UniformSampleCone, y

Percentage Accurate: 57.3% → 98.3%

Time: 12.3s

Alternatives: 12

Speedup: 5.4×

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: 57.3% 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: 98.3% accurate, 0.6× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-sqrt.f32 56.9

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

4. Applied rewrites 56.9%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 0.8

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

7. Applied rewrites 0.8%

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

9. Applied rewrites 97.6%

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

    1. lift-*.f32 N/A

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

    2. lift-*.f32 N/A

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

    3. lift-*.f32 N/A

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

    4. associate-*r* N/A

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

    5. associate-*r* N/A

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

    6. lift-sqrt.f32 N/A

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

    7. lift-sqrt.f32 N/A

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

    8. rem-square-sqrt N/A

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

    9. lift-PI.f32 N/A

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

    10. *-commutative N/A

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

    11. lift-PI.f32 N/A

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

    12. associate-*r* N/A

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

    13. *-commutative N/A

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

    14. lift-*.f32 N/A

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

    15. count-2 N/A

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

    16. lift-*.f32 N/A

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

    17. lift-*.f32 N/A

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

    18. distribute-lft-out N/A

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

    19. lower-*.f32 N/A

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

    20. lower-+.f32 98.5

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

11. Applied rewrites 98.5%

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

12. Final simplification 98.5%

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

Alternative 2: 75.3% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.0%

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

   1. lift-*.f32 N/A

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

   2. lift-PI.f32 N/A

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

   3. add-sqr-sqrt N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*l* N/A

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

   8. lower-*.f32 N/A

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

   9. lower-*.f32 N/A

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

   10. lift-PI.f32 N/A

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

   11. lower-sqrt.f32 N/A

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

   12. lift-PI.f32 N/A

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

   13. lower-sqrt.f32 56.9

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

4. Applied rewrites 56.9%

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

5. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. cancel-sign-sub-inv N/A

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

   4. metadata-eval N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. lower--.f32 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f32 N/A

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

   12. lower-pow.f32 N/A

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

   13. lower--.f32 0.8

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

7. Applied rewrites 0.8%

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

8. Taylor expanded in maxCos around 0

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

10. Applied rewrites 84.5%

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

11. Final simplification 83.3%

    \[\leadsto \sqrt{\left(\left(2 - \mathsf{fma}\left(-2, ux, 2\right) \cdot maxCos\right) - ux\right) \cdot ux} \cdot \sin \left(\left(\left(\sqrt{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{\mathsf{PI}\left(\right)}\right) \]
12. Add Preprocessing

Alternative 3: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\\ \mathbf{if}\;2 \cdot uy \leq 0.0003600000054575503:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(2 - \left({\left(maxCos - 1\right)}^{2} \cdot ux - -2 \cdot maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin t\_0 \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* uy (PI)) 2.0)))
   (if (<= (* 2.0 uy) 0.0003600000054575503)
     (*
      t_0
      (sqrt
       (* (- 2.0 (- (* (pow (- maxCos 1.0) 2.0) ux) (* -2.0 maxCos))) ux)))
     (* (sin t_0) (sqrt (+ (* (- 1.0 ux) ux) ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 3.60000005e-4

   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. Step-by-step derivation

      1. lift-*.f32 N/A

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

      2. lift-PI.f32 N/A

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

      3. add-sqr-sqrt N/A

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

      4. associate-*r* N/A

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

      5. lower-*.f32 N/A

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

      6. lift-*.f32 N/A

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

      7. associate-*l* N/A

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

      8. lower-*.f32 N/A

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

      9. lower-*.f32 N/A

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

      10. lift-PI.f32 N/A

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

      11. lower-sqrt.f32 N/A

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

      12. lift-PI.f32 N/A

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

      13. lower-sqrt.f32 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. +-commutative N/A

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

      6. lower-fma.f32 N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 N/A

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

      12. lower-pow.f32 N/A

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

      13. lower--.f32 1.0

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

   7. Applied rewrites 1.0%

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

   9. Applied rewrites 97.6%

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

   10. Taylor expanded in uy around 0

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

       1. *-commutative N/A

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

       2. lower-*.f32 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f32 N/A

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

       5. lower-PI.f32 98.6

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

   12. Applied rewrites 98.6%

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

   if 3.60000005e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 55.2%

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

   3. 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 37.0

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

   5. Applied rewrites 37.0%

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

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 36.1%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 92.8

