UniformSampleCone, y

Percentage Accurate: 57.0% → 98.1%

Time: 12.1s

Alternatives: 19

Speedup: 5.0×

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: 57.0% 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: 98.1% accurate, 0.4× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

   1. lift-*.f32 N/A

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

   2. sqr-abs-rev N/A

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

   3. lift-+.f32 N/A

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

   4. lift--.f32 N/A

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

   5. associate-+l- N/A

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

   6. fabs-sub N/A

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

   7. lift-+.f32 N/A

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

   8. lift--.f32 N/A

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

   9. associate-+l- N/A

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

   10. fabs-sub N/A

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

   11. sqr-abs N/A

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

   12. lower-*.f32 N/A

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

   13. lower--.f32 N/A

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

   14. lower--.f32 N/A

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

   15. lift-*.f32 N/A

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

   16. *-commutative N/A

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

   17. lower-*.f32 N/A

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

   18. lower--.f32 N/A

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

   19. lower--.f32 61.1

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

   20. lift-*.f32 N/A

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

   21. *-commutative N/A

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

   22. lower-*.f32 61.1

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

4. Applied rewrites 61.1%

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

5. Taylor expanded in maxCos around 0

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

   1. lower--.f32 59.3

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

7. Applied rewrites 59.3%

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

   1. lift-sin.f32 N/A

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

   2. lift-*.f32 N/A

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

   3. *-commutative N/A

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

   4. lift-*.f32 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f32 N/A

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

   7. *-commutative N/A

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

   8. sin-2 N/A

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

   9. lower-*.f32 N/A

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

   10. lower-*.f32 N/A

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

   11. lower-sin.f32 N/A

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

   12. lower-cos.f32 59.3

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

9. Applied rewrites 59.3%

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

10. Taylor expanded in ux around inf

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

    1. *-commutative N/A

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

    2. lower-*.f32 N/A

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

    3. lower--.f32 N/A

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

    4. *-commutative N/A

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

    5. lower-*.f32 N/A

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

    6. lower-/.f32 N/A

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

    7. lower--.f32 N/A

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

    8. lower-pow.f32 N/A

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

    9. lower--.f32 N/A

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

    10. unpow2 N/A

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

    11. lower-*.f32 98.2

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

12. Applied rewrites 98.2%

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

Alternative 2: 51.6% accurate, 0.9× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 52.7

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

5. Applied rewrites 52.7%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-fma.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 51.6

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

   16. lift-+.f32 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lift-fma.f32 51.4

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

7. Applied rewrites 50.8%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 77.8%

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

11. Taylor expanded in uy around inf

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

    1. count-2-rev N/A

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

    2. distribute-lft-in N/A

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

    3. count-2-rev N/A

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

    4. lower-sin.f32 N/A

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

    5. count-2-rev N/A

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

    6. distribute-lft-in N/A

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

    7. count-2-rev N/A

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

    8. *-commutative N/A

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

    9. lower-*.f32 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f32 N/A

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

    12. lower-PI.f32 69.7

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

13. Applied rewrites 69.9%

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

14. Final simplification 69.2%

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

Alternative 3: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011800000356743112:\\ \;\;\;\;\sqrt{1 - maxCos} \cdot \left(\sqrt{ux} \cdot \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011800000356743112)
   (*
    (sqrt (- 1.0 maxCos))
    (* (sqrt ux) (* (sqrt 2.0) (sin (* (* 2.0 uy) (PI))))))
   (*
    (sin (* (* uy 2.0) (PI)))
    (sqrt
     (-
      1.0
      (*
       (+ (- 1.0 ux) (* ux maxCos))
       (* (- (+ (/ 1.0 ux) maxCos) 1.0) ux)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.18000004e-4

   1. Initial program 39.0%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 39.0

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 39.0

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. lower-*.f32 N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 91.4

