UniformSampleCone, y

Percentage Accurate: 57.6% → 90.9%

Time: 10.7s

Alternatives: 19

Speedup: 3.5×

Specification

\[\left(\left(2.328306437 \cdot 10^{-10} \leq ux \land ux \leq 1\right) \land \left(2.328306437 \cdot 10^{-10} \leq uy \land uy \leq 1\right)\right) \land \left(0 \leq maxCos \land maxCos \leq 1\right)\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))))

Initial Program: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0))))))

Alternative 1: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \left(ux - maxCos \cdot ux\right)\\ \mathbf{if}\;ux \leq 8.600000001024455 \cdot 10^{-5}:\\ \;\;\;\;\left({\left({2}^{0.25}\right)}^{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ -1.0 (- ux (* maxCos ux)))))
   (if (<= ux 8.600000001024455e-5)
     (*
      (* (pow (pow 2.0 0.25) 2.0) (sin (* (* (PI) uy) 2.0)))
      (sqrt (* (- 1.0 maxCos) ux)))
     (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_0 t_0)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 8.6e-5

   1. Initial program 36.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 36.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 93.1

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

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.2%

      \[\leadsto \left({\left({2}^{0.25}\right)}^{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right)} \cdot ux} \]

   if 8.6e-5 < ux

   1. Initial program 88.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 88.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 88.6

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

   4. Applied rewrites 88.6%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 1\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 8.600000001024455 \cdot 10^{-5}:\\ \;\;\;\;\left({\left({2}^{0.25}\right)}^{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(-1 + \left(ux - maxCos \cdot ux\right)\right) \cdot \left(-1 + \left(ux - maxCos \cdot ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := -1 + \left(ux - maxCos \cdot ux\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9998250007629395:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - maxCos} \cdot \left(\sqrt{ux} \cdot \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (+ -1.0 (- ux (* maxCos ux)))))
   (if (<= (* t_0 t_0) 0.9998250007629395)
     (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_1 t_1))))
     (*
      (sqrt (- 1.0 maxCos))
      (* (sqrt ux) (* (sqrt 2.0) (sin (* (* 2.0 uy) (PI)))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999825001

   1. Initial program 88.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 88.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 88.6

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

   4. Applied rewrites 88.6%

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

   if 0.999825001 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 36.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 36.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 93.1

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

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.1%

      \[\leadsto \sqrt{1 - maxCos} \cdot \color{blue}{\left(\sqrt{ux} \cdot \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9998250007629395:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(-1 + \left(ux - maxCos \cdot ux\right)\right) \cdot \left(-1 + \left(ux - maxCos \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - maxCos} \cdot \left(\sqrt{ux} \cdot \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := -1 + \left(ux - maxCos \cdot ux\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9998250007629395:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_1 \cdot t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (+ -1.0 (- ux (* maxCos ux)))))
   (if (<= (* t_0 t_0) 0.9998250007629395)
     (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* t_1 t_1))))
     (*
      (* (sqrt (* (- 1.0 maxCos) ux)) (sin (* (* 2.0 uy) (PI))))
      (sqrt 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999825001

   1. Initial program 88.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 88.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 88.6

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

   4. Applied rewrites 88.6%

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

   if 0.999825001 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 36.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 36.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 93.1

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

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.1%

      \[\leadsto \left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9998250007629395:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(-1 + \left(ux - maxCos \cdot ux\right)\right) \cdot \left(-1 + \left(ux - maxCos \cdot ux\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := t\_0 \cdot t\_0\\ \mathbf{if}\;t\_1 \leq 0.9998250007629395:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* t_0 t_0)))
   (if (<= t_1 0.9998250007629395)
     (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 t_1)))
     (*
      (* (sqrt (* (- 1.0 maxCos) ux)) (sin (* (* 2.0 uy) (PI))))
      (sqrt 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999825001

   1. Initial program 88.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   if 0.999825001 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 36.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 36.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 93.1

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

   7. Applied rewrites 93.1%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}} \]
   8. Step-by-step derivation

   9. Applied rewrites 93.1%

      \[\leadsto \left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9998250007629395:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996200203895569:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996200203895569)
     (*
      (sin (* (* uy 2.0) (PI)))
      (sqrt (- 1.0 (* (- (- ux (* maxCos ux)) 1.0) (- ux 1.0)))))
     (*
      (* (sqrt (* (- 1.0 maxCos) ux)) (sin (* (* 2.0 uy) (PI))))
      (sqrt 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99962002