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

   10. Applied rewrites 92.8%

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

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0003600000054575503:\\ \;\;\;\;\left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \sqrt{\left(2 - \left({\left(maxCos - 1\right)}^{2} \cdot ux - -2 \cdot maxCos\right)\right) \cdot ux}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\frac{1 - ux}{maxCos} + ux\right) \cdot ux\\ \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\left(t\_0 - \frac{1}{maxCos} \cdot t\_0\right) + \frac{ux}{maxCos}\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (+ (/ (- 1.0 ux) maxCos) ux) ux)))
   (if (<= maxCos 4.999999873689376e-6)
     (* (sin (* (* uy (PI)) 2.0)) (sqrt (+ (* (- 1.0 ux) ux) ux)))
     (*
      (sqrt
       (*
        (* maxCos maxCos)
        (-
         (/ (/ ux maxCos) maxCos)
         (+ (- t_0 (* (/ 1.0 maxCos) t_0)) (/ ux maxCos)))))
      (* (* 2.0 (PI)) uy)))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-6

   1. Initial program 56.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 49.5

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

   5. Applied rewrites 49.5%

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

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

      3. +-commutative N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.0

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

   10. Applied rewrites 98.0%

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

   if 4.99999987e-6 < maxCos

   1. Initial program 57.6%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 54.2

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

   5. Applied rewrites 54.2%

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

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in maxCos around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 54.9%

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

   12. Applied rewrites 88.8%

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

4. Final simplification 96.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(maxCos \cdot maxCos\right) \cdot \left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\left(\left(\frac{1 - ux}{maxCos} + ux\right) \cdot ux - \frac{1}{maxCos} \cdot \left(\left(\frac{1 - ux}{maxCos} + ux\right) \cdot ux\right)\right) + \frac{ux}{maxCos}\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot ux\\ \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \sqrt{t\_0 + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\left(\frac{ux}{maxCos} \cdot 2 - \frac{ux \cdot ux}{maxCos}\right) - \frac{\frac{t\_0}{maxCos}}{maxCos}\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) ux)))
   (if (<= maxCos 4.999999873689376e-6)
     (* (sin (* (* uy (PI)) 2.0)) (sqrt (+ t_0 ux)))
     (*
      (sqrt
       (*
        (-
         (/ (/ ux maxCos) maxCos)
         (-
          (- (* (/ ux maxCos) 2.0) (/ (* ux ux) maxCos))
          (/ (/ t_0 maxCos) maxCos)))
        (* maxCos maxCos)))
      (* (* 2.0 (PI)) uy)))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 4.99999987e-6

   1. Initial program 56.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 49.5

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

   5. Applied rewrites 49.5%

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

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

      3. +-commutative N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 49.8%

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

   8. Taylor expanded in maxCos around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sqrt.f32 N/A

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

      4. associate-*r* N/A

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

      5. mul-1-neg N/A

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

      6. cancel-sign-sub N/A

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

      7. lower-+.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lower--.f32 N/A

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

      11. lower-sin.f32 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 N/A

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

      16. lower-PI.f32 98.0

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

   10. Applied rewrites 98.0%

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

   if 4.99999987e-6 < maxCos

   1. Initial program 57.6%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 54.2

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

   5. Applied rewrites 54.2%

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

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in maxCos around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 54.9%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 77.7%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 4.999999873689376 \cdot 10^{-6}:\\ \;\;\;\;\sin \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right) \cdot \sqrt{\left(1 - ux\right) \cdot ux + ux}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\left(\frac{ux}{maxCos} \cdot 2 - \frac{ux \cdot ux}{maxCos}\right) - \frac{\frac{\left(1 - ux\right) \cdot ux}{maxCos}}{maxCos}\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010400000028312206:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0010400000028312206)
   (* (sqrt (+ (* (- 1.0 ux) ux) ux)) (* (* 2.0 (PI)) uy))
   (* (sqrt (* 2.0 ux)) (sin (* (* uy (PI)) 2.0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00104