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

   7. Applied rewrites 91.4%

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

   9. Applied rewrites 91.4%

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

   if 1.18000004e-4 < ux

   1. Initial program 89.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 89.6

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

   5. Applied rewrites 89.6%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011800000356743112:\\ \;\;\;\;\sqrt{1 - maxCos} \cdot \left(\sqrt{ux} \cdot \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(\left(\frac{1}{ux} + maxCos\right) - 1\right) \cdot ux\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\sqrt{1 - maxCos} \cdot \left(\sqrt{ux} \cdot \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011999999696854502)
   (*
    (sqrt (- 1.0 maxCos))
    (* (sqrt ux) (* (sqrt 2.0) (sin (* (* 2.0 uy) (PI))))))
   (*
    (sin (* (* uy 2.0) (PI)))
    (sqrt
     (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (+ (- 1.0 ux) (* ux maxCos))))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.19999997e-4

   1. Initial program 39.2%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 39.2

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 39.2

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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. lower-*.f32 N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 91.2

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

   7. Applied rewrites 91.2%

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

   9. Applied rewrites 91.3%

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

   if 1.19999997e-4 < ux

   1. Initial program 89.5%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 89.7

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

      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 89.7

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

   4. Applied rewrites 89.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\sqrt{1 - maxCos} \cdot \left(\sqrt{ux} \cdot \left(\sqrt{2} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= ux 0.00011999999696854502)
   (* (* (sqrt (* (- 1.0 maxCos) ux)) (sin (* (* 2.0 uy) (PI)))) (sqrt 2.0))
   (*
    (sin (* (* uy 2.0) (PI)))
    (sqrt
     (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (+ (- 1.0 ux) (* ux maxCos))))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.19999997e-4

   1. Initial program 39.2%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 39.2

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 39.2

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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. lower-*.f32 N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 91.2

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

   7. Applied rewrites 91.2%

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

   9. Applied rewrites 91.3%

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

   if 1.19999997e-4 < ux

   1. Initial program 89.5%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 89.7

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

      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 89.7

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

   4. Applied rewrites 89.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(ux - maxCos \cdot ux\right) - 1\\ \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - t\_0 \cdot t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (- (- ux (* maxCos ux)) 1.0)))
   (if (<= ux 0.00011999999696854502)
     (* (* (sqrt (* (- 1.0 maxCos) ux)) (sin (* (* 2.0 uy) (PI)))) (sqrt 2.0))
     (* (sin (* (+ uy uy) (PI))) (sqrt (- 1.0 (* t_0 t_0)))))))

Derivation

1. Split input into 2 regimes

2. if ux < 1.19999997e-4

   1. Initial program 39.2%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 39.2

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 39.2

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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. lower-*.f32 N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 91.2

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

   7. Applied rewrites 91.2%

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

   9. Applied rewrites 91.3%

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

   if 1.19999997e-4 < ux

   1. Initial program 89.5%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 89.7

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 89.7

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

   4. Applied rewrites 89.7%

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

      1. lift-*.f32 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f32 89.7

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

   6. Applied rewrites 89.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ux \leq 0.00011999999696854502:\\ \;\;\;\;\left(\sqrt{\left(1 - maxCos\right) \cdot ux} \cdot \sin \left(\left(2 \cdot uy\right) \cdot \mathsf{PI}\left(\right)\right)\right) \cdot \sqrt{2}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy + uy\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(ux - maxCos \cdot ux\right) - 1\right) \cdot \left(\left(ux - maxCos \cdot ux\right) - 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - maxCos \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.006000000052154064)
   (*
    (* (* (PI) 2.0) uy)
    (sqrt
     (*
      (-
       (/ 2.0 ux)
       (- (* (+ -1.0 maxCos) (- maxCos 1.0)) (* -2.0 (/ maxCos ux))))
      (* ux ux))))
   (* (* (sqrt 2.0) (sin (* (* (PI) uy) 2.0))) (sqrt (- ux (* maxCos ux))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00600000005

   1. Initial program 60.4%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 58.0

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

      16. lift-+.f32 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lift-fma.f32 57.8

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 91.1%

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

   12. Applied rewrites 95.4%

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

   if 0.00600000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 63.0%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 63.2

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. lower-*.f32 N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 74.0

