Beckmann Sample, near normal, slope_y

Percentage Accurate: 58.2% → 97.8%

Time: 9.8s

Alternatives: 9

Speedup: 7.7×

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: 58.2% 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: 97.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9850000143051147:\\ \;\;\;\;\sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot {\left(\left(u1 \cdot u1\right) \cdot \left(\left(1 + \left(0.9166666666666666 \cdot u1\right) \cdot u1\right) + u1\right)\right)}^{0.25}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9850000143051147)
   (* (sin (* (+ u2 u2) (PI))) (sqrt (- (log (- 1.0 u1)))))
   (*
    (sin (* (* 2.0 (PI)) u2))
    (pow (* (* u1 u1) (+ (+ 1.0 (* (* 0.9166666666666666 u1) u1)) u1)) 0.25))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.985000014

   1. Initial program 96.2%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 95.2

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

   4. Applied rewrites 95.2%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

   6. Applied rewrites 95.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f32 N/A

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

      8. pow-plus N/A

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

      9. lift-cbrt.f32 N/A

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

      10. metadata-eval N/A

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

      11. rem-cube-cbrt N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. count-2 N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. distribute-lft-out N/A

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

   8. Applied rewrites 96.2%

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

   if 0.985000014 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 49.5%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 84.3

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

   5. Applied rewrites 84.3%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 84.3

         \[\leadsto {\color{blue}{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 84.3%

      \[\leadsto \color{blue}{{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto {\color{blue}{\left({u1}^{2} \cdot \left(1 + u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right)\right)\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(1 + u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right)\right) \cdot {u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto {\color{blue}{\left(\left(1 + u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right)\right) \cdot {u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto {\left(\color{blue}{\left(u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right) + 1\right)} \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto {\left(\left(\color{blue}{\left(1 + \frac{11}{12} \cdot u1\right) \cdot u1} + 1\right) \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto {\left(\color{blue}{\mathsf{fma}\left(1 + \frac{11}{12} \cdot u1, u1, 1\right)} \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{\frac{11}{12} \cdot u1 + 1}, u1, 1\right) \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{11}{12}, u1, 1\right)}, u1, 1\right) \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. unpow2 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{11}{12}, u1, 1\right), u1, 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 49.2

         \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.9166666666666666, u1, 1\right), u1, 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 48.7%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.9166666666666666, u1, 1\right), u1, 1\right) \cdot \left(u1 \cdot u1\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 98.5%

       \[\leadsto {\left(\left(\left(1 + \left(0.9166666666666666 \cdot u1\right) \cdot u1\right) + u1\right) \cdot \left(\color{blue}{u1} \cdot u1\right)\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9850000143051147:\\ \;\;\;\;\sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot {\left(\left(u1 \cdot u1\right) \cdot \left(\left(1 + \left(0.9166666666666666 \cdot u1\right) \cdot u1\right) + u1\right)\right)}^{0.25}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\log \left(1 - u1\right)\\ t_1 := \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;t\_0 \leq 0.00019999999494757503:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (- (log (- 1.0 u1)))) (t_1 (sin (* (+ u2 u2) (PI)))))
   (if (<= t_0 0.00019999999494757503)
     (* (pow (* u1 u1) 0.25) t_1)
     (* t_1 (sqrt t_0)))))

Derivation

1. Split input into 2 regimes

2. if (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1))) < 1.99999995e-4

   1. Initial program 37.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 92.6

         \[\leadsto {\color{blue}{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 92.6%

      \[\leadsto \color{blue}{{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 92.6

         \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 92.6%

       \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f32 N/A

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

       5. count-2 N/A

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

       6. lift-*.f32 N/A

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

       7. lift-*.f32 N/A

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

       8. distribute-lft-out N/A

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

       9. lower-*.f32 N/A

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

       10. lower-+.f32 92.6

           \[\leadsto {\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u2 + u2\right)}\right) \]

   12. Applied rewrites 92.6%

       \[\leadsto {\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]

   if 1.99999995e-4 < (neg.f32 (log.f32 (-.f32 #s(literal 1 binary32) u1)))

   1. Initial program 90.0%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 89.5

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

   4. Applied rewrites 89.5%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

   6. Applied rewrites 89.7%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f32 N/A

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

      8. pow-plus N/A

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

      9. lift-cbrt.f32 N/A

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

      10. metadata-eval N/A

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

      11. rem-cube-cbrt N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. count-2 N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. distribute-lft-out N/A

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

   8. Applied rewrites 90.0%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;-\log \left(1 - u1\right) \leq 0.00019999999494757503:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 96.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9980000257492065:\\ \;\;\;\;\sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(1 + u1\right) \cdot \left(u1 \cdot u1\right)\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9980000257492065)
   (* (sin (* (+ u2 u2) (PI))) (sqrt (- (log (- 1.0 u1)))))
   (* (pow (* (+ 1.0 u1) (* u1 u1)) 0.25) (sin (* (* 2.0 (PI)) u2)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.998000026

