UniformSampleCone, y

Percentage Accurate: 57.8% → 98.3%

Time: 10.9s

Alternatives: 8

Speedup: 4.2×

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: 57.8% 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

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

FPCore

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

Derivation

1. Initial program 58.8%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 59.6

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

5. Applied rewrites 59.6%

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

   1. lift-*.f32 N/A

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

   2. rem-square-sqrt N/A

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

   3. lift-sqrt.f32 N/A

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

   4. lift-sqrt.f32 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f32 N/A

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

   7. associate-*r* N/A

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

   8. lift-*.f32 N/A

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

   9. lift-*.f32 N/A

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

   10. lower-*.f32 59.5

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f32 59.5

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

   14. lift-*.f32 N/A

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

   15. *-commutative N/A

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

   16. lower-*.f32 59.5

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

7. Applied rewrites 59.5%

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

8. Taylor expanded in ux around 0

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. mul-1-neg N/A

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

   5. unsub-neg N/A

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

   6. lower--.f32 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f32 N/A

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

   9. lower-pow.f32 N/A

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

   10. lower--.f32 N/A

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

   11. lower-*.f32 97.6

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

10. Applied rewrites 97.6%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. associate-*r* N/A

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

    5. lift-*.f32 N/A

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

    6. associate-*l* N/A

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

    7. lift-sqrt.f32 N/A

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

    8. lift-sqrt.f32 N/A

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

    9. rem-square-sqrt N/A

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

    10. associate-*l* N/A

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

    11. *-commutative N/A

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

    12. associate-*r* N/A

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

    13. lower-*.f32 N/A

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

    14. lower-*.f32 98.5

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

12. Applied rewrites 98.5%

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

13. Final simplification 98.5%

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

Alternative 2: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{\left(\left(2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0006500000017695129)
   (*
    (sqrt (* (- (- 2.0 (* (pow (- maxCos 1.0) 2.0) ux)) (* maxCos 2.0)) ux))
    (* (* 2.0 (PI)) uy))
   (* (sqrt (+ (* (- 1.0 ux) ux) ux)) (sin (* 2.0 (* uy (PI)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 6.50000002e-4

   1. Initial program 59.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

   7. Applied rewrites 37.3%

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

   8. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. *-commutative N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. sub-neg N/A

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

      9. unpow2 N/A

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

      10. associate-*r* N/A

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

      11. lower-*.f32 N/A

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

   10. Applied rewrites 98.3%

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

   if 6.50000002e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 40.6

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

   5. Applied rewrites 40.6%

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

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

   7. Applied rewrites 18.9%

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

   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 92.5

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

   10. Applied rewrites 92.5%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{\left(\left(2 - {\left(maxCos - 1\right)}^{2} \cdot ux\right) - maxCos \cdot 2\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* 2.0 uy) 0.0006500000017695129)
   (*
    (sqrt
     (-
      (* (* ux ux) (- (/ 2.0 ux) (- (/ maxCos ux) (- maxCos 1.0))))
      (* (* (fma maxCos ux (- 1.0 ux)) maxCos) ux)))
    (* (* 2.0 (PI)) uy))
   (* (sqrt (+ (* (- 1.0 ux) ux) ux)) (sin (* 2.0 (* uy (PI)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 6.50000002e-4

   1. Initial program 59.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

   7. Applied rewrites 35.5%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 97.6

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

   10. Applied rewrites 97.6%

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

   if 6.50000002e-4 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 57.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 40.6

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

   5. Applied rewrites 40.6%

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

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

      14. *-commutative N/A

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

      15. lower-fma.f32 N/A

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

   7. Applied rewrites 18.9%

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

   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 92.5

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

   10. Applied rewrites 92.5%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.0006500000017695129:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(1 - ux\right) \cdot ux + ux} \cdot \sin \left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 88.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \sin t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)))
   (if (<= (* 2.0 uy) 0.03999999910593033)
     (*
      (sqrt
       (-
        (* (* ux ux) (- (/ 2.0 ux) (- (/ maxCos ux) (- maxCos 1.0))))
        (* (* (fma maxCos ux (- 1.0 ux)) maxCos) ux)))
      t_0)
     (* (sqrt (* (fma -2.0 maxCos 2.0) ux)) (sin t_0)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.0399999991