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

   7. Applied rewrites 74.0%

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

   8. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

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

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

      16. distribute-rgt-in N/A

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

      17. *-lft-identity N/A

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

      18. mul-1-neg N/A

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

      19. fp-cancel-sub-sign N/A

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

      20. lower--.f32 N/A

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

      21. lower-*.f32 74.0

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

   10. Applied rewrites 74.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{ux - maxCos \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.006000000052154064)
   (*
    (* (* (PI) 2.0) uy)
    (sqrt
     (*
      (-
       (/ 2.0 ux)
       (- (* (+ -1.0 maxCos) (- maxCos 1.0)) (* -2.0 (/ maxCos ux))))
      (* ux ux))))
   (* (* (sqrt 2.0) (sin (* (* (PI) uy) 2.0))) (sqrt (* (- 1.0 maxCos) ux)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00600000005

   1. Initial program 60.4%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 58.0

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

      16. lift-+.f32 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lift-fma.f32 57.8

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

   7. Applied rewrites 56.6%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 91.1%

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

   12. Applied rewrites 95.4%

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

   if 0.00600000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 63.0%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 63.2

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

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

      14. *-lft-identity N/A

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

      15. metadata-eval N/A

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

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

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

      17. lower-*.f32 N/A

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

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

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

      19. metadata-eval N/A

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

      20. *-lft-identity N/A

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

      21. lower--.f32 74.0

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

   7. Applied rewrites 74.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 90.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.006000000052154064)
   (*
    (* (* (PI) 2.0) uy)
    (sqrt
     (*
      (-
       (/ 2.0 ux)
       (- (* (+ -1.0 maxCos) (- maxCos 1.0)) (* -2.0 (/ maxCos ux))))
      (* ux ux))))
   (* (sin (* (* uy 2.0) (PI))) (sqrt (* ux (* 2.0 (- 1.0 maxCos)))))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00600000005

   1. Initial program 60.4%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 58.0

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

      16. lift-+.f32 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lift-fma.f32 57.8

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

   7. Applied rewrites 57.8%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 91.1%

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

   12. Applied rewrites 95.4%

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

   if 0.00600000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 63.0%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 63.2

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 63.2

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

   4. Applied rewrites 63.2%

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

   5. Taylor expanded in ux around 0

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

      1. count-2-rev N/A

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

      2. distribute-lft-out N/A

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

      3. count-2-rev N/A

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

      4. metadata-eval N/A

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

      5. lower-*.f32 N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. metadata-eval N/A

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

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

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

      10. lower-*.f32 N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f32 73.8

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

   7. Applied rewrites 73.8%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{ux \cdot \left(2 \cdot \left(1 - maxCos\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 89.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (* uy 2.0) 0.006000000052154064)
   (*
    (* (* (PI) 2.0) uy)
    (sqrt
     (*
      (-
       (/ 2.0 ux)
       (- (* (+ -1.0 maxCos) (- maxCos 1.0)) (* -2.0 (/ maxCos ux))))
      (* ux ux))))
   (* (sin (* (* (PI) uy) 2.0)) (sqrt (* (fma -2.0 maxCos 2.0) ux)))))

Derivation

1. Split input into 2 regimes

2. if (*.f32 uy #s(literal 2 binary32)) < 0.00600000005

   1. Initial program 60.4%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 59.3

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

   5. Applied rewrites 59.3%

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

      1. lift-*.f32 N/A

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

      2. lift-+.f32 N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. lift-*.f32 N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f32 N/A

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

      8. lower-*.f32 N/A

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

      9. lift-+.f32 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f32 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f32 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f32 58.0

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

      16. lift-+.f32 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f32 N/A

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

      19. *-commutative N/A

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

      20. lift-fma.f32 57.8

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

   7. Applied rewrites 57.4%

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

   8. Taylor expanded in ux around inf

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

   10. Applied rewrites 91.1%

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

   12. Applied rewrites 95.4%

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

   if 0.00600000005 < (*.f32 uy #s(literal 2 binary32))

   1. Initial program 63.0%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 64.6

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

   5. Applied rewrites 64.6%

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

   6. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. lower-sin.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f32 N/A

