UniformSampleCone, y

Percentage Accurate: 57.7% → 98.3%

Time: 14.2s

Alternatives: 19

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: 57.7% 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 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 ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 58.8

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

5. Applied rewrites 58.8%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)}} \]
6. 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. lift-*.f32 58.8

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

   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 58.8

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

   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 58.8

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

7. Applied rewrites 58.8%

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

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. *-commutative N/A

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

   12. lower-*.f32 97.5

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

10. Applied rewrites 97.5%

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

    1. lift-*.f32 N/A

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

    2. *-commutative N/A

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

    3. lift-*.f32 N/A

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

    4. lift-*.f32 N/A

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

    5. associate-*l* N/A

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

    6. *-commutative N/A

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

    7. associate-*r* N/A

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

    8. lift-sqrt.f32 N/A

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

    9. lift-sqrt.f32 N/A

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

    10. rem-square-sqrt N/A

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

    11. associate-*r* N/A

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

    12. lower-*.f32 N/A

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

    13. lower-*.f32 98.2

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

12. Applied rewrites 98.2%

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

13. Final simplification 98.2%

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

Alternative 2: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ t_1 := \left(1 - ux\right) + maxCos \cdot ux\\ \mathbf{if}\;1 - t\_1 \cdot t\_1 \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(1 - ux\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 uy) (PI)))) (t_1 (+ (- 1.0 ux) (* maxCos ux))))
   (if (<= (- 1.0 (* t_1 t_1)) 0.00015999999595806003)
     (* (sqrt (* (* (- maxCos 1.0) ux) -2.0)) t_0)
     (* (sqrt (- 1.0 (* (- (+ (* maxCos ux) 1.0) ux) (- 1.0 ux)))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))) < 1.59999996e-4

   1. Initial program 34.2%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      7. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

      8. pow2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

      9. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

      10. rem-log-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

      11. pow-to-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

      12. pow2 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      14. lower-expm1.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   4. Applied rewrites 48.7%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]
   5. Step-by-step derivation

      1. lift-log1p.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + ux \cdot \left(-1 + maxCos\right)\right)} \cdot 2\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{ux \cdot \left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(ux \cdot -1 + ux \cdot maxCos\right)}\right) \cdot 2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \left(ux \cdot -1 + \color{blue}{ux \cdot maxCos}\right)\right) \cdot 2\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + ux \cdot -1\right) + ux \cdot maxCos\right)} \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{-1 \cdot ux}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      9. sub-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot 2\right)} \]

      10. associate-+l- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot 2\right)} \]

      11. flip3-- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\frac{{1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}}{1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)}\right)} \cdot 2\right)} \]

      12. log-div N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

      13. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

   6. Applied rewrites 91.2%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left(1 - {\left(ux - maxCos \cdot ux\right)}^{3}\right) - \mathsf{log1p}\left({\left(ux - maxCos \cdot ux\right)}^{2} + 1 \cdot \left(ux - maxCos \cdot ux\right)\right)\right)} \cdot 2\right)} \]

   7. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 93.4

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

   9. Applied rewrites 93.4%

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

   if 1.59999996e-4 < (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))

   1. Initial program 89.4%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lower-+.f32 N/A

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

      7. lower-neg.f32 N/A

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

      8. lower-+.f32 89.3

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

      9. lift-*.f32 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f32 89.3

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

   4. Applied rewrites 89.3%

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

   5. Taylor expanded in maxCos around 0

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

      1. lower--.f32 87.1

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

   7. Applied rewrites 87.1%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(\left(1 - ux\right) + maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) + maxCos \cdot ux\right) \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(maxCos \cdot ux + 1\right) - ux\right) \cdot \left(1 - ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ t_1 := \left(1 - ux\right) + maxCos \cdot ux\\ \mathbf{if}\;1 - t\_1 \cdot t\_1 \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot t\_1} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 uy) (PI)))) (t_1 (+ (- 1.0 ux) (* maxCos ux))))
   (if (<= (- 1.0 (* t_1 t_1)) 0.00015999999595806003)
     (* (sqrt (* (* (- maxCos 1.0) ux) -2.0)) t_0)
     (* (sqrt (- 1.0 (* (- 1.0 ux) t_1))) t_0))))

Derivation

1. Split input into 2 regimes

2. if (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)))) < 1.59999996e-4

   1. Initial program 34.2%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      7. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

      8. pow2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

      9. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

      10. rem-log-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

      11. pow-to-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

      12. pow2 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      14. lower-expm1.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   4. Applied rewrites 48.7%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]
   5. Step-by-step derivation