   1. Initial program 58.9%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 56.9

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

   5. Applied rewrites 56.9%

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

      1. lift--.f32 N/A

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

      2. lift-*.f32 N/A

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

      3. lift-+.f32 N/A

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

      4. distribute-rgt-in N/A

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

      5. associate--r+ N/A

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

      6. lower--.f32 N/A

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

      7. lower--.f32 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f32 N/A

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

      10. lift-+.f32 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f32 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f32 N/A

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

      15. *-commutative N/A

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

      16. lift-*.f32 N/A

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

      17. *-commutative N/A

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

   7. Applied rewrites 29.2%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f32 N/A

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

      7. +-commutative N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f32 N/A

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

      11. lower-/.f32 N/A

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

      12. lower--.f32 N/A

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

      13. unpow2 N/A

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

      14. lower-*.f32 90.5

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

   10. Applied rewrites 89.9%

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

   if 0.0399999991 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 58.3%

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

   3. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower--.f32 N/A

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

      4. +-commutative N/A

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

      5. lower-+.f32 N/A

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

      6. lower-/.f32 59.3

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

   5. Applied rewrites 59.3%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-sin.f32 N/A

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

      6. lower-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. lower-PI.f32 N/A

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

      10. lower-sqrt.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-commutative N/A

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

      14. lower-*.f32 N/A

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

      15. +-commutative N/A

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

      16. lower-fma.f32 7.0

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

   8. Applied rewrites 7.0%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;2 \cdot uy \leq 0.03999999910593033:\\ \;\;\;\;\sqrt{\left(ux \cdot ux\right) \cdot \left(\frac{2}{ux} - \left(\frac{maxCos}{ux} - \left(maxCos - 1\right)\right)\right) - \left(\mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot maxCos\right) \cdot ux} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux} \cdot \sin \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.8% accurate, 1.8× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.8%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 52.2

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

5. Applied rewrites 52.2%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

7. Applied rewrites 30.6%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. associate-*r/ N/A

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

   5. metadata-eval N/A

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

   6. lower-/.f32 N/A

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

   7. +-commutative N/A

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

   8. mul-1-neg N/A

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

   9. unsub-neg N/A

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

   10. lower--.f32 N/A

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

   11. lower-/.f32 N/A

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

   12. lower--.f32 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f32 81.2

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

10. Applied rewrites 81.0%

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

11. Final simplification 81.2%

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

Alternative 6: 73.3% accurate, 1.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.8%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 52.2

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

5. Applied rewrites 52.2%

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

   1. lift--.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. lift-+.f32 N/A

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

   4. distribute-rgt-in N/A

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

   5. associate--r+ N/A

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

   6. lower--.f32 N/A

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

   7. lower--.f32 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f32 N/A

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

   10. lift-+.f32 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f32 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f32 N/A

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

   15. *-commutative N/A

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

   16. lift-*.f32 N/A

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

   17. *-commutative N/A

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

7. Applied rewrites 28.3%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 77.5%

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

11. Final simplification 77.5%

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

Alternative 7: 77.1% 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 58.8%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 52.2

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

5. Applied rewrites 52.2%

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

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

   14. *-commutative N/A

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

   15. lower-fma.f32 N/A

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

7. Applied rewrites 20.1%

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

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 77.6

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

10. Applied rewrites 77.6%

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

11. Final simplification 77.6%

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

Alternative 8: 7.1% accurate, 5.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 58.8%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
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 52.2

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

5. Applied rewrites 52.2%

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

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

   14. *-commutative N/A

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

   15. lower-fma.f32 N/A

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

7. Applied rewrites 20.1%

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

8. Taylor expanded in ux around 0

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

10. Applied rewrites 7.1%

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

11. Final simplification 7.1%

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

Reproduce

herbie shell --seed 2024295 
(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)))))))