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

      8. lower-PI.f32 N/A

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

      9. lower-sqrt.f32 N/A

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

      10. metadata-eval N/A

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

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

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

      12. *-commutative N/A

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

      13. lower-*.f32 N/A

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

      14. +-commutative N/A

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

      15. lower-fma.f32 8.7

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

   8. Applied rewrites 8.7%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;uy \cdot 2 \leq 0.006000000052154064:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - \left(\left(-1 + maxCos\right) \cdot \left(maxCos - 1\right) - -2 \cdot \frac{maxCos}{ux}\right)\right) \cdot \left(ux \cdot ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 81.1% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 52.7

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

5. Applied rewrites 52.7%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-fma.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 51.6

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

   16. lift-+.f32 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lift-fma.f32 51.4

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

7. Applied rewrites 51.1%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 77.8%

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

12. Applied rewrites 81.1%

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

13. Final simplification 81.1%

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

Alternative 12: 45.7% accurate, 2.0× speedup

Math

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

FPCore

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

Derivation

1. Initial program 61.0%

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

3. Taylor expanded in uy around 0

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

   1. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f32 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f32 N/A

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

   6. lower-PI.f32 52.7

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

5. Applied rewrites 52.7%

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

   1. lift-*.f32 N/A

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

   2. lift-+.f32 N/A

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

   3. distribute-rgt-in N/A

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

   4. +-commutative N/A

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

   5. lift-*.f32 N/A

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

   6. associate-*l* N/A

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

   7. lower-fma.f32 N/A

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

   8. lower-*.f32 N/A

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

   9. lift-+.f32 N/A

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

   10. +-commutative N/A

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

   11. lift-*.f32 N/A

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

   12. *-commutative N/A

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

   13. lift-fma.f32 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f32 51.6

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

   16. lift-+.f32 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f32 N/A

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

   19. *-commutative N/A

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

   20. lift-fma.f32 51.4

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

7. Applied rewrites 50.8%

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

8. Taylor expanded in ux around inf

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

   1. *-commutative N/A

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

   2. lower-*.f32 N/A

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

10. Applied rewrites 77.8%

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

11. Final simplification 77.8%

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

Alternative 13: 75.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9994000196456909:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (if (<= (+ (- 1.0 ux) (* ux maxCos)) 0.9994000196456909)
   (*
    (* (* (PI) 2.0) uy)
    (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (- 1.0 ux)))))
   (* (* (* (* (sqrt 2.0) (PI)) uy) 2.0) (sqrt (* (- 1.0 maxCos) ux)))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99940002

   1. Initial program 91.7%

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

   3. Taylor expanded in uy around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f32 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f32 N/A

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

      6. lower-PI.f32 76.4

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

   5. Applied rewrites 76.4%

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

   6. Taylor expanded in maxCos around 0

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

      1. lower--.f32 74.6

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

   8. Applied rewrites 74.6%

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

      1. lift-+.f32 N/A

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

      2. lift--.f32 N/A

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

      3. associate-+l- N/A

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

      4. lift-*.f32 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f32 N/A

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

      7. lift--.f32 N/A

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

      8. lower--.f32 74.6

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

   10. Applied rewrites 74.6%

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

   if 0.99940002 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 42.3%

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

      1. lift-*.f32 N/A

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

      2. sqr-abs-rev N/A

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

      3. lift-+.f32 N/A

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

      4. lift--.f32 N/A

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

      5. associate-+l- N/A

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

      6. fabs-sub N/A

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

      7. lift-+.f32 N/A

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

      8. lift--.f32 N/A

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

      9. associate-+l- N/A

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

      10. fabs-sub N/A

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

      11. sqr-abs N/A

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

      12. lower-*.f32 N/A

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

      13. lower--.f32 N/A

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

      14. lower--.f32 N/A

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

      15. lift-*.f32 N/A

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

      16. *-commutative N/A

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

      17. lower-*.f32 N/A

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

      18. lower--.f32 N/A

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

      19. lower--.f32 42.4

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

      20. lift-*.f32 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f32 42.4

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

   4. Applied rewrites 42.4%

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

   5. Taylor expanded in ux around 0

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

      1. *-commutative N/A

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

      2. lower-*.f32 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f32 N/A

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

      5. lower-sqrt.f32 N/A

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

      6. lower-sin.f32 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f32 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f32 N/A

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

      11. lower-PI.f32 N/A

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

      12. lower-sqrt.f32 N/A

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

      13. *-commutative N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]

