Beckmann Sample, near normal, slope_y

Percentage Accurate: 57.8% → 98.3%

Time: 10.6s

Alternatives: 11

Speedup: 8.9×

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: 57.8% 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, 1.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.2%

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

   1. lift-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 98.5

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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f32 N/A

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

   8. add-cube-cbrt N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lower-cbrt.f32 N/A

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

   19. *-commutative N/A

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

   20. 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. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f32 N/A

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

   5. pow-plus N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(2 + 1\right)}} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lift-cbrt.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{\left(2 + 1\right)} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   7. metadata-eval N/A

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

   8. rem-cube-cbrt N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f32 N/A

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

   13. lower-+.f32 98.5

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

8. Applied rewrites 98.5%

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

Alternative 2: 96.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;u1 \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.25, u1, 0.3333333333333333\right), u1, 0.5\right), u1, 1\right) \cdot u1} \cdot \sin \left(\mathsf{PI}\left(\right) \cdot \left(u2 + u2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (if (<= u1 0.03999999910593033)
   (*
    (sqrt (* (fma (fma (fma 0.25 u1 0.3333333333333333) u1 0.5) u1 1.0) u1))
    (sin (* (PI) (+ u2 u2))))
   (*
    (sqrt (- (log (- 1.0 u1))))
    (*
     (+
      (fma (* (* u2 u2) -1.3333333333333333) (* (* (PI) (PI)) (PI)) (PI))
      (PI))
     u2))))

Derivation

1. Split input into 2 regimes

2. if u1 < 0.0399999991

   1. Initial program 49.7%

      \[\sqrt{-\log \left(1 - u1\right)} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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 98.6

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

   4. Applied rewrites 98.6%

      \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
   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. *-commutative N/A

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

      3. lift-*.f32 N/A

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

      4. count-2-rev N/A

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

      5. distribute-lft-in N/A

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

      6. *-commutative N/A

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

      7. lift-PI.f32 N/A

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

      8. add-cube-cbrt N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f32 N/A

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

      12. pow2 N/A

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

      13. lower-pow.f32 N/A

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

      14. lift-PI.f32 N/A

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

      15. lower-cbrt.f32 N/A

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

      16. lower-*.f32 N/A

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

      17. lift-PI.f32 N/A

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

      18. lower-cbrt.f32 N/A

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

      19. *-commutative N/A

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

      20. 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. Step-by-step derivation

      1. lift-fma.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. associate-*r* N/A

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

      4. lift-pow.f32 N/A

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

      5. pow-plus N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(2 + 1\right)}} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

      6. lift-cbrt.f32 N/A

         \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{\left(2 + 1\right)} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

      7. metadata-eval N/A

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

      8. rem-cube-cbrt N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-*.f32 N/A

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

      13. lower-+.f32 98.6

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

   8. Applied rewrites 98.6%

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

   9. Taylor expanded in u1 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f32 N/A

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

       6. +-commutative N/A

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

       7. *-commutative N/A

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

       8. lower-fma.f32 N/A

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

       9. +-commutative N/A

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

       10. lower-fma.f32 98.6

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

   11. Applied rewrites 98.6%

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

   if 0.0399999991 < u1

   1. Initial program 96.9%

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

   3. Taylor expanded in u2 around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

   5. Applied rewrites 90.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \color{blue}{\left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, {\mathsf{PI}\left(\right)}^{3}, \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 90.1%

      \[\leadsto \sqrt{-\log \left(1 - u1\right)} \cdot \left(\left(\mathsf{fma}\left(\left(u2 \cdot u2\right) \cdot -1.3333333333333333, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \mathsf{PI}\left(\right), \mathsf{PI}\left(\right)\right) + \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 93.3% accurate, 1.6× speedup

Math

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

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 (* (PI) (+ u2 u2)))))

Derivation

1. Initial program 56.2%

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

   1. lift-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 98.5

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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f32 N/A

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

   8. add-cube-cbrt N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lower-cbrt.f32 N/A

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

   19. *-commutative N/A

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

   20. 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. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f32 N/A

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

   5. pow-plus N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(2 + 1\right)}} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lift-cbrt.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{\left(2 + 1\right)} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   7. metadata-eval N/A

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

   8. rem-cube-cbrt N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f32 N/A

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

   13. lower-+.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in u1 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. +-commutative N/A

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

    7. *-commutative N/A

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

    8. lower-fma.f32 N/A

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

    9. +-commutative N/A

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

    10. lower-fma.f32 94.2

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

11. Applied rewrites 94.2%

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

Alternative 4: 91.6% accurate, 1.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.2%

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

   1. lift-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 98.5

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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f32 N/A

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

   8. add-cube-cbrt N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lower-cbrt.f32 N/A

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

   19. *-commutative N/A

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

   20. 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. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f32 N/A

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

   5. pow-plus N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(2 + 1\right)}} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lift-cbrt.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{\left(2 + 1\right)} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   7. metadata-eval N/A

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

   8. rem-cube-cbrt N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f32 N/A

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

   13. lower-+.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in u1 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-fma.f32 N/A

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

    6. +-commutative N/A

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

    7. lower-fma.f32 92.5

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

11. Applied rewrites 92.5%

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

Alternative 5: 88.0% 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 56.2%

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

   1. lift-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 98.5

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

4. Applied rewrites 98.5%

   \[\leadsto \color{blue}{\sqrt{-\mathsf{log1p}\left(-u1\right)}} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot u2\right) \]
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. *-commutative N/A