   1. Initial program 89.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 89.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \color{blue}{\left(ux - 1\right)}} \]
   6. Step-by-step derivation

      1. lower--.f32 87.3

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

   7. Applied rewrites 87.3%

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

   if 0.99962002 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 38.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 38.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 38.8

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 91.9

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

   7. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}} \]
   8. Step-by-step derivation

   9. Applied rewrites 92.0%

      \[\leadsto \left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \color{blue}{\sqrt{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9996200203895569:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996200203895569:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996200203895569)
     (*
      (sin (* (* uy 2.0) (PI)))
      (sqrt (- 1.0 (* (- (- ux (* maxCos ux)) 1.0) (- ux 1.0)))))
     (*
      (* (sqrt 2.0) (sin (* (* (PI) uy) 2.0)))
      (sqrt (* (- 1.0 maxCos) ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99962002

   1. Initial program 89.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 89.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \color{blue}{\left(ux - 1\right)}} \]
   6. Step-by-step derivation

      1. lower--.f32 87.3

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

   7. Applied rewrites 87.3%

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

   if 0.99962002 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 38.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 38.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 38.8

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 91.9

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

   7. Applied rewrites 91.9%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9996200203895569:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996200203895569:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(ux - 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sin (* (* uy 2.0) (PI)))))
   (if (<= (* t_0 t_0) 0.9996200203895569)
     (* t_1 (sqrt (- 1.0 (* (- (- ux (* maxCos ux)) 1.0) (- ux 1.0)))))
     (* t_1 (sqrt (* ux (* 2.0 (- 1.0 maxCos))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99962002

   1. Initial program 89.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 89.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 89.8

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

   4. Applied rewrites 89.8%

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

   5. Taylor expanded in maxCos around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \color{blue}{\left(ux - 1\right)}} \]
   6. Step-by-step derivation

      1. lower--.f32 87.3

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

   7. Applied rewrites 87.3%

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

   if 0.99962002 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 38.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 38.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 38.8

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}} \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(1 - maxCos\right) + ux \cdot \left(1 - maxCos\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)}} \]

      3. count-2-rev N/A

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

      4. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \left(1 - maxCos\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot \left(1 - maxCos\right)\right)}} \]

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right)\right)} \]

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

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 \cdot \left(1 + -1 \cdot maxCos\right)\right)}} \]

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

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)}\right)} \]

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f32 91.7

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

   7. Applied rewrites 91.7%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996200203895569:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (sin (* (* uy 2.0) (PI)))))
   (if (<= (* t_0 t_0) 0.9996200203895569)
     (* t_1 (sqrt (- 1.0 (* t_0 (- 1.0 ux)))))
     (* t_1 (sqrt (* ux (* 2.0 (- 1.0 maxCos))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.99962002

   1. Initial program 89.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in maxCos around 0

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

      1. lower--.f32 87.3

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   5. Applied rewrites 87.3%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   if 0.99962002 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 38.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 38.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 38.8

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}} \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(1 - maxCos\right) + ux \cdot \left(1 - maxCos\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)}} \]

      3. count-2-rev N/A

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

      4. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \left(1 - maxCos\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot \left(1 - maxCos\right)\right)}} \]

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right)\right)} \]

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

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 \cdot \left(1 + -1 \cdot maxCos\right)\right)}} \]

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

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)}\right)} \]

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f32 91.7

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

   7. Applied rewrites 91.7%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;t\_0 \cdot t\_0 \leq 0.9996600151062012:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (* t_0 t_0) 0.9996600151062012)
     (*
      (* (* (PI) 2.0) uy)
      (sqrt
       (*
        (-
         (/ 2.0 ux)
         (fma (/ maxCos ux) 2.0 (* (- maxCos 1.0) (- maxCos 1.0))))
        (* ux ux))))
     (* (sin (* (* uy 2.0) (PI))) (sqrt (* ux (* 2.0 (- 1.0 maxCos))))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))) < 0.999660015

   1. Initial program 89.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 77.2

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 77.2%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(ux \cdot maxCos\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)}} \]

      5. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)} + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux\right) \cdot maxCos} + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux\right) \cdot maxCos + \color{blue}{\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\right)} \]