      14. *-lft-identity N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{1 \cdot maxCos}\right) \cdot ux} \]

      15. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{\left(\mathsf{neg}\left(-1\right)\right)} \cdot maxCos\right) \cdot ux} \]

      16. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right)} \cdot ux} \]

      17. lower-*.f32 N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 + -1 \cdot maxCos\right) \cdot ux}} \]

      18. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - \left(\mathsf{neg}\left(-1\right)\right) \cdot maxCos\right)} \cdot ux} \]

      19. metadata-eval N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{1} \cdot maxCos\right) \cdot ux} \]

      20. *-lft-identity N/A

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - \color{blue}{maxCos}\right) \cdot ux} \]

      21. lower--.f32 89.5

          \[\leadsto \left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right)} \cdot ux} \]

   7. Applied rewrites 89.5%

      \[\leadsto \color{blue}{\left(\sqrt{2} \cdot \sin \left(\left(\mathsf{PI}\left(\right) \cdot uy\right) \cdot 2\right)\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}} \]

   8. Taylor expanded in uy around 0

      \[\leadsto \left(2 \cdot \left(uy \cdot \left(\mathsf{PI}\left(\right) \cdot \sqrt{2}\right)\right)\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]
   9. Step-by-step derivation

   10. Applied rewrites 76.2%

       \[\leadsto \left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\color{blue}{\left(1 - maxCos\right) \cdot ux}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9994000196456909:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt{2} \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right) \cdot 2\right) \cdot \sqrt{\left(1 - maxCos\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 75.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9994000196456909:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= (+ (- 1.0 ux) (* ux maxCos)) 0.9994000196456909)
     (* t_0 (sqrt (- 1.0 (* (- 1.0 (- ux (* maxCos ux))) (- 1.0 ux)))))
     (* t_0 (sqrt (* (- 2.0 (* maxCos 2.0)) ux))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99940002

   1. Initial program 91.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 76.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 74.6

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 74.6%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   9. Step-by-step derivation

      1. lift-+.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(1 - ux\right)} \]

      2. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(1 - ux\right)} + ux \cdot maxCos\right) \cdot \left(1 - ux\right)} \]

      3. associate-+l- N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - ux \cdot maxCos\right)\right)} \cdot \left(1 - ux\right)} \]

      4. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{ux \cdot maxCos}\right)\right) \cdot \left(1 - ux\right)} \]

      5. *-commutative N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(1 - ux\right)} \]

      6. lift-*.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - \color{blue}{maxCos \cdot ux}\right)\right) \cdot \left(1 - ux\right)} \]

      7. lift--.f32 N/A

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \color{blue}{\left(ux - maxCos \cdot ux\right)}\right) \cdot \left(1 - ux\right)} \]

      8. lower--.f32 74.6

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(1 - ux\right)} \]

   10. Applied rewrites 74.6%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - \left(ux - maxCos \cdot ux\right)\right)} \cdot \left(1 - ux\right)} \]

   if 0.99940002 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 42.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 38.2

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. 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 37.5

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 37.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   9. Taylor expanded in ux around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

       4. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

       6. lower-fma.f32 72.9

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

   11. Applied rewrites 72.7%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.0%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9994000196456909:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - \left(ux - maxCos \cdot ux\right)\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 75.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 - ux\right) + ux \cdot maxCos\\ t_1 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;t\_0 \leq 0.9994000196456909:\\ \;\;\;\;t\_1 \cdot \sqrt{1 - t\_0 \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_1 \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (+ (- 1.0 ux) (* ux maxCos))) (t_1 (* (* (PI) 2.0) uy)))
   (if (<= t_0 0.9994000196456909)
     (* t_1 (sqrt (- 1.0 (* t_0 (- 1.0 ux)))))
     (* t_1 (sqrt (* (- 2.0 (* maxCos 2.0)) ux))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99940002

   1. Initial program 91.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 76.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 74.6

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 74.6%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   if 0.99940002 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 42.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 38.2

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. 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 37.5

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 37.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   9. Taylor expanded in ux around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