   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 58.4

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

   5. Applied rewrites 58.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. 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. +-commutative N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 54.8%

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

   8. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 89.0

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

   10. Applied rewrites 89.0%

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

   if 0.00104 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 54.2%

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

   3. 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 33.9

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

   5. Applied rewrites 33.9%

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

   6. Taylor expanded in ux around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      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 46.8

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

   8. Applied rewrites 46.8%

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

   9. Taylor expanded in maxCos around 0

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

   11. Applied rewrites 46.8%

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

   12. Taylor expanded in uy around inf

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

       1. lower-sin.f32 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f32 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f32 N/A

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

       6. lower-PI.f32 75.9

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

   14. Applied rewrites 75.9%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0010400000028312206:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot ux} \cdot \sin \left(\left(uy \cdot \mathsf{PI}\left(\right)\right) \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot ux\\ t_1 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ \mathbf{if}\;maxCos \leq 1.4999999853326784 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{t\_0 + ux} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\left(\frac{ux}{maxCos} \cdot 2 - \frac{ux \cdot ux}{maxCos}\right) - \frac{\frac{t\_0}{maxCos}}{maxCos}\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) ux)) (t_1 (* (* 2.0 (PI)) uy)))
   (if (<= maxCos 1.4999999853326784e-10)
     (* (sqrt (+ t_0 ux)) t_1)
     (*
      (sqrt
       (*
        (-
         (/ (/ ux maxCos) maxCos)
         (-
          (- (* (/ ux maxCos) 2.0) (/ (* ux ux) maxCos))
          (/ (/ t_0 maxCos) maxCos)))
        (* maxCos maxCos)))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.49999999e-10

   1. Initial program 57.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 49.4

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

   5. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. 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. +-commutative N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 49.7%

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

   8. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 80.4

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

   10. Applied rewrites 80.4%

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

   if 1.49999999e-10 < maxCos

   1. Initial program 56.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 52.8

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

   5. Applied rewrites 52.8%

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

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in maxCos around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 65.6%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 80.1%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.4999999853326784 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\left(\frac{ux}{maxCos} \cdot 2 - \frac{ux \cdot ux}{maxCos}\right) - \frac{\frac{\left(1 - ux\right) \cdot ux}{maxCos}}{maxCos}\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) \cdot ux\\ t_1 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ \mathbf{if}\;maxCos \leq 1.4999999853326784 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{t\_0 + ux} \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\frac{ux}{maxCos} \cdot 2 - \frac{\frac{t\_0}{maxCos}}{maxCos}\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- 1.0 ux) ux)) (t_1 (* (* 2.0 (PI)) uy)))
   (if (<= maxCos 1.4999999853326784e-10)
     (* (sqrt (+ t_0 ux)) t_1)
     (*
      (sqrt
       (*
        (-
         (/ (/ ux maxCos) maxCos)
         (- (* (/ ux maxCos) 2.0) (/ (/ t_0 maxCos) maxCos)))
        (* maxCos maxCos)))
      t_1))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.49999999e-10

   1. Initial program 57.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 49.4

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

   5. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   6. 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. +-commutative N/A

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 49.7%

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

   8. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 80.4

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

   10. Applied rewrites 80.4%

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

   if 1.49999999e-10 < maxCos

   1. Initial program 56.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 52.8

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

   5. Applied rewrites 52.8%

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

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 42.1%

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

   8. Taylor expanded in maxCos around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 65.6%

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

   11. Taylor expanded in ux around 0

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

   13. Applied rewrites 78.9%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.4999999853326784 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\frac{\frac{ux}{maxCos}}{maxCos} - \left(\frac{ux}{maxCos} \cdot 2 - \frac{\frac{\left(1 - ux\right) \cdot ux}{maxCos}}{maxCos}\right)\right) \cdot \left(maxCos \cdot maxCos\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 79.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ \mathbf{if}\;maxCos \leq 1.1000000085914508 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{4 \cdot \left(maxCos \cdot maxCos\right) - 4}{-2 \cdot maxCos - 2} \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)))
   (if (<= maxCos 1.1000000085914508e-5)
     (* (sqrt (+ (* (- 1.0 ux) ux) ux)) t_0)
     (*
      (sqrt
       (* (/ (- (* 4.0 (* maxCos maxCos)) 4.0) (- (* -2.0 maxCos) 2.0)) ux))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.10000001e-5