      8. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

      9. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux}, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      10. lift-+.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\left(\left(1 - ux\right) + \color{blue}{ux \cdot maxCos}\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\left(\left(1 - ux\right) + \color{blue}{maxCos \cdot ux}\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      14. lift-fma.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

      16. lower-*.f32 75.1

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

   7. Applied rewrites 74.9%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)}} \]

   8. Taylor expanded in ux around inf

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

   10. Applied rewrites 79.8%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(ux \cdot ux\right)}} \]

   if 0.999660015 < (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))

   1. Initial program 38.2%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 38.2

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 38.2

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

   4. Applied rewrites 38.2%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}} \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(1 - maxCos\right) + ux \cdot \left(1 - maxCos\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)}} \]

      3. count-2-rev N/A

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

      4. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \left(1 - maxCos\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot \left(1 - maxCos\right)\right)}} \]

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right)\right)} \]

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

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 \cdot \left(1 + -1 \cdot maxCos\right)\right)}} \]

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

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)}\right)} \]

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f32 92.1

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

   7. Applied rewrites 92.1%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) \leq 0.9996600151062012:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.0001900000061141327:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.0001900000061141327)
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* ux (* 2.0 (- 1.0 maxCos)))))
   (* (sin (* (* (PI) uy) 2.0)) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.90000006e-4

   1. Initial program 38.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 38.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 38.8

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{2 \cdot \left(ux \cdot \left(1 - maxCos\right)\right)}} \]
   6. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(1 - maxCos\right) + ux \cdot \left(1 - maxCos\right)}} \]

      2. distribute-lft-out N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(1 - maxCos\right) + \left(1 - maxCos\right)\right)}} \]

      3. count-2-rev N/A

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

      4. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot \left(1 - maxCos\right)\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{ux \cdot \left(\left(\mathsf{neg}\left(-2\right)\right) \cdot \left(1 - maxCos\right)\right)}} \]

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right)\right)} \]

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

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 + -1 \cdot maxCos\right)}\right)} \]

      10. lower-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 \cdot \left(1 + -1 \cdot maxCos\right)\right)}} \]

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

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)}\right)} \]

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f32 91.7

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

   7. Applied rewrites 91.7%

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

   if 1.90000006e-4 < ux

   1. Initial program 89.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 77.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 75.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 75.3%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   9. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]
   10. Step-by-step derivation

       1. lower--.f32 75.0

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]

   11. Applied rewrites 75.0%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]

   12. Taylor expanded in uy around inf

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

       1. count-2-rev N/A

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

       2. distribute-lft-in N/A

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

       3. count-2-rev N/A

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

       4. lower-sin.f32 N/A

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

       5. count-2-rev N/A

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

       6. distribute-lft-in N/A

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

       7. count-2-rev N/A

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

       8. *-commutative N/A

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

       9. lower-*.f32 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f32 N/A

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

       12. lower-PI.f32 87.0

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

   14. Applied rewrites 87.0%

       \[\leadsto \color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)} \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 67.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;uy \leq 0.0015999999595806003:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= uy 0.0015999999595806003)
     (*
      t_0
      (sqrt
       (*
        (-
         (/ 2.0 ux)
         (fma (/ maxCos ux) 2.0 (* (- maxCos 1.0) (- maxCos 1.0))))
        (* ux ux))))
     (* (sin t_0) (sqrt (* (fma -2.0 maxCos 2.0) ux))))))

Derivation

1. Split input into 2 regimes

2. if uy < 0.00159999996

   1. Initial program 58.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 58.1

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      3. distribute-lft-in N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(ux \cdot maxCos\right)\right)}} \]

      4. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)}} \]

      5. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)} + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)} \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux\right) \cdot maxCos} + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux\right) \cdot maxCos + \color{blue}{\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\right)} \]

      8. lower-fma.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

      9. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux}, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      10. lift-+.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      11. lift-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\left(\left(1 - ux\right) + \color{blue}{ux \cdot maxCos}\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      12. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\left(\left(1 - ux\right) + \color{blue}{maxCos \cdot ux}\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      13. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      14. lift-fma.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

      15. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

      16. lower-*.f32 56.7

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

   7. Applied rewrites 54.8%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)}} \]

   8. Taylor expanded in ux around inf

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

      2. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

   10. Applied rewrites 89.9%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(ux \cdot ux\right)}} \]

   if 0.00159999996 < uy

   1. Initial program 58.1%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(\left(maxCos + \frac{1}{ux}\right) - 1\right) \cdot ux\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(\left(maxCos + \frac{1}{ux}\right) - 1\right) \cdot ux\right)}} \]

