

    5. lift-pow.f32 N/A

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

    6. unpow2 N/A

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

    7. cube-unmult N/A

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

    8. lift-cbrt.f32 N/A

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

    9. rem-cube-cbrt N/A

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

    10. lift-*.f32 N/A

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

    11. *-commutative N/A

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

    12. distribute-lft-out N/A

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

    13. lower-*.f32 N/A

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

    14. lower-+.f32 87.5

        \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u2 + u2\right)}\right) \]

11. Applied rewrites 87.5%

    \[\leadsto \sqrt{\mathsf{fma}\left(0.5, u1, 1\right) \cdot u1} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]
12. Add Preprocessing

Alternative 8: 76.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (sqrt u1) (sin (* (* 2.0 (PI)) u2))))

Derivation

1. Initial program 57.9%

   \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-neg.f32 N/A

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

   2. lift-log.f32 N/A

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

   3. neg-log N/A

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

   4. lift--.f32 N/A

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

   5. flip-- N/A

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

   6. associate-/r/ N/A

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

   7. log-prod N/A

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

   8. lower-+.f32 N/A

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

   9. log-rec N/A

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

   10. lower-neg.f32 N/A

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

   11. metadata-eval N/A

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

   12. fp-cancel-sub-sign-inv N/A

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

   13. lower-log1p.f32 N/A

       \[\leadsto \sqrt{\left(-\color{blue}{\mathsf{log1p}\left(\left(\mathsf{neg}\left(u1\right)\right) \cdot u1\right)}\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   14. lower-*.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\color{blue}{\left(\mathsf{neg}\left(u1\right)\right) \cdot u1}\right)\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   15. lower-neg.f32 N/A

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\color{blue}{\left(-u1\right)} \cdot u1\right)\right) + \log \left(1 + u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   16. lower-log1p.f32 98.4

       \[\leadsto \sqrt{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \color{blue}{\mathsf{log1p}\left(u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

4. Applied rewrites 98.4%

   \[\leadsto \sqrt{\color{blue}{\left(-\mathsf{log1p}\left(\left(-u1\right) \cdot u1\right)\right) + \mathsf{log1p}\left(u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

5. Taylor expanded in u1 around 0

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

   1. lower-sqrt.f32 76.2

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

7. Applied rewrites 76.2%

   \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024358 
(FPCore (cosTheta_i u1 u2)
  :name "Beckmann Sample, near normal, slope_y"
  :precision binary32
  :pre (and (and (and (> cosTheta_i 0.9999) (<= cosTheta_i 1.0)) (and (<= 2.328306437e-10 u1) (<= u1 1.0))) (and (<= 2.328306437e-10 u2) (<= u2 1.0)))
  (* (sqrt (- (log (- 1.0 u1)))) (sin (* (* 2.0 (PI)) u2))))