   1. Initial program 94.3%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*l* N/A

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

      4. *-commutative N/A

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

      5. lift-PI.f32 N/A

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

      6. add-cube-cbrt N/A

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

      7. associate-*l* N/A

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

      8. associate-*l* N/A

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

      9. lower-*.f32 N/A

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

      10. pow2 N/A

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

      11. lower-pow.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-cbrt.f32 N/A

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

      14. lower-*.f32 N/A

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

      15. lower-*.f32 N/A

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

      16. lift-PI.f32 N/A

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

      17. lower-cbrt.f32 93.6

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

   4. Applied rewrites 93.6%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-*.f32 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f32 N/A

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

      8. unpow2 N/A

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

      9. associate-*r* N/A

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

      10. lift-*.f32 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f32 N/A

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

   6. Applied rewrites 93.8%

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

      1. lift-*.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. lift-pow.f32 N/A

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

      8. pow-plus N/A

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

      9. lift-cbrt.f32 N/A

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

      10. metadata-eval N/A

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

      11. rem-cube-cbrt N/A

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

      12. lift-PI.f32 N/A

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

      13. lift-PI.f32 N/A

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

      14. *-commutative N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. count-2 N/A

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

      18. lift-*.f32 N/A

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

      19. lift-*.f32 N/A

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

      20. distribute-lft-out N/A

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

   8. Applied rewrites 94.3%

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

   if 0.998000026 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 44.8%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 87.6

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

   5. Applied rewrites 87.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 87.6

         \[\leadsto {\color{blue}{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 87.6%

      \[\leadsto \color{blue}{{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

      \[\leadsto {\color{blue}{\left({u1}^{2} \cdot \left(1 + u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right)\right)\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto {\color{blue}{\left(\left(1 + u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right)\right) \cdot {u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      2. lower-*.f32 N/A

         \[\leadsto {\color{blue}{\left(\left(1 + u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right)\right) \cdot {u1}^{2}\right)}}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      3. +-commutative N/A

         \[\leadsto {\left(\color{blue}{\left(u1 \cdot \left(1 + \frac{11}{12} \cdot u1\right) + 1\right)} \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      4. *-commutative N/A

         \[\leadsto {\left(\left(\color{blue}{\left(1 + \frac{11}{12} \cdot u1\right) \cdot u1} + 1\right) \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      5. lower-fma.f32 N/A

         \[\leadsto {\left(\color{blue}{\mathsf{fma}\left(1 + \frac{11}{12} \cdot u1, u1, 1\right)} \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      6. +-commutative N/A

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{\frac{11}{12} \cdot u1 + 1}, u1, 1\right) \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      7. lower-fma.f32 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{11}{12}, u1, 1\right)}, u1, 1\right) \cdot {u1}^{2}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      8. unpow2 N/A

         \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{11}{12}, u1, 1\right), u1, 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}\right)}^{\frac{1}{4}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

      9. lower-*.f32 51.5

         \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.9166666666666666, u1, 1\right), u1, 1\right) \cdot \color{blue}{\left(u1 \cdot u1\right)}\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 51.5%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.9166666666666666, u1, 1\right), u1, 1\right) \cdot \left(u1 \cdot u1\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   11. Taylor expanded in u1 around 0

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

   13. Applied rewrites 98.0%

       \[\leadsto {\left(\left(1 + u1\right) \cdot \left(\color{blue}{u1} \cdot u1\right)\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9980000257492065:\\ \;\;\;\;\sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(1 + u1\right) \cdot \left(u1 \cdot u1\right)\right)}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= (- 1.0 u1) 0.9998000264167786)
   (* (* (* 2.0 (PI)) u2) (sqrt (- (log (- 1.0 u1)))))
   (* (pow (* u1 u1) 0.25) (sin (* (+ u2 u2) (PI))))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999800026

   1. Initial program 90.0%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.4

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

   5. Applied rewrites 76.4%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 37.0%

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

   3. Taylor expanded in u1 around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f32 92.6

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

   5. Applied rewrites 92.6%

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

      1. lift-sqrt.f32 N/A

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

      2. pow1/2 N/A

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

      3. metadata-eval N/A

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

      4. pow-sqr N/A

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

      5. pow-prod-down N/A

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

      6. lower-pow.f32 N/A

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

      7. lower-*.f32 92.6

         \[\leadsto {\color{blue}{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   7. Applied rewrites 92.6%

      \[\leadsto \color{blue}{{\left(\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)\right)}^{0.25}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   8. Taylor expanded in u1 around 0

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

      1. unpow2 N/A

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

      2. lower-*.f32 92.6

         \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   10. Applied rewrites 92.6%

       \[\leadsto {\color{blue}{\left(u1 \cdot u1\right)}}^{0.25} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   11. Step-by-step derivation