      3. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\color{blue}{\left(\left(maxCos + \frac{1}{ux}\right) - 1\right)} \cdot ux\right)} \]

      4. +-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{ux} + maxCos\right)} - 1\right) \cdot ux\right)} \]

      5. lower-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\color{blue}{\left(\frac{1}{ux} + maxCos\right)} - 1\right) \cdot ux\right)} \]

      6. lower-/.f32 58.8

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\left(\color{blue}{\frac{1}{ux}} + maxCos\right) - 1\right) \cdot ux\right)} \]

   5. Applied rewrites 58.8%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)}} \]

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]

      2. lower-*.f32 N/A

         \[\leadsto \color{blue}{\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]

      3. count-2-rev N/A

         \[\leadsto \sin \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right) + uy \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \sin \color{blue}{\left(uy \cdot \left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      5. count-2-rev N/A

         \[\leadsto \sin \left(uy \cdot \color{blue}{\left(2 \cdot \mathsf{PI}\left(\right)\right)}\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      6. lower-sin.f32 N/A

         \[\leadsto \color{blue}{\sin \left(uy \cdot \left(2 \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      7. count-2-rev N/A

         \[\leadsto \sin \left(uy \cdot \color{blue}{\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right)}\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      8. distribute-lft-in N/A

         \[\leadsto \sin \color{blue}{\left(uy \cdot \mathsf{PI}\left(\right) + uy \cdot \mathsf{PI}\left(\right)\right)} \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      9. count-2-rev N/A

         \[\leadsto \sin \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      10. *-commutative N/A

          \[\leadsto \sin \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      11. associate-*r* N/A

          \[\leadsto \sin \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      12. lower-*.f32 N/A

          \[\leadsto \sin \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      13. *-commutative N/A

          \[\leadsto \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      14. lower-*.f32 N/A

          \[\leadsto \sin \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      15. lower-PI.f32 N/A

          \[\leadsto \sin \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - 2 \cdot maxCos\right)} \]

      16. metadata-eval N/A

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

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

          \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

      18. lower-sqrt.f32 N/A

          \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \color{blue}{\sqrt{ux \cdot \left(2 + -2 \cdot maxCos\right)}} \]

      19. *-commutative N/A

          \[\leadsto \sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

   8. Applied rewrites 10.0%

      \[\leadsto \color{blue}{\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \leq 0.0015999999595806003:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 76.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 + \left(ux - maxCos \cdot ux\right)\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_1 \cdot t\_1} \leq 0.013199999928474426:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ -1.0 (- ux (* maxCos ux)))) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_1 t_1))) 0.013199999928474426)
     (* (* (* (* (sqrt 2.0) (PI)) uy) 2.0) (sqrt (* (- 1.0 maxCos) ux)))
     (* (* (* (PI) 2.0) uy) (sqrt (- 1.0 (* t_0 t_0)))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0132

   1. Initial program 36.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 36.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.1%

       \[\leadsto \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]

   if 0.0132 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 88.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 76.5

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      6. fabs-sub N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      7. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(ux - ux \cdot maxCos\right) - 1\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(ux - ux \cdot maxCos\right) - 1\right|} \]

      9. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(ux - ux \cdot maxCos\right) - 1\right|} \]

      10. fabs-sub N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(ux - ux \cdot maxCos\right) - 1\right|} \]

      11. mul-fabs N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left|\left(\left(ux - ux \cdot maxCos\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)\right|}} \]

      12. lower-fabs.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left|\left(\left(ux - ux \cdot maxCos\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)\right|}} \]

      13. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)}\right|} \]

      14. lower--.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right)} \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)\right|} \]

      15. lower--.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(\color{blue}{\left(ux - ux \cdot maxCos\right)} - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)\right|} \]

      16. lift-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(\left(ux - \color{blue}{ux \cdot maxCos}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)\right|} \]

      17. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)\right|} \]

      18. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(\left(ux - \color{blue}{maxCos \cdot ux}\right) - 1\right) \cdot \left(\left(ux - ux \cdot maxCos\right) - 1\right)\right|} \]

      19. lower--.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left|\left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \color{blue}{\left(\left(ux - ux \cdot maxCos\right) - 1\right)}\right|} \]