      1. lift-log1p.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + ux \cdot \left(-1 + maxCos\right)\right)} \cdot 2\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{ux \cdot \left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(ux \cdot -1 + ux \cdot maxCos\right)}\right) \cdot 2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \left(ux \cdot -1 + \color{blue}{ux \cdot maxCos}\right)\right) \cdot 2\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + ux \cdot -1\right) + ux \cdot maxCos\right)} \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{-1 \cdot ux}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      9. sub-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot 2\right)} \]

      10. associate-+l- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot 2\right)} \]

      11. flip3-- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\frac{{1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}}{1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)}\right)} \cdot 2\right)} \]

      12. log-div N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

      13. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

   6. Applied rewrites 91.2%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left(1 - {\left(ux - maxCos \cdot ux\right)}^{3}\right) - \mathsf{log1p}\left({\left(ux - maxCos \cdot ux\right)}^{2} + 1 \cdot \left(ux - maxCos \cdot ux\right)\right)\right)} \cdot 2\right)} \]

   7. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 93.4

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

   9. Applied rewrites 93.4%

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

   if 1.59999996e-4 < (-.f32 #s(literal 1 binary32) (*.f32 (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))))

   1. Initial program 89.4%

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

   3. Taylor expanded in maxCos around 0

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

      1. lower--.f32 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;1 - \left(\left(1 - ux\right) + maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) + maxCos \cdot ux\right) \leq 0.00015999999595806003:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + maxCos \cdot ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ t_1 := \left(1 - ux\right) + maxCos \cdot ux\\ \mathbf{if}\;t\_1 \leq 0.9999200105667114:\\ \;\;\;\;\sqrt{1 - t\_1 \cdot t\_1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 uy) (PI)))) (t_1 (+ (- 1.0 ux) (* maxCos ux))))
   (if (<= t_1 0.9999200105667114)
     (* (sqrt (- 1.0 (* t_1 t_1))) t_0)
     (* (sqrt (* (* (- maxCos 1.0) ux) -2.0)) t_0))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999920011

   1. Initial program 89.4%

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

   if 0.999920011 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 34.2%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      7. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

      8. pow2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

      9. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

      10. rem-log-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

      11. pow-to-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

      12. pow2 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      14. lower-expm1.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   4. Applied rewrites 48.7%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]
   5. Step-by-step derivation

      1. lift-log1p.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + ux \cdot \left(-1 + maxCos\right)\right)} \cdot 2\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{ux \cdot \left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(ux \cdot -1 + ux \cdot maxCos\right)}\right) \cdot 2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \left(ux \cdot -1 + \color{blue}{ux \cdot maxCos}\right)\right) \cdot 2\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + ux \cdot -1\right) + ux \cdot maxCos\right)} \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{-1 \cdot ux}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      9. sub-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot 2\right)} \]

      10. associate-+l- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot 2\right)} \]

      11. flip3-- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\frac{{1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}}{1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)}\right)} \cdot 2\right)} \]

      12. log-div N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

      13. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

   6. Applied rewrites 91.2%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left(1 - {\left(ux - maxCos \cdot ux\right)}^{3}\right) - \mathsf{log1p}\left({\left(ux - maxCos \cdot ux\right)}^{2} + 1 \cdot \left(ux - maxCos \cdot ux\right)\right)\right)} \cdot 2\right)} \]

   7. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 93.4

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

   9. Applied rewrites 93.4%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + maxCos \cdot ux \leq 0.9999200105667114:\\ \;\;\;\;\sqrt{1 - \left(\left(1 - ux\right) + maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) + maxCos \cdot ux\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 58.8

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

5. Applied rewrites 58.8%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)}} \]
6. 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. lift-*.f32 58.8

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

   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 58.8

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

   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 58.8

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

7. Applied rewrites 58.8%

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

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. *-commutative N/A

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

   12. lower-*.f32 97.5

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

10. Applied rewrites 97.5%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 84.5%

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

14. Final simplification 82.6%

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

Alternative 6: 90.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{if}\;ux \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(1 - ux\right) + maxCos \cdot ux\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (sin (* (* 2.0 uy) (PI)))))
   (if (<= ux 7.999999797903001e-5)
     (* (sqrt (* (* (- maxCos 1.0) ux) -2.0)) t_0)
     (*
      (sqrt
       (- 1.0 (* (+ (- 1.0 ux) (* maxCos ux)) (- 1.0 (- ux (* maxCos ux))))))
      t_0))))

Derivation

1. Split input into 2 regimes

2. if ux < 7.9999998e-5

   1. Initial program 34.2%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      7. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