       1. lift-*.f32 N/A

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

       2. lift-*.f32 N/A

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

       3. associate-*l* N/A

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

       4. lift-*.f32 N/A

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

       5. count-2 N/A

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

       6. lift-*.f32 N/A

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

       7. lift-*.f32 N/A

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

       8. distribute-lft-out N/A

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

       9. lower-*.f32 N/A

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

       10. lower-+.f32 92.6

           \[\leadsto {\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\left(u2 + u2\right)}\right) \]

   12. Applied rewrites 92.6%

       \[\leadsto {\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(u1 \cdot u1\right)}^{0.25} \cdot \sin \left(\left(u2 + u2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\\ \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;t\_0 \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) u2)))
   (if (<= (- 1.0 u1) 0.9998000264167786)
     (* t_0 (sqrt (- (log (- 1.0 u1)))))
     (* (sqrt u1) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) u1) < 0.999800026

   1. Initial program 90.0%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.4

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

   5. Applied rewrites 76.4%

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

   if 0.999800026 < (-.f32 #s(literal 1 binary32) u1)

   1. Initial program 37.0%

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

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

   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 92.6

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

   7. Applied rewrites 92.6%

      \[\leadsto \color{blue}{\sqrt{u1}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - u1 \leq 0.9998000264167786:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \cdot \sqrt{-\log \left(1 - u1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{u1} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.3% 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 59.3%

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

4. Applied rewrites 14.9%

   \[\leadsto \color{blue}{{\left({\left(\mathsf{log1p}\left(u1\right)\right)}^{0.25}\right)}^{2}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]

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 75.3

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

7. Applied rewrites 75.3%

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

Alternative 7: 66.1% accurate, 4.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.3

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

5. Applied rewrites 75.3%

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

   1. lift-*.f32 N/A

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

   2. lift-sin.f32 N/A

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

   3. lift-*.f32 N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. sin-2 N/A

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

   7. associate-*r* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-cos.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f32 N/A

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

   15. lower-sin.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 75.3

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

7. Applied rewrites 75.3%

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

8. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower-PI.f32 65.2

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

10. Applied rewrites 65.2%

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

    1. +-lft-identity N/A

       \[\leadsto \left(\sqrt{\color{blue}{0 + \left(-\left(-u1\right)\right)}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

    2. flip-+ N/A

       \[\leadsto \left(\sqrt{\color{blue}{\frac{0 \cdot 0 - \left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)}{0 - \left(-\left(-u1\right)\right)}}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

    3. lower-/.f32 N/A

       \[\leadsto \left(\sqrt{\color{blue}{\frac{0 \cdot 0 - \left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)}{0 - \left(-\left(-u1\right)\right)}}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

    4. metadata-eval N/A

       \[\leadsto \left(\sqrt{\frac{\color{blue}{0} - \left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)}{0 - \left(-\left(-u1\right)\right)}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

    5. lower--.f32 N/A

       \[\leadsto \left(\sqrt{\frac{\color{blue}{0 - \left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

    6. lower-*.f32 N/A

       \[\leadsto \left(\sqrt{\frac{0 - \color{blue}{\left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)}}{0 - \left(-\left(-u1\right)\right)}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

    7. lower--.f32 65.2

       \[\leadsto \left(\sqrt{\frac{0 - \left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)}{\color{blue}{0 - \left(-\left(-u1\right)\right)}}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

12. Applied rewrites 65.2%

    \[\leadsto \left(\sqrt{\color{blue}{\frac{0 - \left(-\left(-u1\right)\right) \cdot \left(-\left(-u1\right)\right)}{0 - \left(-\left(-u1\right)\right)}}} \cdot 2\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]

13. Final simplification 65.2%

    \[\leadsto \left(u2 \cdot \mathsf{PI}\left(\right)\right) \cdot \left(\sqrt{\frac{\left(-\left(-u1\right)\right) \cdot \left(-u1\right)}{-u1}} \cdot 2\right) \]
14. Add Preprocessing

Alternative 8: 66.1% accurate, 7.7× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u1 around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f32 75.3

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

5. Applied rewrites 75.3%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 65.2

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

8. Applied rewrites 65.2%

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

9. Final simplification 65.2%

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

Alternative 9: 4.6% accurate, 8.3× speedup

Math

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

FPCore

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

Derivation

1. Initial program 59.3%

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

3. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. unpow2 N/A

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

   3. rem-square-sqrt N/A

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

   4. mul-1-neg N/A

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

   5. lower-neg.f32 N/A

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

   6. lower-sqrt.f32 4.0

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

5. Applied rewrites 4.0%

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

6. Taylor expanded in u2 around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 4.7

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

8. Applied rewrites 4.7%

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

9. Final simplification 4.7%

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

Reproduce

herbie shell --seed 2024271 
(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))))