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

   3. lift-*.f32 N/A

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

   4. count-2-rev N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lift-PI.f32 N/A

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

   8. add-cube-cbrt N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. lower-fma.f32 N/A

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

   12. pow2 N/A

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

   13. lower-pow.f32 N/A

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

   14. lift-PI.f32 N/A

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

   15. lower-cbrt.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lift-PI.f32 N/A

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

   18. lower-cbrt.f32 N/A

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

   19. *-commutative N/A

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

   20. 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. Step-by-step derivation

   1. lift-fma.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. associate-*r* N/A

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

   4. lift-pow.f32 N/A

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

   5. pow-plus N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left(\color{blue}{{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}^{\left(2 + 1\right)}} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   6. lift-cbrt.f32 N/A

      \[\leadsto \sqrt{-\mathsf{log1p}\left(-u1\right)} \cdot \sin \left({\color{blue}{\left(\sqrt[3]{\mathsf{PI}\left(\right)}\right)}}^{\left(2 + 1\right)} \cdot u2 + u2 \cdot \mathsf{PI}\left(\right)\right) \]

   7. metadata-eval N/A

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

   8. rem-cube-cbrt N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. distribute-rgt-out N/A

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

   12. lower-*.f32 N/A

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

   13. lower-+.f32 98.5

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

8. Applied rewrites 98.5%

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

9. Taylor expanded in u1 around 0

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. +-commutative N/A

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

    4. lower-fma.f32 88.6

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

11. Applied rewrites 88.6%

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

Alternative 6: 76.5% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.2%

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

   1. lift-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. flip-+ N/A

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

   8. lift--.f32 N/A

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

   9. frac-times N/A

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

   10. log-div N/A

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

   11. lower--.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-log.f32 N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. distribute-lft-in N/A

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

   5. count-2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

Alternative 7: 66.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \left(2 \cdot {\left(u1 \cdot u1\right)}^{0.25}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \end{array} \]

FPCore

(FPCore (cosTheta_i u1 u2)
 :precision binary32
 (* (* 2.0 (pow (* u1 u1) 0.25)) (* (PI) u2)))

Derivation

1. Initial program 56.2%

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

   1. lift-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. flip-+ N/A

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

   8. lift--.f32 N/A

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

   9. frac-times N/A

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

   10. log-div N/A

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

   11. lower--.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-log.f32 N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. distribute-lft-in N/A

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

   5. count-2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 68.1%

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

12. Applied rewrites 68.1%

    \[\leadsto \left(2 \cdot {\left(u1 \cdot u1\right)}^{0.25}\right) \cdot \left(\mathsf{PI}\left(\right) \cdot u2\right) \]
13. Add Preprocessing

Alternative 8: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.2%

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

   1. lift-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. flip-+ N/A

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

   8. lift--.f32 N/A

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

   9. frac-times N/A

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

   10. log-div N/A

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

   11. lower--.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-log.f32 N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. distribute-lft-in N/A

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

   5. count-2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 68.1%

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

12. Applied rewrites 68.1%

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

Alternative 9: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.2%

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

   1. lift-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. flip-+ N/A

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

   8. lift--.f32 N/A

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

   9. frac-times N/A

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

   10. log-div N/A

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

   11. lower--.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-log.f32 N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. distribute-lft-in N/A

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

   5. count-2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 68.1%

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

12. Applied rewrites 68.1%

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

Alternative 10: 66.4% accurate, 8.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 56.2%

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

   1. lift-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. flip-+ N/A

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

   8. lift--.f32 N/A

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

   9. frac-times N/A

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

   10. log-div N/A

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

   11. lower--.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-log.f32 N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. distribute-lft-in N/A

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

   5. count-2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 68.1%

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

Alternative 11: 25.3% accurate, 11.0× speedup

Math

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

FPCore

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

C

float code(float cosTheta_i, float u1, float u2) {
	return (2.0f * u2) * sqrtf(u1);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(4) function code(costheta_i, u1, u2)
use fmin_fmax_functions
    real(4), intent (in) :: costheta_i
    real(4), intent (in) :: u1
    real(4), intent (in) :: u2
    code = (2.0e0 * u2) * sqrt(u1)
end function

Julia

function code(cosTheta_i, u1, u2)
	return Float32(Float32(Float32(2.0) * u2) * sqrt(u1))
end

MATLAB

function tmp = code(cosTheta_i, u1, u2)
	tmp = (single(2.0) * u2) * sqrt(u1);
end

Derivation

1. Initial program 56.2%

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

   1. lift-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. flip-+ N/A

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

   8. lift--.f32 N/A

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

   9. frac-times N/A

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

   10. log-div N/A

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

   11. lower--.f32 N/A

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

   12. lower-log.f32 N/A

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

   13. lower-*.f32 N/A

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

   14. metadata-eval N/A

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

   15. lower--.f32 N/A

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

   16. lower-*.f32 N/A

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

   17. lower-log.f32 N/A

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

4. Applied rewrites 56.1%

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

5. Taylor expanded in u1 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. count-2-rev N/A

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

   4. distribute-lft-in N/A

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

   5. count-2-rev N/A

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

   6. lower-sin.f32 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. lower-*.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-PI.f32 N/A

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

   13. lower-sqrt.f32 77.7

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

7. Applied rewrites 77.7%

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

8. Taylor expanded in u2 around 0

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

10. Applied rewrites 68.1%

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

12. Applied rewrites 25.4%

    \[\leadsto \left(2 \cdot u2\right) \cdot \sqrt{u1} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024359 
(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))))