       4. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

       6. lower-fma.f32 72.9

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

   11. Applied rewrites 72.7%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.0%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9994000196456909:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 75.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\\ \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9994000196456909:\\ \;\;\;\;t\_0 \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux}\\ \end{array} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (let* ((t_0 (* (* (PI) 2.0) uy)))
   (if (<= (+ (- 1.0 ux) (* ux maxCos)) 0.9994000196456909)
     (* t_0 (sqrt (- 1.0 (* (- 1.0 ux) (- 1.0 ux)))))
     (* t_0 (sqrt (* (- 2.0 (* maxCos 2.0)) ux))))))

Derivation

1. Split input into 2 regimes

2. if (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos)) < 0.99940002

   1. Initial program 91.7%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in uy around 0

      \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      3. lower-*.f32 N/A

         \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      5. lower-*.f32 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

      6. lower-PI.f32 76.4

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]
   7. Step-by-step derivation

      1. lower--.f32 74.6

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 74.6%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   9. Taylor expanded in maxCos around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]
   10. Step-by-step derivation

       1. lower--.f32 74.4

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]

   11. Applied rewrites 74.4%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(1 - ux\right)} \cdot \left(1 - ux\right)} \]

   if 0.99940002 < (+.f32 (-.f32 #s(literal 1 binary32) ux) (*.f32 ux maxCos))

   1. Initial program 42.3%

      \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
   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 38.2

         \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. 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 37.5

         \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   8. Applied rewrites 37.5%

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

   9. Taylor expanded in ux around 0

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
   10. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

       2. fp-cancel-sign-sub-inv N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

       3. *-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

       4. lower-*.f32 N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

       5. +-commutative N/A

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

       6. lower-fma.f32 72.9

          \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

   11. Applied rewrites 72.7%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
   12. Step-by-step derivation

   13. Applied rewrites 76.0%

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - ux\right) + ux \cdot maxCos \leq 0.9994000196456909:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(1 - ux\right) \cdot \left(1 - ux\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 76.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (* (PI) 2.0) uy) (sqrt (* (- (/ 2.0 ux) 1.0) (* ux ux)))))

Derivation

1. Initial program 61.0%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lower-PI.f32 52.7

      \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
6. Step-by-step derivation

   1. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

   2. lift-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}} \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   4. +-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\left(\left(ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   5. lift-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{\left(ux \cdot maxCos\right)} \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right) + \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   6. associate-*l* N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\color{blue}{ux \cdot \left(maxCos \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} + \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   7. lower-fma.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(ux, maxCos \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right), \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)}} \]

   8. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, \color{blue}{maxCos \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   9. lift-+.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)}, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   10. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)}, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   11. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right), \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   12. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \left(\color{blue}{maxCos \cdot ux} + \left(1 - ux\right)\right), \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   13. lift-fma.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right)}, \left(1 - ux\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)\right)} \]

   14. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

   15. lower-*.f32 51.6

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(1 - ux\right)}\right)} \]

   16. lift-+.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \color{blue}{\left(\left(1 - ux\right) + ux \cdot maxCos\right)} \cdot \left(1 - ux\right)\right)} \]

   17. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \color{blue}{\left(ux \cdot maxCos + \left(1 - ux\right)\right)} \cdot \left(1 - ux\right)\right)} \]

   18. lift-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \left(\color{blue}{ux \cdot maxCos} + \left(1 - ux\right)\right) \cdot \left(1 - ux\right)\right)} \]

   19. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \left(\color{blue}{maxCos \cdot ux} + \left(1 - ux\right)\right) \cdot \left(1 - ux\right)\right)} \]

   20. lift-fma.f32 51.4

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \color{blue}{\mathsf{fma}\left(maxCos, ux, 1 - ux\right)} \cdot \left(1 - ux\right)\right)} \]

7. Applied rewrites 50.6%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \color{blue}{\mathsf{fma}\left(ux, maxCos \cdot \mathsf{fma}\left(maxCos, ux, 1 - ux\right), \mathsf{fma}\left(maxCos, ux, 1 - ux\right) \cdot \left(1 - ux\right)\right)}} \]

8. Taylor expanded in ux around inf

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{{ux}^{2} \cdot \left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right)}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