   7. Applied rewrites 76.9%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left|\left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 1\right)\right|}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.013199999928474426:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(-1 + \left(ux - maxCos \cdot ux\right)\right) \cdot \left(-1 + \left(ux - maxCos \cdot ux\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 76.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.013199999928474426:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.013199999928474426)
     (* (* (* (* (sqrt 2.0) (PI)) uy) 2.0) (sqrt (* (- 1.0 maxCos) ux)))
     (*
      (* (+ (PI) (PI)) uy)
      (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) t_0)))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0132

   1. Initial program 36.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 36.6

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 36.6

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

   4. Applied rewrites 36.6%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 93.1

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

   7. Applied rewrites 93.1%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]
   9. Step-by-step derivation

   10. Applied rewrites 77.1%

       \[\leadsto \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]

   if 0.0132 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 88.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 76.5

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower--.f32 76.6

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - ux \cdot maxCos\right)}\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      8. lower-*.f32 76.6

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   7. Applied rewrites 76.6%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 76.6%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.013199999928474426:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_0 \cdot t\_0} \leq 0.019500000402331352:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_0 t_0))) 0.019500000402331352)
     (* (* (* (* (sqrt 2.0) (PI)) uy) 2.0) (sqrt (* (- 1.0 maxCos) ux)))
     (*
      (* (* (PI) 2.0) uy)
      (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.0195000004

   1. Initial program 38.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

      2. sqr-abs-rev N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(1 - ux\right) + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|}} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      4. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right| \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      6. fabs-sub N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|} \cdot \left|\left(1 - ux\right) + ux \cdot maxCos\right|} \]

      7. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right) + ux \cdot maxCos}\right|} \]

      8. lift--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right|} \]

      9. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \left|\color{blue}{1 - \left(ux - ux \cdot maxCos\right)}\right|} \]

      10. fabs-sub N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left|\left(ux - ux \cdot maxCos\right) - 1\right| \cdot \color{blue}{\left|\left(ux - ux \cdot maxCos\right) - 1\right|}} \]

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 38.8

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 38.8

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

   4. Applied rewrites 38.8%

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

   5. Taylor expanded in ux around 0

      \[\leadsto \color{blue}{\sqrt{ux \cdot \left(1 - maxCos\right)} \cdot \left(\sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

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

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in uy around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.5%

       \[\leadsto \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]

   if 0.0195000004 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 89.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 77.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 77.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 75.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 75.3%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   9. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(1 - ux\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(1 - ux\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(1 - ux\right)} \]

      4. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(1 - ux\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(1 - ux\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(1 - ux\right)} \]

      7. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - maxCos \cdot ux\right)}\right) \cdot \left(1 - ux\right)} \]

      8. lower--.f32 75.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(1 - ux\right)} \]

   10. Applied rewrites 75.3%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(1 - ux\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.019500000402331352:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ t_1 := \left(1 - ux\right) + ux \cdot maxCos\\ \mathbf{if}\;\sqrt{1 - t\_1 \cdot t\_1} \leq 0.006949999835342169:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)) (t_1 (+ (- 1.0 ux) (* ux maxCos))))
   (if (<= (sqrt (- 1.0 (* t_1 t_1))) 0.006949999835342169)
     (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* t_0 (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))) < 0.00694999984

   1. Initial program 33.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 30.1

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower--.f32 30.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - ux \cdot maxCos\right)}\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      8. lower-*.f32 30.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   7. Applied rewrites 30.3%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   8. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

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

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

      6. lower-fma.f32 72.9

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

   10. Applied rewrites 58.1%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

   if 0.00694999984 < (sqrt.f32 (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))))

   1. Initial program 86.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 74.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 72.5

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 72.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   9. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(1 - ux\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(1 - ux\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(1 - ux\right)} \]

      4. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(1 - ux\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(1 - ux\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(1 - ux\right)} \]

      7. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - maxCos \cdot ux\right)}\right) \cdot \left(1 - ux\right)} \]

      8. lower--.f32 72.5

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(1 - ux\right)} \]

   10. Applied rewrites 72.5%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(1 - ux\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \leq 0.006949999835342169:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 76.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (*
  (* (* (PI) 2.0) uy)
  (sqrt
   (*
    (- (/ 2.0 ux) (fma (/ maxCos ux) 2.0 (* (- maxCos 1.0) (- maxCos 1.0))))
    (* ux ux)))))