      8. pow2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

      9. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

      10. rem-log-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

      11. pow-to-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

      12. pow2 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      14. lower-expm1.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   4. Applied rewrites 48.7%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]
   5. Step-by-step derivation

      1. lift-log1p.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + ux \cdot \left(-1 + maxCos\right)\right)} \cdot 2\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{ux \cdot \left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(ux \cdot -1 + ux \cdot maxCos\right)}\right) \cdot 2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \left(ux \cdot -1 + \color{blue}{ux \cdot maxCos}\right)\right) \cdot 2\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + ux \cdot -1\right) + ux \cdot maxCos\right)} \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{-1 \cdot ux}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      9. sub-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot 2\right)} \]

      10. associate-+l- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot 2\right)} \]

      11. flip3-- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\frac{{1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}}{1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)}\right)} \cdot 2\right)} \]

      12. log-div N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

      13. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

   6. Applied rewrites 91.2%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left(1 - {\left(ux - maxCos \cdot ux\right)}^{3}\right) - \mathsf{log1p}\left({\left(ux - maxCos \cdot ux\right)}^{2} + 1 \cdot \left(ux - maxCos \cdot ux\right)\right)\right)} \cdot 2\right)} \]

   7. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 93.4

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

   9. Applied rewrites 93.4%

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

   if 7.9999998e-5 < ux

   1. Initial program 89.4%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lower--.f32 N/A

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

      5. lower--.f32 89.5

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

      6. lift-*.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 89.5

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

   4. Applied rewrites 89.5%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 7.999999797903001 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(1 - ux\right) + maxCos \cdot ux\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.6% accurate, 1.0× speedup

Math

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

FPCore

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

Derivation

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 ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 58.8

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

5. Applied rewrites 58.8%

   \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)}} \]
6. 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. lift-*.f32 58.8

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

   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 58.8

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

   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 58.8

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

7. Applied rewrites 58.8%

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

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. *-commutative N/A

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

   12. lower-*.f32 97.5

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

10. Applied rewrites 97.5%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 92.6%

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

14. Final simplification 92.6%

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

Alternative 8: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(maxCos - 1\right) \cdot ux\\ \mathbf{if}\;\left(1 - ux\right) + maxCos \cdot ux \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{1 - \left(t\_0 + 1\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0 \cdot -2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (- maxCos 1.0) ux)))
   (if (<= (+ (- 1.0 ux) (* maxCos ux)) 0.9998499751091003)
     (*
      (sqrt (- 1.0 (* (+ t_0 1.0) (- 1.0 (- ux (* maxCos ux))))))
      (* (* 2.0 (PI)) uy))
     (* (sqrt (* t_0 -2.0)) (sin (* (* 2.0 uy) (PI)))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.999849975

   1. Initial program 90.0%

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

   3. Taylor expanded in 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 90.0

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

   5. Applied rewrites 90.0%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 77.2

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

   8. Applied rewrites 77.2%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lower--.f32 77.1

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

   10. Applied rewrites 77.1%

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

   12. Applied rewrites 77.2%

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

   if 0.999849975 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 35.6%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. neg-sub0 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{\left(0 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + 1} \]

      5. associate-+l- N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      6. lower--.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) - 1\right)}} \]

      7. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} - 1\right)} \]

      8. pow2 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}} - 1\right)} \]

      9. pow-to-exp N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(\color{blue}{e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}} - 1\right)} \]

      10. rem-log-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\color{blue}{\log \left(e^{\log \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot 2}\right)}} - 1\right)} \]

      11. pow-to-exp N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left({\left(\left(1 - ux\right) + ux \cdot maxCos\right)}^{2}\right)}} - 1\right)} \]

      12. pow2 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      13. lift-*.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \left(e^{\log \color{blue}{\left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} - 1\right)} \]

      14. lower-expm1.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \color{blue}{\mathsf{expm1}\left(\log \left(\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)\right)}} \]

   4. Applied rewrites 49.4%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{\color{blue}{0 - \mathsf{expm1}\left(\mathsf{log1p}\left(ux \cdot \left(-1 + maxCos\right)\right) \cdot 2\right)}} \]
   5. Step-by-step derivation

      1. lift-log1p.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\log \left(1 + ux \cdot \left(-1 + maxCos\right)\right)} \cdot 2\right)} \]

      2. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{ux \cdot \left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      3. lift-+.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + ux \cdot \color{blue}{\left(-1 + maxCos\right)}\right) \cdot 2\right)} \]

      4. distribute-lft-in N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \color{blue}{\left(ux \cdot -1 + ux \cdot maxCos\right)}\right) \cdot 2\right)} \]