   2. lower-*.f32 N/A

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 \cdot \frac{1}{ux} - \left(-1 \cdot \left(maxCos - 1\right) + \left(2 \cdot \frac{maxCos}{ux} + maxCos \cdot \left(maxCos - 1\right)\right)\right)\right) \cdot {ux}^{2}}} \]

10. Applied rewrites 77.8%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(\frac{2}{ux} - \mathsf{fma}\left(\frac{maxCos}{ux}, 2, \left(maxCos - 1\right) \cdot \left(maxCos + -1\right)\right)\right) \cdot \left(ux \cdot ux\right)}} \]

11. Taylor expanded in maxCos around 0

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 \cdot \frac{1}{ux} - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]
12. Step-by-step derivation

13. Applied rewrites 77.9%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(\color{blue}{ux} \cdot ux\right)} \]

14. Final simplification 77.9%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(\frac{2}{ux} - 1\right) \cdot \left(ux \cdot ux\right)} \]
15. Add Preprocessing

Alternative 18: 66.1% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (* (PI) 2.0) uy) (sqrt (* (- 2.0 (* maxCos 2.0)) ux))))

Derivation

1. Initial program 61.0%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lower-PI.f32 52.7

      \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

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 51.6

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

8. Applied rewrites 51.6%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

9. Taylor expanded in ux around 0

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
10. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

    2. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

    3. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

    4. lower-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

    5. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

    6. lower-fma.f32 62.1

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

11. Applied rewrites 62.1%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]
12. Step-by-step derivation

13. Applied rewrites 64.2%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux} \]

14. Final simplification 64.2%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\left(2 - maxCos \cdot 2\right) \cdot ux} \]
15. Add Preprocessing

Alternative 19: 63.5% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{2 \cdot ux} \end{array} \]

FPCore

(FPCore (ux uy maxCos)
 :precision binary32
 (* (* (* (PI) 2.0) uy) (sqrt (* 2.0 ux))))

Derivation

1. Initial program 61.0%

   \[\sin \left(\left(uy \cdot 2\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
2. Add Preprocessing

3. Taylor expanded in uy around 0

   \[\leadsto \color{blue}{\left(2 \cdot \left(uy \cdot \mathsf{PI}\left(\right)\right)\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(2 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot uy\right)}\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   3. lower-*.f32 N/A

      \[\leadsto \color{blue}{\left(\left(2 \cdot \mathsf{PI}\left(\right)\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   4. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   5. lower-*.f32 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{PI}\left(\right) \cdot 2\right)} \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

   6. lower-PI.f32 52.7

      \[\leadsto \left(\left(\color{blue}{\mathsf{PI}\left(\right)} \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

5. Applied rewrites 52.7%

   \[\leadsto \color{blue}{\left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right)} \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \left(\left(1 - ux\right) + ux \cdot maxCos\right)} \]

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 51.6

      \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

8. Applied rewrites 51.6%

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{1 - \left(\left(1 - ux\right) + ux \cdot maxCos\right) \cdot \color{blue}{\left(1 - ux\right)}} \]

9. Taylor expanded in ux around 0

   \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{ux \cdot \left(2 - 2 \cdot maxCos\right)}} \]
10. Step-by-step derivation

    1. metadata-eval N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \left(2 - \color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot maxCos\right)} \]

    2. fp-cancel-sign-sub-inv N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{ux \cdot \color{blue}{\left(2 + -2 \cdot maxCos\right)}} \]

    3. *-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

    4. lower-*.f32 N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(2 + -2 \cdot maxCos\right) \cdot ux}} \]

    5. +-commutative N/A

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\left(-2 \cdot maxCos + 2\right)} \cdot ux} \]

    6. lower-fma.f32 62.1

       \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right)} \cdot ux} \]

11. Applied rewrites 62.0%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(-2, maxCos, 2\right) \cdot ux}} \]

12. Taylor expanded in maxCos around 0

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{2 \cdot ux} \]
13. Step-by-step derivation

14. Applied rewrites 62.1%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{2 \cdot ux} \]

15. Final simplification 62.1%

    \[\leadsto \left(\left(\mathsf{PI}\left(\right) \cdot 2\right) \cdot uy\right) \cdot \sqrt{2 \cdot ux} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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)))))))