Derivation

1. Initial program 58.3%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lower-PI.f32 50.8

      \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

   3. distribute-lft-in N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(ux \cdot maxCos\right)\right)}} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(ux \cdot maxCos\right) + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)}} \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(ux \cdot maxCos\right)} + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)} \]

   6. associate-*r* N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux\right) \cdot maxCos} + \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)\right)} \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux\right) \cdot maxCos + \color{blue}{\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\right)} \]

   8. lower-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   9. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot ux}, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   10. lift-+.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   11. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\left(\left(1 - ux\right) + \color{blue}{ux \cdot maxCos}\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   12. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\left(\left(1 - ux\right) + \color{blue}{maxCos \cdot ux}\right) \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   13. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\left(maxCos \cdot ux + \left(1 - ux\right)\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   14. lift-fma.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \cdot ux, maxCos, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   15. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

   16. lower-*.f32 49.8

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

7. Applied rewrites 48.0%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot ux, maxCos, \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)}} \]

8. Taylor expanded in ux around inf

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

10. Applied rewrites 76.5%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(ux \cdot ux\right)}} \]

11. Final simplification 76.5%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos - 1\right)\right)\right) \cdot \left(ux \cdot ux\right)} \]
12. Add Preprocessing

Alternative 17: 58.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;ux \leq 2.4000000848900527 \cdot 10^{-5}:\\ \;\;\;\;t\_0 \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= ux 2.4000000848900527e-5)
     (* t_0 (sqrt (* (fma -2.0 maxCos 2.0) ux)))
     (* t_0 (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (- 1.0 ux))))))))

Derivation

1. Split input into 2 regimes

2. if ux < 2.40000008e-5

   1. Initial program 33.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 30.1

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower--.f32 30.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - ux \cdot maxCos\right)}\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      8. lower-*.f32 30.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   7. Applied rewrites 30.3%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   8. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

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

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

      6. lower-fma.f32 72.9

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

   10. Applied rewrites 69.0%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

   if 2.40000008e-5 < ux

   1. Initial program 86.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 74.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 72.5

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 72.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 2.4000000848900527 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 58.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 2.4000000848900527 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 2.4000000848900527e-5)
   (* (* (* (PI) 2.0) uy) (sqrt (* (fma -2.0 maxCos 2.0) ux)))
   (* (* (+ (PI) (PI)) uy) (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 2.40000008e-5

   1. Initial program 33.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 30.1

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. lower--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower--.f32 30.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - ux \cdot maxCos\right)}\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      7. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      8. lower-*.f32 30.3

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   7. Applied rewrites 30.3%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   8. Taylor expanded in ux around 0

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

      1. metadata-eval N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

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

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

      3. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

      4. lower-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

      5. +-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

      6. lower-fma.f32 72.9

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

   10. Applied rewrites 70.2%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

   if 2.40000008e-5 < ux

   1. Initial program 86.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 74.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 74.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 72.5

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 72.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   9. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]
   10. Step-by-step derivation

       1. lower--.f32 72.3

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]

   11. Applied rewrites 72.3%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]
   12. Step-by-step derivation

   13. Applied rewrites 72.3%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 2.4000000848900527 \cdot 10^{-5}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) + \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.5% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (* (PI) 2.0) uy) (sqrt (* (fma -2.0 maxCos 2.0) ux))))

Derivation

1. Initial program 58.3%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lower-PI.f32 50.8

      \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 50.8%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. lift-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. lift--.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. associate-+l- N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. lower--.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower--.f32 51.0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - ux \cdot maxCos\right)}\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   7. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   8. lower-*.f32 51.0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

7. Applied rewrites 51.0%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

8. Taylor expanded in ux around 0

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

   1. metadata-eval N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

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

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

   3. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

   4. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

   5. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

   6. lower-fma.f32 62.4

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

10. Applied rewrites 55.6%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

11. Final simplification 62.4%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024339 
(FPCore (ux uy maxCos)
  :name "UniformSampleCone, y"
  :precision binary32
  :pre (and (and (and (<= 2.328306437e-10 ux) (<= ux 1.0)) (and (<= 2.328306437e-10 uy) (<= uy 1.0))) (and (<= 0.0 maxCos) (<= maxCos 1.0)))
  (* (sin (* (* uy 2.0) (PI))) (sqrt (- 1.0 (* (+ (- 1.0 ux) (* ux maxCos)) (+ (- 1.0 ux) (* ux maxCos)))))))