      5. lift-*.f32 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(1 + \left(ux \cdot -1 + \color{blue}{ux \cdot maxCos}\right)\right) \cdot 2\right)} \]

      6. associate-+r+ N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\left(1 + ux \cdot -1\right) + ux \cdot maxCos\right)} \cdot 2\right)} \]

      7. *-commutative N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{-1 \cdot ux}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      8. neg-mul-1 N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\left(1 + \color{blue}{\left(\mathsf{neg}\left(ux\right)\right)}\right) + ux \cdot maxCos\right) \cdot 2\right)} \]

      9. sub-neg N/A

         \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot 2\right)} \]

      10. associate-+l- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot 2\right)} \]

      11. flip3-- N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\log \color{blue}{\left(\frac{{1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}}{1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)}\right)} \cdot 2\right)} \]

      12. log-div N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

      13. lower--.f32 N/A

          \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left({1}^{3} - {\left(ux - ux \cdot maxCos\right)}^{3}\right) - \log \left(1 \cdot 1 + \left(\left(ux - ux \cdot maxCos\right) \cdot \left(ux - ux \cdot maxCos\right) + 1 \cdot \left(ux - ux \cdot maxCos\right)\right)\right)\right)} \cdot 2\right)} \]

   6. Applied rewrites 90.5%

      \[\leadsto \sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{0 - \mathsf{expm1}\left(\color{blue}{\left(\log \left(1 - {\left(ux - maxCos \cdot ux\right)}^{3}\right) - \mathsf{log1p}\left({\left(ux - maxCos \cdot ux\right)}^{2} + 1 \cdot \left(ux - maxCos \cdot ux\right)\right)\right)} \cdot 2\right)} \]

   7. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower--.f32 92.7

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

   9. Applied rewrites 92.7%

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

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + maxCos \cdot ux \leq 0.9998499751091003:\\ \;\;\;\;\sqrt{1 - \left(\left(maxCos - 1\right) \cdot ux + 1\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(maxCos - 1\right) \cdot ux\right) \cdot -2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 71.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 6.799999852091787 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) + maxCos \cdot ux\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 6.799999852091787e-8)
   (* (sqrt (* (fma maxCos -2.0 2.0) ux)) (sin (* (* 2.0 uy) (PI))))
   (*
    (sqrt
     (-
      1.0
      (* (* (- (+ (/ 1.0 ux) maxCos) 1.0) ux) (+ (- 1.0 ux) (* maxCos ux)))))
    (* (* 2.0 (PI)) uy))))

Derivation

1. Split input into 2 regimes

2. if ux < 6.79999985e-8

   1. Initial program 12.0%

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

   3. Taylor expanded in ux around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f32 39.4

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

   5. Applied rewrites 39.4%

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

   if 6.79999985e-8 < ux

   1. Initial program 74.9%

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

   3. Taylor expanded in 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 75.2

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

   5. Applied rewrites 75.2%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 65.8

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

   8. Applied rewrites 65.8%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 6.799999852091787 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(maxCos, -2, 2\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right) \cdot \left(\left(1 - ux\right) + maxCos \cdot ux\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.2% accurate, 2.3× speedup

Math

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

FPCore

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

Derivation

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 ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 58.8

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

5. Applied rewrites 58.8%

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

6. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 52.0

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

8. Applied rewrites 52.0%

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

9. Final simplification 52.0%

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

Alternative 11: 52.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := \left(maxCos - 1\right) \cdot ux + 1\\ \mathbf{if}\;ux \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), t\_1, 1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - t\_1 \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)) (t_1 (+ (* (- maxCos 1.0) ux) 1.0)))
   (if (<= ux 3.999999975690116e-8)
     (* (sqrt (fma (- ux (fma maxCos ux 1.0)) t_1 1.0)) t_0)
     (* (sqrt (- 1.0 (* t_1 (- 1.0 (- ux (* maxCos ux)))))) t_0))))

Derivation

1. Split input into 2 regimes

2. if ux < 3.99999998e-8

   1. Initial program 7.8%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 11.5

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

   5. Applied rewrites 11.5%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 11.5

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

   8. Applied rewrites 11.5%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f32 N/A

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

   10. Applied rewrites 16.1%

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

   if 3.99999998e-8 < ux

   1. Initial program 73.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 64.4

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

   8. Applied rewrites 64.4%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lower--.f32 64.4

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

   10. Applied rewrites 64.4%

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

   12. Applied rewrites 64.4%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \left(maxCos - 1\right) \cdot ux + 1, 1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(\left(maxCos - 1\right) \cdot ux + 1\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ t_1 := \left(1 - ux\right) + maxCos \cdot ux\\ \mathbf{if}\;ux \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \left(maxCos - 1\right) \cdot ux + 1, 1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - t\_1 \cdot t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)) (t_1 (+ (- 1.0 ux) (* maxCos ux))))
   (if (<= ux 3.999999975690116e-8)
     (*
      (sqrt (fma (- ux (fma maxCos ux 1.0)) (+ (* (- maxCos 1.0) ux) 1.0) 1.0))
      t_0)
     (* t_0 (sqrt (- 1.0 (* t_1 t_1)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 3.99999998e-8