   1. Initial program 57.2%

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

   3. 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 49.9

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

   5. Applied rewrites 49.9%

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

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 50.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 80.8

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

   10. Applied rewrites 80.8%

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

   if 1.10000001e-5 < maxCos

   1. Initial program 56.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in ux around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      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 48.6

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

   8. Applied rewrites 48.6%

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

   10. Applied rewrites 48.6%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\mathsf{fma}\left(maxCos \cdot maxCos, 4, -4\right) \cdot \frac{1}{\mathsf{fma}\left(-2, maxCos, -2\right)}\right) \cdot ux} \]
   11. Step-by-step derivation

   12. Applied rewrites 70.0%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\frac{4 - 4 \cdot \left(maxCos \cdot maxCos\right)}{2 - -2 \cdot maxCos} \cdot ux} \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.1000000085914508 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{4 \cdot \left(maxCos \cdot maxCos\right) - 4}{-2 \cdot maxCos - 2} \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ \mathbf{if}\;maxCos \leq 1.1000000085914508 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)))
   (if (<= maxCos 1.1000000085914508e-5)
     (* (sqrt (+ (* (- 1.0 ux) ux) ux)) t_0)
     (* (sqrt (* (+ (* -2.0 maxCos) 2.0) ux)) t_0))))

Derivation

1. Split input into 2 regimes

2. if maxCos < 1.10000001e-5

   1. Initial program 57.2%

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

   3. 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 49.9

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

   5. Applied rewrites 49.9%

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

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

      4. lift--.f32 N/A

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

      5. sub-neg N/A

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

      6. lift-neg.f32 N/A

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

      7. associate-+r+ N/A

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

      8. lift-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-+.f32 N/A

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

      13. distribute-lft-in N/A

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

      14. lower-+.f32 N/A

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

   7. Applied rewrites 50.0%

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

   8. Taylor expanded in maxCos around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. cancel-sign-sub N/A

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

      4. lower-+.f32 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f32 N/A

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

      7. lower--.f32 80.8

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

   10. Applied rewrites 80.8%

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

   if 1.10000001e-5 < maxCos

   1. Initial program 56.4%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 52.8

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

   5. Applied rewrites 52.8%

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

   6. Taylor expanded in ux around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      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 48.6

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

   8. Applied rewrites 48.6%

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

   10. Applied rewrites 70.0%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;maxCos \leq 1.1000000085914508 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(-2 \cdot maxCos + 2\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 77.2% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \end{array} \]

FPCore

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

Derivation

1. Initial program 57.0%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.3

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

5. Applied rewrites 50.3%

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

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

   4. lift--.f32 N/A

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

   5. sub-neg N/A

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

   6. lift-neg.f32 N/A

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

   7. associate-+r+ N/A

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

   8. lift-*.f32 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-+.f32 N/A

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

   13. distribute-lft-in N/A

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

   14. lower-+.f32 N/A

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

7. Applied rewrites 47.5%

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

8. Taylor expanded in maxCos around 0

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

   1. associate-*r* N/A

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

   2. mul-1-neg N/A

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

   3. cancel-sign-sub N/A

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

   4. lower-+.f32 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f32 N/A

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

   7. lower--.f32 75.8

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

10. Applied rewrites 75.8%

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

11. Final simplification 75.8%

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

Alternative 12: 63.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (+ uy uy) (PI)) (sqrt (* 2.0 ux))))

Derivation

1. Initial program 57.0%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 50.3

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

5. Applied rewrites 50.3%

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

6. Taylor expanded in ux around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   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 63.4

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

8. Applied rewrites 63.4%

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

9. Taylor expanded in maxCos around 0

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

11. Applied rewrites 63.4%

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

13. Applied rewrites 63.4%

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

14. Final simplification 63.4%

    \[\leadsto \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{2 \cdot ux} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024251 
(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)))))))