   1. Initial program 7.8%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 11.5

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

   5. Applied rewrites 11.5%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 11.5

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

   8. Applied rewrites 11.5%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f32 N/A

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

   10. Applied rewrites 16.1%

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

   if 3.99999998e-8 < ux

   1. Initial program 73.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 64.3

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

   5. Applied rewrites 64.3%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \left(maxCos - 1\right) \cdot ux + 1, 1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + maxCos \cdot ux\right) \cdot \left(\left(1 - ux\right) + maxCos \cdot ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\\ \mathbf{if}\;ux \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \left(maxCos - 1\right) \cdot ux + 1, 1\right)} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* 2.0 (PI)) uy)))
   (if (<= ux 3.999999975690116e-8)
     (*
      (sqrt (fma (- ux (fma maxCos ux 1.0)) (+ (* (- maxCos 1.0) ux) 1.0) 1.0))
      t_0)
     (* (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 (- ux (* maxCos ux)))))) t_0))))

Derivation

1. Split input into 2 regimes

2. if ux < 3.99999998e-8

   1. Initial program 7.8%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 11.5

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

   5. Applied rewrites 11.5%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 11.5

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

   8. Applied rewrites 11.5%

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

      1. lift--.f32 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-*.f32 N/A

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

      5. lift-+.f32 N/A

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

      6. lift--.f32 N/A

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

      7. lift-*.f32 N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f32 N/A

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

   10. Applied rewrites 16.1%

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

   if 3.99999998e-8 < ux

   1. Initial program 73.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 73.3

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

   5. Applied rewrites 73.3%

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

   6. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 64.4

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

   8. Applied rewrites 64.4%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lower--.f32 64.4

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

   10. Applied rewrites 64.4%

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

   11. Taylor expanded in maxCos around 0

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

   13. Applied rewrites 63.3%

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

4. Final simplification 52.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 3.999999975690116 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(ux - \mathsf{fma}\left(maxCos, ux, 1\right), \left(maxCos - 1\right) \cdot ux + 1, 1\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \left(1 - ux\right) \cdot \left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.2% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

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 ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

   3. lower--.f32 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f32 N/A

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

   6. lower-/.f32 58.8

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

5. Applied rewrites 58.8%

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

6. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 52.0

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

8. Applied rewrites 52.0%

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

   1. lift-+.f32 N/A

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

   2. lift--.f32 N/A

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

   3. associate-+l- N/A

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

   4. lift-*.f32 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f32 N/A

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

   7. lift--.f32 N/A

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

   8. lower--.f32 52.0

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

10. Applied rewrites 52.0%

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

11. Taylor expanded in maxCos around 0

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

13. Applied rewrites 50.3%

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

14. Final simplification 50.3%

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

Alternative 15: 49.2% accurate, 3.3× speedup

Math

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

FPCore

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

Derivation

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 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in maxCos around 0

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

   1. lower--.f32 50.3

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

8. Applied rewrites 50.3%

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

9. Final simplification 50.3%

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

Alternative 16: 25.8% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

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 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 25.7%

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

9. Final simplification 25.7%

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

Alternative 17: 20.2% accurate, 3.5× speedup

Math

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

FPCore

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

Derivation

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 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in maxCos around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. lower-*.f32 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f32 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f32 20.6

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

8. Applied rewrites 20.6%

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

9. Final simplification 20.6%

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

Alternative 18: 63.2% accurate, 4.2× speedup

Math

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

FPCore

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

Derivation

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 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around 0

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

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

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

   2. metadata-eval N/A

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

   3. *-commutative N/A

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

   4. lower-*.f32 N/A

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

   5. +-commutative N/A

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

   6. lower-fma.f32 63.6

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

8. Applied rewrites 63.6%

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

9. Final simplification 63.6%

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

Alternative 19: 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 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 51.1

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

5. Applied rewrites 51.1%

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

6. Taylor expanded in ux around 0

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

8. Applied rewrites 7.1%

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

9. Final simplification 7.1%

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

Reproduce

herbie shell --seed 2024243 
(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)))))))